Ready to Go On? Chapters Intervention

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1 Ready to Go On? Chapters 11 1 Intervention A. Perimeter and Area You can apply formulas for perimeter, circumference, and area to find and compare measures of geometric figures. To find perimeters and areas of objects for which we do not have specific formulas, we can sometimes break the object into smaller, nonoverlapping regions for which we do have formulas. If you compute the area in two different ways the answers are the same. 1. Find the Area of Squares, Rectangles, Parallelograms, and Triangles (Lesson 11.1) Find the area of nabc. Method 1: Use } BC as the base. The base is extended to measure the height AE. So, b 5 6 and h 5 4. Area 5 } 1 b p h 5 } 1 6(4) 5 1 square units. Method : Use } AC as the base. The base is extended to measure the height BD. So, b 5 8 and h 5 3. Area 5 } 1 b p h 5 } 1 8(3) 5 1 square units. Use the diagram to find the areas of the figures given in Exercises T BCE. T ADF 3. T ACB 4. ~ABCD 5. ` ABEF 4 A E D 8 A F 0 11 C 6 C D B B E 15 Intervention Ch11 1 A. Perimeter and Area Use the diagram to find the areas of the figures given in Exercises E B F 6. hbicf 7. h ABCD 8. hefgh 9. T ABC 10. T BIC A I 6 C H D G Holt McDougal Geometry 19

2 Ready to Go On? Chapters 11 1 Intervention cont d Intervention Ch11 1 A. Perimeter and Area Commit the formulas for area to memory.. Find the Areas of Trapezoids, Rhombuses, and Kites (Lesson 11.) Some wooden tiles in a parquet floor form the geometric figure shown. a. Find the area of the rhombus ABCD. b. Find the area of the trapezoid ACGF. a. Step 1: Find the length of each diagonal. The diagonals of a rhombus bisect each other, so BE 5 ED and AE 5 EC. Thus, BD 5 BE 1 ED 5 Ï } 3 1 Ï } Ï } 3 AC 5 AE 1 EC Step : Find the area of the rhombus. Let d 1 represent BD and d represent AC. Then, A 5 1 } d 1 p d 5 } Ï } 3 (4) Substitute. 5 8 Ï } 3 Simplify. F Formula for area of a rhombus. The area of the rhombus is 8 Ï } 3 square inches. A B Îã3 in. in. b. The height of the trapezoid is ED 5 EB 5 Ï } 3. The lengths of the bases are AC and FG. Now, AC 5 4, as found above, and FG 5 FD 1 DG where FD 5 DG 5 BC. Since T BEC is a right triangle with legs of length and Ï } 3, the length of the hypotenuse BC is found to be 4, by the Pythagorean Theorem. Thus, FG We have A 5 } 1 h(b 1 b ) Formula for area of a trapezoid } 1 Ï } 3 (4 1 8) Substitution. 5 1 Ï } 3 Simplify. The area of the trapezoid is 1 Ï } 3 square inches. Find the area of each trapezoid, rhombus, or kite. E D C G Holt McDougal Geometry

3 Ready to Go On? Chapters 11 1 Intervention cont d Area is measured in units so we must square the ratio of the perimeters to get the ratio of the areas. 3. Use Ratios of Similar Polygons (Lesson 11.3) A large Jello mold is in the shape of a right triangle, with base length 9 inches and height 1 inches. A smaller mold is similar to the larger mold. The area of the smaller mold is 4 square inches. Find the height of the smaller mold. What is the ratio of the perimeter of the smaller mold to that of the larger? First draw a diagram to represent the problem. Label dimensions and areas. Then use Theorem If the area ratio is a : b, then the length ratio is a : b. Write ratio of known areas, then simplify: 4 : : 9. Thus, Length in smaller mold : Length in larger mold 5 : 3. 1 in. 1 9 in. A 5 (9) (1) 5 54 in.? in. 1 A 5 (6) (8) 5 4 in. Any length in the smaller mold is } of the corresponding length in the larger mold. 3 So the height of the smaller mold is } (1 inches) 5 8 inches. Since the ratio : 3 holds 3 for each pair of corresponding sides, the ratio of the perimeter of the smaller to that of the larger is also : 3. Corresponding lengths in similar figures are given. Find the ratios of the perimeters and areas (smaller to larger). Also, find the unknown area. Intervention Ch11 1 A. Perimeter and Area in. 18. in. 3 ft A 5? 5 ft A 5 4 in. A 5 4 ft Holt McDougal Geometry 131

4 Ready to Go On? Chapters 11 1 Intervention cont d yd 0. 5 yd 8 yd 4 yd 3 yd 6 yd A 5 16 yd A 5?.5 m 4 m A 5? A m Intervention Ch11 1 A. Perimeter and Area Vocabulary You can often use the Pythagorean Theorem to find the apothem. 4. Find the Area of a Regular Polygon (Lesson 11.6) The center of a polygon is the center of its circumscribed circle. The apothem of a polygon is the height to a base of an isosceles triangle that has two radii as legs. You are tilling soil for a garden which is to be shaped like a regular pentagon, 6 feet on a side, 6 ft with a radius of about 5.1 feet. What is the area of the garden you will till? R 5.1 ft a Step 1: Find the perimeter P of the garden. A pentagon has 5 sides, so P 5 5 p feet. Step : Find the apothem a. The apothem is height RS of P S Q TPQR. Because TPQR is isosceles, altitude } RS bisects } PQ. So, QS 5 } 1 (QP) 5 } 1 (6) 5 3 feet. To find RS, use the Pythagorean Theorem for TRQS: a 5 RS ø Ï } Ï } ø Step 3: Find the area A of the garden. A 5 } 1 ap Formula for area of a regular polygon. ø } 1 (4.14)(30) Substitute Simplify. The area of the garden to be tilled is about square feet. Find the area of a regular polygon with the given number of sides n, the given side length s, and the radius r. 1. n 5 3, s 5, r n 5 4, s 5 5, r n 5 5, s 5 7, r n 5 6, s 5 8, r n 5 8, s 5, r n 5 1, s 5 10, r Holt McDougal Geometry

5 Ready to Go On? Chapters 11 1 Intervention cont d Vocabulary 5. Use Length to Find a Geometric Probability, Use Area to Find a Geometric Probability (Lesson 11.7) The probability of an event is a measure of the likelihood that the event will occur. A geometric probability is a ratio that involves a geometric measure such as length or area. a. Find the probability that a point chosen at random on } PQ is on } RS. The probability that an event occurs plus the probability that the event does not occur is 1. b. Find the probability that a point chosen at random on PQTU is in RSTU. 4 P U R 3 a. P(Point is on } RS ) 5 } RS PQ 5 } 3 ø 0.43, or about 43%. 7 b. Find the ratio of the shaded area to the area of the rectangle PQTU. Let A T 5 area of trapezoid RSTU and A R 5 area of rectangle PQTU. Then, P(Point is in the trapezoid RSTU) 5 A T } 5 A about 71%. R 7 S 1 } p 4(3 1 7) } 4 p 7 Q T 5 } } 5 ø 0.71, or 7 Find the probability that a randomly chosen point in the figure lies in the shaded region. Intervention Ch11 1 A. Perimeter and Area Ï 1 Holt McDougal Geometry 133

6 Ready to Go On? Chapters 11 1 Intervention cont d Check Intervention Ch11 1 A. Perimeter and Area Use the figure and the fact that GH 1 to find quantities in Exercises 1 6. A H B F G C E D 1. The area of T FBE.. The area of kite HCDE. 3. The area of parallelogram ABCG. 4. The area of trapezoid ABCD. 5. The ratio of the perimeters and areas of ABGF and BCEF. 6. The probability that a randomly chosen point in the figure also lies in TFGE. 7. Find the area of a regular decagon of side length Holt McDougal Geometry

7 Ready to Go On? Chapters 11 1 Intervention cont d B. Circumference and Area of Circles Throughout this section, you should use the π key on your calculator, then round to the hundredths place unless instructed otherwise. Vocabulary For a fascinating introduction to the number π, read A History of π by Petr Beckmann. 1. Find a Circumference (Lesson 11.4) The circumference of a circle is the distance around the circle. For all circles, the ratio of the circumference to the diameter is the constant value π. Find the indicated measure. a. The circumference of a circle with radius 19 inches. b. The radius of a circle with circumference 6 centimeters. a. C 5 π p r Write circumference formula. 5 p π p 19 Substitute 19 for r. 5 38π Simplify. < Use a calculator. The circumference is about inches. b. C 5 π p r Write circumference formula. 6 5 p π p r Substitute 6 for C. 6 } 5 r π Divide each side by π < r Use a calculator. The radius is about 9.87 centimeters. Intervention Ch11 1 B. Circumference and Area Find the circumference of the circle with the given radius r. 1. r 5 5 feet. r yards 3. r 5 18 miles 4. r mm Find the radius of the circle with the given circumference C. 5. C 5 4 miles 6. C 5 9 meters 7. C 5 } π inches 8. C 5 10 units Holt McDougal Geometry 135

8 Ready to Go On? Chapters 11 1 Intervention cont d. Find an Arc Length from an Angle Measure (Lesson 11.4) Vocabulary An arc length is a portion of the circumference of a circle. Intervention Ch11 1 B. Circumference and Area The ratio of the length of an arc to the circumference of a circle is equal to the ratio of the arc s central angle to 360. Find the length of each arc which does not contain the point S. a. b. c. A S 6 ft A S P 45 in. B B Q 135 a. Arc length of C AB 5 } 45 p π() < 1.57 inches. 360 b. Arc length of C AB 5 } 135 p π(6) < feet. 360 c. Arc length of C AB 5 } 30 p π(3) < 1.57 m. 360 A 30 R 3 m B S Find the length of an arc of a circle of radius r whose central angle is P. 9. r 5 5 yds, m/p r 5 7 km, m/p r 5 15 ft, m/p r 5 light-years, m/p Find the Area of a Circle (Lesson 11.5) Find the indicated measure. The area of a circle of radius r is proportional to the area of a square of side length r. a. Area b. Diameter r cm A cm 136 Holt McDougal Geometry

9 Ready to Go On? Chapters 11 1 Intervention cont d a. A 5 π p r Write formula for the area of a circle. 5 π (3.5) Substitute 3.5 for r π Simplify. < Use a calculator. The area of the circle is about centimeters. b. A 5 π p r Write formula for the area of a circle. Vocabulary π p r Substitute for A } π 5 r Divide both sides by π < r Find the positive square root of each side. The radius of the circle is about 9.95 centimeters, so the diameter is about 19.9 centimeters. Find the area of the circle with the given radius r. 13. r 5 5 feet 14. r yards 15. r 5 18 miles 16. r mm Find the radius of the circle with the given area A. 17. A 5 4 miles 18. A 5 9 meters 19. A 5 π } inches 0. A 5 10 units 4. Find the Area of a Sector (Lesson 11.5) A sector of a circle is the region bounded by two radii of the circle and their intercepted arc. Find the area of the sectors formed by UTV. Intervention Ch11 1 B. Circumference and Area Use the π key on your calculator and round at the end. S T 9 90 U V Step 1: Find the measures of the minor and major arcs. Because m/utv 5 90, m C UV 5 90, and m C USV = 70. Holt McDougal Geometry 137

10 Ready to Go On? Chapters 11 1 Intervention cont d Intervention Ch11 1 B. Circumference and Area Step : Find the areas of the small C and large sectors. Area of small sector 5 } m UV 360 p π p r Write formula for area of a sector. 5 } p π p 9 Substitute. < 63.6 C Use a calculator. Area of large sector 5 } m USV 360 p π p r Write formula for area of a sector. 5 } p π p 9 Substitute. < Use a calculator. The areas of the small and large sectors are about 63.6 square units and square units, respectively. Find the area of the sector of a circle with the given radius and intercepted arc measure. 1. r 5 8, 80. r 5 6, r 5 16, r 5 4, r 5 10, r 5 7, 180 Check Round all answers to the nearest hundredth. 1. Find the circumference of a circle with radius 10 inches.. Find the radius of a circle with circumference 1 inches. 3. Find the arc length and area of a sector of a circle with radius 6 meters whose central angle is Find the area of a circle with radius 1 cm. 5. Find the radius of a circle with area 1 cm. 6. Find the circumference of a circle with area 1 cm. 7. The diameter of the sphere is 30, and m C AB = 10. Find the two distances between A and B along the great circle. A 30 P B C 138 Holt McDougal Geometry

11 Ready to Go On? Chapters 11 1 Intervention cont d C. Solids Many objects in the physical world can be idealized as polyhedrons, so we study such solids and their properties. Vocabulary A polyhedron may be neither a prism nor a pyramid. 1. Identify and Name Solids (Lesson 1.1) A polyhedron is a solid that is bounded by polygons, called faces, that enclose a single region of space. An edge of a polyhedron is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet. Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. a. b. Intervention Ch11 1 C. Solids c. a. The solid is formed by polygons, so it is a polyhedron. The two bases are congruent hexagons, so it is a hexagonal prism. It has 8 faces, 1 vertices, and 18 edges. b. The solid is formed by slicing the tip off a cone, so it is not a polyhedron. c. The solid is formed by polygons, so it is a polyhedron. The base is a pentagon, so it is a pentagonal pyramid. It has 6 faces, consisting of 1 base, 3 visible triangular faces, and non-visible triangular faces. The polyhedron has 6 faces, 6 vertices, and 10 edges. Holt McDougal Geometry 139

12 Ready to Go On? Chapters 11 1 Intervention cont d Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges Intervention Ch11 1 C. Solids Vocabulary. Use Euler s Theorem (Lesson 1.1) A polyhedron is regular if all of its faces are congruent regular polygons. Find the number of faces, vertices, and edges of the regular tetrahedron. Check your answer using Euler s Theorem. Use Euler s formula to check if your sketch of a solid is accurate. By counting on the diagram, the tetrahedron has 4 faces, 4 vertices, and 6 edges. Use Euler s Theorem to check. F 1 V 5 E 1 Euler s Theorem Substitute This is a true statement. So the solution checks. 140 Holt McDougal Geometry

13 Ready to Go On? Chapters 11 1 Intervention cont d You found the number of faces, vertices, and edges of the polyhedra in the previous section. Now check your answers using Euler s theorem Vocabulary 3. Describe Cross Sections (Lesson 1.1) The intersection of a plane and a solid is called a cross section. Describe the shaped formed by the intersection of the plane P and the right prism whose bases are equilateral triangles. Intervention Ch11 1 C. Solids Shading different faces in varying shades of gray helps to make your sketches more understandable. a. P b. P c. P a. The cross section is an equilateral triangle. b. The cross section is a rectangle. c. The cross section is an isosceles triangle. Holt McDougal Geometry 141

14 Ready to Go On? Chapters 11 1 Intervention cont d Describe the intersection of the plane and the solid. 9. P 10. P Intervention Ch11 1 C. Solids 11. P 1. P Vocabulary 4. Use the Scale Factor of Similar Solids (Lesson 1.7) Two solids of the same type with equal ratios of corresponding linear measures, such as heights or radii, are called similar solids. The common ratio is called the scale factor of one solid to the other solid. Area varies as the square of the length, volume varies as the cube of the length. The lidless boxes shown are similar with a scale factor of 63 : 100. Find the surface area and volume of the smaller box. Let S 1 and V 1 denote the surface area and volume, respectively, of the smaller box. Use Theorem 1.13 to write two proportions: S 1 } S 5 V } a 1 } b V 5 } a3 b 3 S 5 47 cm V cm 3 S 1 } 47 5 V } 63 1 } } Solving, we obtain S 1 ø and V 1 ø So, the surface area of the smaller box is about square centimeters, and the volume of the smaller box is about cubic centimeters. 14 Holt McDougal Geometry

15 Ready to Go On? Chapters 11 1 Intervention cont d You are given the scale factor for a pair of similar solids and the surface area S and volume V of the smaller solid. Find the surface area S 1 and volume V 1 of the larger solid in each case : 4, S 5 10 cm, V 5 36 cm : 40, S 5 10 m, V m : 4, S 5 56 in., V 5 89 in : 45, S 5.56 miles, V miles 3 Check 1. Find the values of F, E, and V in Euler s Formula for a hexagonal prism.. Find the values of F, E, and V in Euler s Formula for a decagonal pyramid. 3. Describe the possible cross-sections of a cube. 4. A plane intersects a pentagonal prism parallel to its base in such a way that the ratio of volumes is 8 : 7. a. What is the scale factor? b. If the surface area of the smaller prism is 400, then what is the surface area of the larger prism? Intervention Ch11 1 C. Solids Holt McDougal Geometry 143

16 Ready to Go On? Chapters 11 1 Intervention cont d Intervention Ch11 1 D. Surface Area of Solids Vocabulary Look for congruent faces to save work when finding surface area. D. Surface Area of Solids 1. Use a Net (Lesson 1.) Imagine that you cut along some faces of a polyhedron so that you can unfold it and spread it flat on a table. The two-dimensional representation of the faces is called a net. Find the surface area of an equilateral triangular prism with height 6 centimeters and triangle side length 4 centimeters. Step 1: Sketch the prism. Imagine unfolding it to make a net. 4 cm 4 cm 6 cm 6 cm 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm Step : Find the areas of the rectangles and triangles that form the faces of the prism. Congruent faces Dimensions Area of each face Top and bottom faces 4 cm by Ï } 1 3 cm } (4) 1 Ï } Ï } 3 cm Side faces 4 cm by 6 cm 4(6) 5 4 cm Step 3: Add the areas of all the faces to find the surface area of the prism. The surface area is S 5 14 Ï } 3 1 3(4) 5 8 Ï } < cm. Describe the solid formed by the net. Then find its surface area. Round your answer to two decimal places ft 8 ft 1. in..3 in. 1. in. 144 Holt McDougal Geometry

17 Ready to Go On? Chapters 11 1 Intervention cont d ft 4. 5 ft Vocabulary Even though it takes some effort to find the apothem of a regular polygon, it makes light work of finding the surface area of the corresponding prism.. Find the Surface Area of a Right Prism (Lesson 1.) In a right prism, each lateral face is perpendicular to both bases. Find the surface area of the right trapezoidal prism Intervention Ch11 1 D. Surface Area of Solids Step 1: Find the perimeter P and area B of a base x x The perimeter is P feet. Each base is a trapezoid whose height x can be computed using the Pythagorean Theorem: x 5 Ï } ø.60 ft. Using the formula for area of a trapezoid we obtain B ø 1 } ( )(.6) ft. Step : Use the formula for the surface area that uses the area of the base in Theorem 1.. S 5 B 1 Ph ø (1.58) 1.6(10) Simplify. Surface area of a right prism. Substitute known values. The surface area of the right trapezoidal prism is about feet. Holt McDougal Geometry 145

18 Ready to Go On? Chapters 11 1 Intervention cont d Find the surface area of the right prism of height h whose base is a regular polygon with n sides each of length s. 5. h 5 10, n 5 3, s h 5 1, n 5 4, s h 5 3, n 5 5, s h 5 100, n 5 6, s 5 1 Intervention Ch11 1 D. Surface Area of Solids Vocabulary Notice how similar the surface area formulas for prisms and cylinders are. Vocabulary 3. Find the Surface Area of a Right Cylinder (Lesson 1.) A cylinder is a solid with congruent circular bases that lie in parallel planes. You are wrapping a stack of 1 cookies in tin foil. Each cookie is a short cylinder with height 0.5 inches and radius 1.75 inches. What is the minimum amount of tin foil needed to cover the stack of cookies? The 1 cookies are stacked, so that the height of the stack will be 1(0.5) 5 3 inches. The radius is 1.75 inches. The minimum amount of tin foil needed will be equal to the surface area of the stack of cookies. S 5 π p r 1 π p r p h Surface area of a cylinder. 5 π(1.75) 1 π(1.75)(3) Substitute known values. ø 5.3 Use a calculator. You will need at least 5.3 square inches, or about 0.36 square feet, of tin foil. Find the surface area of the right cylinder with given radius r and height h. Round your answer to two decimal places. 9. r 5 3 ft, h 5 4 ft 10. r 5 1 m, h 5 40 m 11. r cm, h 5 4 cm 1. r 5 5 yd, h 5 7 yd 13. r in., h 5.7 in. 14. r 5 7 ft, h 5 85 ft 4. Find the Surface Area of a Pyramid (Lesson 1.3) A pyramid is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex, called the vertex of the pyramid. The height is the perpendicular distance between the base and the vertex. The slant height of a regular pyramid is the height of a lateral face. Any polygon can be used as the base of a pyramid. Find the surface area of the regular pentagonal pyramid. First, find the area of the base using the formula for the area of a regular polygon, } 1 a p P. The apothem a of the pentagon is about 6.88 feet and the perimeter P is 5 p feet. So, the area of the base B is about 1 }(6.88)(50) ø 17 square feet. 10 ft 8 ft 6.88 ft 146 Holt McDougal Geometry

19 Ready to Go On? Chapters 11 1 Intervention cont d Now, find the surface area: S 5 B 1 } 1 Pl Formula for surface area of a regular pyramid. ø 17 1 } 1 (50)(8) Substitute known values Simplify. The surface area of the regular pentagonal pyramid is about 37 square feet. Vocabulary Find the surface area of the regular pyramid with slant height whose base has n sides each of length s. Round your answers to the nearest hundredth. 15. l 5 10, n 5 4, s l 5 18, n 5 6, s l 5 0, n 5 8, s l 5 4, n 5 1, s Find the Surface Area of a Cone (Lesson 1.3) The lateral surface of a right cone is the set of all segments that connect the vertex with points on the base edge of the cone. The slant height of a right cone is the distance between the vertex and a point on the base edge. What is the surface area of the right cone? 1 ft Intervention Ch11 1 D. Surface Area of Solids 10 ft A cone be thought of as a regular pyramid with a base of infinitely many sides. To find the slant height, use the Pythagorean Theorem. l 5 h 1 r Write formula. l Substitute. l 5 Ï } 61 Find positive square root. Use the formula for the surface area of a right cone. S 5 π p r 1 π p r p l Formula for surface area of a right cone. 5 π (1) 1 π(1)( Ï } 61 ) Substitute known values. 51π (1 1 Ï } 61 ) Simplify. ø The surface area of the right cone is approximately square feet. Holt McDougal Geometry 147

20 Ready to Go On? Chapters 11 1 Intervention cont d Find the surface area of the right cone with the given height h and radius r. Round your answers to the nearest hundredth. 19. h 5 10, r h 5 6, r h 5 0, r 5 8. h 5 1, r h 5 π, r 5 π 4. h 5 1, r 5 1 Intervention Ch11 1 D. Surface Area of Solids Vocabulary A circle is to two dimensions as a sphere is to three dimensions. 6. Find the Surface Area of a Sphere (Lesson 1.6) A sphere is the set of all points in space equidistant from a given point, called the center of the sphere. Find the surface area of the sphere. S 5 4 π p r 5 4π(3) Substitute 3 for r. 5 36π Simplify. ø ft Formula for surface area of a sphere. Use a calculator. The surface area of the sphere is approximately square feet. Either the surface area S or the radius r of a sphere is given. Find the other value. Round your answers to the nearest hundredth. 5. S 5 56 cm 6. S m 7. r 5 19 inches 8. r 5 π yards 9. r 5 } 3 miles 30. S 5 π feet 8 Check Exercises 1 4 refer to a right prism whose base is a regular hexagon of side length 6 units and whose height is 1 units. 1. Describe and enumerate the congruent faces and state the area of each.. What is the surface area of this prism? 3. Consider a right cylinder of radius 6 and height 1. Is its surface area more or less than the prism? Explain. 4. Suppose a pyramid is constructed with a base congruent to the base of the prism. What should the slant height be so that its surface area equals that of the prism? 5. Find the surface area of a cone of radius 3 cm, height 4 cm. 6. What is the radius of a sphere with surface area 4π units? 148 Holt McDougal Geometry

21 Ready to Go On? Chapters 11 1 Intervention cont d E. Volumes of Solids Just as length is measured in one dimension, and area is a measure made in two dimensions, a solid in space can be measured in terms of its volume. We apply known formulas for calculating the volume of some common three dimensional objects. Vocabulary 1. Find the Volume of Prisms and Cylinders (Lesson 1.4) The volume of a solid is the number of cubic units contained in its interior. Volume is always measured in cubic units, such as cubic centimeters (cm 3 ). Find the volume of the solid. Make sure you remember how to find areas of polygons before attempting to find volumes of prisms or pyramids. a. 4 cm 5 cm b. 7 ft 9 ft Intervention Ch11 1 E. Volumes of Solids 3 cm a. The area of the base is 1 } (3)(4) 5 6 cm and h 5 5 cm. V 5 B p h 5 6 p cm 3. b. The area of the base is π p π ft. Use h 5 9 ft to find the volume. V 5 B p h 5 49π p π ø ft 3. Assume each solid is a right prism or cylinder and find its volume cm. 1 cm cm 3 cm 6 ft ft 3. 5 m 4. 6 cm 3.5 m cm m Holt McDougal Geometry 149

22 Ready to Go On? Chapters 11 1 Intervention cont d 5. cm 6. 4 cm 6 cm 6 cm 8 cm Intervention Ch11 1 E. Volumes of Solids Cavalieri s Principle allows us to use the same formula to find the volume of either a right or an oblique prism.. Find the Volume of an Oblique Cylinder (Lesson 1.4) Find the volume of the oblique cylinder. Cavalieri s Principle allows us to use Theorem 1.7 to find the volume of the oblique cylinder. V 5 π p r p h Formula for volume of a cylinder. 5 π(3 )(5) Substitute known values. 5 45π Simplify. ø Use a calculator. The volume of the oblique cylinder is about m 3. Find the volume of each oblique cylinder. 3 m 5 m 7. m 8. 1 m 1 m m 150 Holt McDougal Geometry

23 Ready to Go On? Chapters 11 1 Intervention cont d 9. 3 cm 10. ft 10 cm 9 ft The volume formulas given in Theorems 1.9 and 1.10 apply to both right and oblique pyramids and cones. 3. Find the Volume of a Pyramid and Cone (Lesson 1.5) Find the volume of the solid. a. b. 8 m 7. ft 3.8 ft 3 m 4 m Intervention Ch11 1 E. Volumes of Solids a. V 5 1 } 3 B p h b. V 5 1 } 3 B p h 5 1 } 3 (3 p 4)(8) 5 1 } 3 (π p r )h 5 3 m } 3 (π p 3.8 )(7.) π ø ft 3 Find the volume of each pyramid or cone cm ft 6.3 ft ft 4 cm 3 cm 4. ft Holt McDougal Geometry 151

24 Ready to Go On? Chapters 11 1 Intervention cont d mm mm 16 m 16 m Intervention Ch11 1 E. Volumes of Solids in. 3 in. 16. p p A hemisphere results from cutting a sphere precisely in half. 4. Find the Volume of a Sphere (Lesson 1.6) A spherical scoop of ice cream has a diameter of.5 inches. What is the volume of two scoops? The diameter of the scoop is.5 inches, so the radius is } inches. Find the volume of each scoop: V 5 } 4 3 π p r3 Formula for volume of a sphere. 5 4 } 3 π(1.5)3 Substitute. ø 8.18 Use a calculator to simplify. There are two scoops, so the total volume of ice cream is about p cubic inches. Find the volume of the sphere with the given radius. Round your answers to the nearest hundredth. 17. r 5 ft 18. r in. 19. r mm 0. r 5 π miles 1. r cm. r 5 7 m 15 Holt McDougal Geometry

25 Ready to Go On? Chapters 11 1 Intervention cont d Check Round your answers to the nearest hundredth. 1. Find the volume of the right prism, whose base is a rhombus Find the volume of the oblique cylinder. Intervention Ch11 1 E. Volumes of Solids 5 3. Find the volume of the right pyramid whose base is a kite Find the radius of a sphere whose volume is 64π m 3. Holt McDougal Geometry 153

26 Name Date Ready to Go On? Chapters 11-1 Quiz Perimeter and Area (Lessons 11.1, 11., 11.3, 11.6, 11.7) Find the area of the polygon A parallelogram has a base of 6 cm and a height of 4 cm. What is its area? A. 1 cm B. 4 cm C. 36 cm D. 48 cm 4. The length of the hypotenuse of a right triangle is 5 cm and the length of leg is 15 cm. a. Find the area of the triangle. b. Find the perimeter of the triangle. 5. A rectangle has an area of 43 m and its length is triple its width. a. Find the length. b. Find the perimeter. 6. Find the area of the trapezoid Answers a. 4b. 5a. 5b a. 9b. 9c. Quizzes A rhombus has an area of 45 cm, and the length of one diagonal is 9 cm. What is the length of the other diagonal? A. 5 cm B. 15 cm C. 0 cm D. 10 cm 8. The lengths of the diagonals of a kite are 16 cm and 1 cm. What is the area? A. 84 cm B. 168 cm C. 336 cm D. 67 cm 9. The lengths of the bases of two similar right triangles are 6 inches and 1 inches. The smaller triangle has an area of 4 square inches. a. Find the height of the smaller triangle. b. Find the area of the larger triangle. c. Find the perimeter of each triangle. Round to one decimal place. Holt McDougal Geometry 193

27 Name Date Ready to Go On? Chapters 11-1 Quiz cont d LESSON Quizzes More Copy if needed 10. The areas of two similar trapezoids are 4 m and 18 m. What is the ratio of the perimeter of the larger trapezoid to the perimeter of the smaller trapezoid? A. 4 : 3 B. 4 : Ï } 3 C. : Ï } 3 D. : The side length of a regular hexagon is 11 feet. a. Find the apothem. Round to one decimal place. b. Find the area. Round to one decimal place. 1. Find the probability that a point chosen at random on } AD is on } BC. Express your answer as a percent. A B C D What is the probability that a randomly chosen point in the figure lies in the shaded region? 7 5 Answers a. 11b A. } 5 B. 5 } 8 C. 3 } 8 D. 3 } 5 Circumference and Area of Circles (Lessons 11.4, 11.5) 14. Find the circumference of a circle with diameter 11 inches. Round to two decimal places. 15. Find the approximate radius of a circle with circumference 3 meters. A m B. 7.3 m C m D. 7.6 m 16. Find the length of C RS. Q 80 R 17. Find the area of the shaded region. 9 cm S 15 cm 194 Holt McDougal Geometry 5 cm

28 Name Date Ready to Go On? Chapters 11-1 Quiz cont d 18. What is the area of the shaded region? Answers Quizzes A π B. 16 π C. 8 1 π D Find the area of the shaded region Solids (Lessons 1.1, 1.7) 0. Which of the following solids has 1 edges? A. A rectangular prism B. A triangular prism C. A triangular cylinder D. A square pyramid 1. Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges.. A solid has 1 edges and 6 faces. How many vertices does it have? A. 6 B. 8 C. 10 D How many faces and edges does a hexagonal pyramid have? A. 6 faces, 6 edges B. 7 faces, 1 edges C. 4 faces, 14 edges D. 6 faces, 8 edges 4. What is the shape of the cross section formed by the plane perpendicular to the base through the top vertex of the square pyramid? Holt McDougal Geometry 195

29 Name Date Ready to Go On? Chapters 11-1 Quiz cont d LESSON Quizzes More Copy if needed 5. The ratio of volumes of two spheres is 1 : 7. What is the ratio of their surface areas? A. 1 : 3 B. 1 : 81 C. 1 : 9 D. 3 : Two similar solids have a scale factor of 1 : 3. Find the volume of the smaller solid if the volume of the larger solid is 54 cubic centimeters. 7. Two triangular prisms are similar. The surface area of one prism is 70 square inches. The surface area of the other prism is 105 square inches. Find the scale factor of the smaller prism to the larger prism. Surface Areas of Solids (Lessons 1., 1.3, 1.6) 8. Find the surface area of the solid formed by the net. Round your answer to two decimal places. Answers cm 3. 5 cm 9. What is the surface area of a right rectangular prism with height 3 inches, length 6 inches, and width 1 inches? A. 5 in. B. 108 in. C. 180 in. D. 16 in. 30. Find the surface area of the right hexagonal prism. Round your answer to two decimal places. 6 ft 7 ft 31. What is the surface area of a right cylinder with a radius of 3 cm and a height of 6 cm? A. 7π cm B. 81π cm C. 7π cm D. 54π cm 3. The surface area of a right cylinder is 50 square inches. The radius of the cylinder is 4 inches. Find the height of the cylinder. Round your answer to two decimal places. 196 Holt McDougal Geometry

30 Name Date Ready to Go On? Chapters 11-1 Quiz cont d 33. What is the area of a lateral face of a regular square pyramid with lateral edges of length 17 inches and base edges of length 10 inches? A in. B. 85 in. C in. D. 170 in. 34. What is the surface area of a regular square pyramid with a base edge length of 1 inches and a height of 10 inches? A. 144 in. B. 360 in. C in. D in. 35. Find the surface area of the triangular pyramid shown. Round your answer to two decimal places. 8 cm Answers Quizzes 6 cm 36. What is the surface area of the right cone? 1 m A. 60π m B. 65π m C. 85π m D. 90π m 37. Find the surface area of the solid. The cones are right. Round your answer to two decimal places. 5 m 8 ft 3 ft 5 ft 38. The diameter of a sphere is 9 inches. Find the surface area of the sphere. Round your answer to two decimal places. 39. The surface area of a sphere is 9π square meters. What is the radius of the sphere? A. 1.5 m B. 3 m C. 4 m D. 6 m Holt McDougal Geometry 197

31 Name Date Ready to Go On? Chapters 11-1 Quiz cont d LESSON Quizzes More Copy if needed Volumes of Solids (Lessons 1.4, 1.5, 1.6) 40. What is the volume of a rectangular prism with height 3 inches, length 5 inches, and width 9 inches? A. 135 in. 3 B. 108 in. 3 C. 180 in. 3 D. 16 in A right cylinder has a volume of 8300 cubic centimeters and a height of 15 centimeters. Find the radius. Round your answer to two decimal places. 4. Find the volume of the oblique cylinder. Round your answer to two decimal places. 3 cm 8 cm Answers a. 46b. 43. Find the volume of the pyramid. 17 cm 18 cm 14 cm 44. What is the volume of a regular square pyramid with base edges of length 48 in. and a height of 10 in.? A in. 3 B. 3,040 in. 3 C in. 3 D. 560 in A cone-shaped bottle has a radius of 18 centimeters at its base. If the bottle holds 6400 cubic centimeters when completely filled, what is its height? A cm B cm C. 6.6 cm D. 6.3 cm 46. A sphere has a volume of 1316 cubic meters. Round your answers to two decimal places. a. Find the radius. b. Find the surface area. 198 Holt McDougal Geometry

32 CHAPTER 1 Ready to Go On? Cumulative Review For use after Chapter 1 Find the values of x and y. (Lesson.7) 1.. (5y 6)8 (3x 1 5)8 (5x 19)8 (4y 1 7)8 (y 11)8 3x8 (4x 5)8 (3y 69)8 3. Graph the triangle with vertices X(, 5), Y(1, 3), and Z(3, 4). Then graph a triangle congruent to nxyz. (Lesson 4.) Find the value of x. Round your answers to the nearest tenth. (Lessons 7.5 and 7.6) 4. 9 x x 6. x Find the sum of the measures of the interior angles of the indicated convex polygon. (Lesson 8.1) gon 8. octagon 9. -gon 10. pentagon Points P, Q, R, and S are the vertices of a quadrilateral. Give the most specific name for PQRS. Justify your answer. (Lessons 8.4, 8.5, and 8.6) 11. P(0, 1), Q(, 1), R(5, 6), S(0, 3) 1. P(5, 1), Q(1, 5), R(1, 3), S(3, 1) 13. P(, 4), Q(6, 6), R(9, 3), S(5, 1) 14. P(1, ), Q(, 4), R(1, 6), S(0, 4) Describe the composition of transformations. (Lesson 9.5) 15. y 16. C9 B9 y C G 4 A9 F9 F G9 H H9 H0 G0 F0 x C 0 1 B 0 1 A B x Review A0 Holt McDougal Geometry 1

33 CHAPTER 1 Ready to Go On? Cumulative Review cont d For use after Chapter 1 How many lines of symmetry does the figure have? (Lesson 9.6) 17. A rectangle 18. An isosceles trapezoid 19. A regular octagon Use the given information to write the standard equation of the circle. (Lesson 10.7) 0. The center is (3, ) and a point on the circle is (5, ). 1. The center is (6, 1) and a point on the circle is (1, 6). The ratio of the areas of two similar figures is given. Write the ratio of the lengths of the corresponding sides. (Lesson 11.3). Ratio of areas 5 5:9 3. Ratio of areas : 5 4. Ratio of areas 5 3:4 Find the areas of the sectors formed by DFE. Round your answers to the nearest tenth. (Lesson 11.5) 5. G D 6. D F ft E G F 13 cm 1608 E Find the surface area and the volume for each right solid. Round your answers to two decimal places, if necessary. (Lessons ) 7. 8 cm m 3.8 m 5 ft 5 cm 7 ft 3 ft Review in ft 3. 7 in. 9 in 9 in. 4 in. Holt McDougal Geometry

34 CHAPTER 1 Ready to Go On? Enrichment Enrichment Three-Dimensional Figures Answer each question. 1. The cube at the right is built from 64 smaller cubes. The cube is spray painted on all sides. How many of the cubes are painted on three faces? How many cubes are painted on two faces? How many cubes are painted on only one face?. A triangle has vertices J(4, 1, 3), K(, 6, 7), and L(10, 4, 1). Classify the triangle by the length of its sides. Find the perimeter of the triangle. 3. A polygon is called a Platonic Solid if all of its faces are regular polygons and if each vertex is the point of intersection of the same number of edges. The Platonic Solids are pictured at the right. Complete the table below. Verify that Euler s Formula works for the Platonic Solids. Tetrahedron Name Shape of face Number of edges at each vertex Number of Faces Number of Edges Number of Vertices Tetrahedron Hexahedron Octahedron Octahedron Dodecahedron Dodecahedron Icosahedron Icosahedron Does Euler s Formula work for each Platonic Solid? 34 Holt McDougal Geometry

35 Answers continued 3. J9(5, ), K9(1, 5) 4. slope of y 5 x is 1; 5 () slope of JJ9 5 } (5) 5 } 1 5. J 0(5, ), 1 K 0(1, 5) 6. F9(4, ), E9(1, ), D9(3, 3) 7. F0(4, ), E0(1, ), D0(3, 3) Check student graphs. Coordinates are given. 8. S9(0, 5), T9(4, ), U9(, 1) 9. S-(5, 0), T -(, 4), U-(1, ) 10. S0(0, 5), T 0(4, ), U0(, 1) 11. T 9(, 3), U9(5, 5), V9(1, 6) 1. T 0(, 3), U0(5, 5), V0(1, 6) 13. T -(3, ), U-(5, 5), V-(6, 1) 14. M9(18, 15), N9(9, 6), O9(3, 1) 15. B9(8, 4), C9(6, 6) 16. A0(1, ), B0(1, 4), C 0(3, 1) 17. A0(3, 1), B0(1, 3), C 0(4, 5) 18. D0(, 3), E0(1, 6), F 0(3, 7) 19. D0(, 4), E0(1, 1), F 0(3, 0) Check 1. R9(5, 6), S9(3, 8). R0(, 5), S0(4, 3) 3. G9(, 4) 4. G0(, 4) 5. E9(0, 6), F9(6, 1), G9(4, ) 6. D9(, 5), E9(4, ) C. Symmetry 1. One line of symmetry. Five lines of symmetry 3. Four lines of symmetry 4. One line of symmetry 5. Check student s triangles must be scalene triangle 6. No 7. Yes; Yes; 180 Check True 4. Yes D. Circles and Special Segments 1. J. } JK or } JL 3. } LK 4. } KO 5. ###$ KP MN 7. radius is 3 units 8. Yes, x SY J } 3 Check M } LN. } KL or } KJ OM E. Angle Relationships in Circles (x 1 1) 1 (y ) (x 4) 1 (y 1 ) x 1 y (x 1) 1 (y ) 5 5 Check (x 1 6) 1 (y ) 5 36 Ready to Go On? Chapters 11 1 Intervention A. Perimeter and Area : 3; 4 : 9; 9 in : 5; 9 : 5; 8.64 ft : ; 1 : 4; 64 yds 0. 5 : 8; 5 : 64; 10.7 m 1. Ï } 3 ø 1.7 units. 5 units units Ï } 3 ø units units units % 8. 75% 9. 50% % L K Answer Key Holt McDougal Geometry 45

36 Answers continued Answer Key Check perimeter: 1 : Ï } ; area: 1 : 6. } 1 ø 16.7% units 6 B. Circumference and Area of Circles feet yards miles mm miles meters inches units yards km feet light-years ft yards miles mm miles m inches units units units units units units units Check in in. 3. area: 9.4 m ; arclength: 3.14 m cm cm cm 7. 10p and 0p C. Solids 1. Yes, it is a triangular prism. 5 faces, 6 vertices, 9 edges.. Yes, it is a polyhedron, but it is neither a prism nor a pyramid. 5 faces, 6 vertices, 9 edges. 3. Yes, it is a trapezoidal prism. 6 faces, 8 vertices, 1 edges. 4. Yes, it is a trapezoidal pyramid. 5 faces, 5 vertices, 8 edges a circle 10. an ellipse 11. an equilateral triangle 1. a square 13. S cm, V cm S m, V m S in., V in S miles, V miles 3 Check 1. F 5 8, E 5 18, V 5 1. F 5 11, E 5 0, V Point, square, rectangle, equilateral triangle, isosceles triangle, scalene triangle, line 4a. : 3 4b. 900 D. Surface Area of Solids 1. a right pentagonal prism; S ft. a right cylinder; S in. 3. a cube; S ft 4. a triangular pyramid (tetrahedron); S ft units 6. 4 units units units ft m cm yd in ft units units units units units units units units units units 5. r cm 6. r m 7. S in. 8. S yds 9. S miles 30. r feet Check 1. Two hexagons, each of area 54 Ï } 3 units. Six rectangles, each of area 7 units units 3. More; the hexagon can be inscribed in the circle, so the prism fits entirely inside the cylinder units 5. 4π cm 6. Ï } 6 units E. Volumes of Solids cm ft m cm cm cm m m cm ft cm ft mm m in units ft in mm miles ,778,684. cm 3. 1,563, m 3 Check 1. 4 units units units m 46 Holt McDougal Geometry

37 Answers continued Answer Key Ready to Go On? Chapters 11 1 Quiz B 4a. 150 cm 4b. 60 cm 5a. 7 m 5b. 7 m D 8. B 9a. 14 in. 9b. 168 in. 9c. 35. in.; 70.5 in. 10. C 11a. 9.5 ft 11b ft 1. 5% 13. D in. 15. A 16. 4π cm π cm 18. D 19. } 343π < A 1. Polyhedron; Pentagonal pyramid; 6 faces, 6 vertices, 10 edges. B 3. B 4. Isosceles triangle 5. C 6. cm 3 7. Ï } : Ï } cm 9. A ft 31. D in. 33. A 34. C cm 36. D cm in. 39. A 40. A cm cm cm C 45. A 46a m 46b m 50 Holt McDougal Geometry

38 Answers continued 18.F G 19.F G ; yes 1. r x x 5105; y k 5 6; n m ; n x x 5 Ready to Go On? Chapter 11 Cumulative Review m m 3. Sample answer: } Sample answer: 1,, Ï } 5 5. m M 5 368, m P 5 368, m N Not a right triangle 7. right triangle; altitude ø right triangle; altitude ø Perimeter 5 78 units 3x 1 58 M 7z 7 4z 3 P N x x x 5 1. x 5 11 A9 B9 C9 D9 E9 F9 13. F G F G W9 X9 Y9 Z F G y 1 P 0(3, 3) R0(6, 3) y P 0(3, 3) 0(5, 4) x x R0(1, 3) 17. Q0(5, 1) 1 R0(4, 0) 1 P 0(4, 3) y x 19. No; radii are different. 0. Yes; both arcs have congruent central angles and congruent radii. 1. x 5 4. x x ø π Ready to Go On? Chapter 1 Cumulative Review 1. x 5 1; y x 5 5; y Sample answer: y X 6 A Y 4. x x x Kite; } PQ > } PS and } QR ø } SR are pairs of consecutive sides 1. Trapezoid; } PQ i } SR and } PS is not i to } QR 13. Parallelogram; } QR i } PS and } RS i QP } 14. Rhombus; } PQ i } RS, } PS i } QR, and } QS } PR 15. Translate down 5 and left, then reflect over the y-axis. 16. Rotate 908 counterclockwise about the origin, then translate down 6 units (x 1 3) 1 (y ) (x 6) 1 (y 1 1) : : Ï } 3 : ft ; ft cm ; 95.0 cm 7. S cm ; V cm 3 8. S 5 14 ft ; V ft 3 9. S m ; V m S in. ; V in S ft ; V ft 3 3. S in. ; V in. 3 Z 4 B C x Answer Key 0(9, 6) Holt McDougal Geometry 55

39 Name Date CHAPTER 9 Ready to Go On? Enrichment CHAPTER 10 Ready to Go On? Enrichment Applying Geometric Formulas 1. The radius of the circle is 9 yd. Find the probability that a point chosen randomly inside the circle is in the shaded area. Round to the nearest hundredth Find the probability that a point chosen randomly inside the hexagon is in the shaded area Find the probability that a point chosen randomly inside the circle is in the shaded area Estimate the shaded area inside the figure. Round to the nearest hundredth units 5. Find the area and perimeter of the octagon. P 17.9 units, A 4 unit s 3 m y A y 5 in. x x Enrichment Enrichment Angles and Segments in Chords Use the corresponding diagrams to answer each question. 1. In the circle at the right, m DAB 197, m BCD 163, and m AD 10. Find the following: m DAB 81.5 m BCD 98.5 m DC 7 m CB 91 m BA 77. A photographer is standing at point C taking a group picture at a high school reunion. His camera has a 73 field of vision. The radius of the circle is 14 ft. What is the length of _ BD, the line along which the group of people are standing? Round your answer to the nearest tenth of a foot. 6.8 ft 3. Write a paragraph proof to show that the sum of an angle formed by two tangents to a circle and the minor arc that is intercepted, is 180. m EAD 1 m ECD m DE, m ECD 360 m DE. A 96 B D 73 A C D 84 By substitution m EAD 1 (360 m DE m DE) 1 (360 m DE); Applying the Distributive Property, m EAD 180 m DE. Add m DE to both sides: m EAD m DE Find x and the length of the secant segment. x, AB 8 (x 8 is a solution to the quadratic but since x represents a length, it is not applicable.) A E 6 E 4 x C A B D C B B C Holt McDougal Geometry 31 3 Holt McDougal Geometry Name Date Name Date CHAPTER 11 Ready to Go On? Enrichment CHAPTER 1 Ready to Go On? Enrichment Lines and Arcs in Circles Use your knowledge of lines and arcs in circles to answer each question. 1. The picture at the right shows the part of a windshield covered by the windshield wiper. The arc of the wiper is 115. The length of the entire wiper is 18 in. and the blade is 1 in. Find the area of the windshield covered by the wiper. Round your answer to the nearest inch. 89 in.. Find the area of the shaded region in the figure at the right. Round your answer to the nearest hundredth c m B 115 E D 9 cm 80 C Enrichment Enrichment Three-Dimensional Figures Answer each question. 1. The cube at the right is built from 64 smaller cubes. The cube is spray painted on all sides. How many of the cubes are painted on three faces? 8 How many cubes are painted on two faces? 4 How many cubes are painted on only one face? 4. A triangle has vertices J(4, 1, 3), K(, 6, 7), and L(10, 4, 1). Classify the triangle by the length of its sides. Isosceles Find the perimeter of the triangle. P A polygon is called a Platonic Solid if all of its faces are regular polygons and if each vertex is the point of intersection of the same number of edges. The Platonic Solids are pictured at the right. Complete the table below. Verify that Euler s Formula works for the Platonic Solids. Tetrahedron 3. Janice is planting a garden in the shape of the sector at the right. How much edging should she buy? Round your answer to the nearest foot. 35 ft 4. The straight edges of the sector at the right are joined together to form a cone. What is the circumference of the base of the cone? ft Name Tetrahedron Shape of face equilateral triangle Number of edges at each vertex Number of Faces Number of Edges Number of Vertices Hexahedron square Octahedron What is the radius base of the cone? C 9.77 in in. r in A clock has a minute hand that is 7 in. long. How far has the tip of the minute hand traveled between 10:5 A.M. and 10:50 A.M.? Round your answer to the nearest tenth of an inch in. Octahedron Dodecahedron Icosahedron equilateral triangle regular pentagon equilateral triangle Does Euler s Formula work for each Platonic Solid? Yes Dodecahedron Icosahedron Holt McDougal Geometry Holt McDougal Geometry Copyright by Holt McDougal. 58 Holt McDougal Geometry All rights reserved.