Chapter 11 Areas of Polygons and Circles

Transcription

1 Section 11-1: Areas of Parallelograms and Triangles SOL: G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; d) solve real-world problems about similar geometric objects. Objective: Find perimeters and areas of parallelograms Find perimeters and areas of triangles Vocabulary: None New Key Concepts: If a parallelogram has an area of A square units, a base of b units and a height of h units then A = bh If a triangle has an area of A square units, a base of b units and a height of h units then A = ½ bh Area and Perimeter Parallelogram Area A = h b h is height b is base (AB or CD) (similar to a rectangle) A h b B C Triangle Area A = ½ * b * h = ½ * ST * RW h is height (altitude) b is base ( to h) R h D Perimeter Add length of all sides P = AB + DB + DC + CA (or P = 2AB + 2BD) P = RS + RT + ST S Postulate 11.1 Area Addition Postulate: The area of a region is the sum of the areas of its nonoverlapping parts. Postulate 11.2 Area Congruence Postulate: If two figures are congruent, then they have the same area. Eample 1: Find the perimeter and area of parallelogram ABCD W T

2 Eample 2: Find the perimeter and area of parallelogram MNOP Eample 3: Find the perimeter and area of parallelogram DEFG Eample 4: Find the area of triangle RST Eample 5: Find the area of triangle RST Concept Summary: The area of a parallelogram is the product of the base and the height The area of a triangle is one-half the product of the base and the height The formula for the area of a triangle can be used to find the areas of many different figures. Congruent figures have equal areas. Homework: pg ; 1-4, 6-9, 13-14

3 Section 11-2: Areas of Trapezoids, Rhombi and Kites SOL: G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; d) solve real-world problems about similar geometric objects. Objective: Find areas of trapezoids Find areas of rhombi and kites Vocabulary: None New Key Concepts: If a trapezoid has an area of A square units, a bases of b 1 and b 2 units and a height of h units then A = ½ (b 1 + b 2 )h If a rhombus or a kite has an area of A square units and diagonals of d 1 and d 2 units, then A = ½ d 1 d 2 Area of Trapezoids & Rhombi & Kites J L N h b 1 b 2 M Rhombus Area A = ½ * d 1 * d 2 = ½ * AD * BC d 1 and d 2 are diagonals K Trapezoid Area A = ½* h* (b 1 + b 2 ) = ½ * LN * (JK + LM) h is height (altitude) b 1 and b 2 are bases (JK & LM) (bases are parallel sides) C A d 2 d 1 D B d 1 d 2 Kite Area A = ½ * d 1 * d 2 = ½ * ST * RW d 1 and d 2 are diagonals Eample 1: Find the area of trapezoid JKLM:

4 Eample 2: Find the area of rhombus RST: Eample 3: Find the area of the kite: Eample 4: Rhombus RSTU has an area of 64 square inches. Find US, if RT = 8 inches. Eample 5: Trapezoid DEFG has an area of 120 square feet. Find the height of DEFG. Concept Summary: The areas of common polygons can be found using formulas on the Geometry formula sheet The formulas can be used to find missing pieces if given the area Homework: pg ; 1-3, 5-7, 9-11, 48-52

5 Section 11-3: Areas of Circles and Sectors SOL: G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; d) solve real-world problems about similar geometric objects. Objective: Find areas of circles Find areas of sectors of circles Vocabulary: Sector of a circle is bounded by a central angle and its intercepted arc Segment region of a circle bounded by and arc and a chord Theorems: None Key Concepts: If a circle has an area of A square units and radius of r units, then A = πr 2 Area of Circles S Center Radius (r) T Circle Area A = π * r 2 = π * (ST) 2 r is a radius (ST) S is the Center

6 Eample 1: Find the area of circle S Eample 2: Find the area of circle S, if RT = 20 Eample 3: Find the area of the sector Eample 4: An outdoor accessories company manufactures circular covers for outdoor umbrellas. If the cover is 8 inches longer than the umbrella on each side, find the area of the cover in square yards Concept Summary: The area of a circle of radius r units is πr 2 square units The area of a sector is a percentage of the area of the circle determined by the angle of the sector Homework: Pg ; 1, 5, 15-19, 33, 50

7 Section 11-4: Areas of Regular Polygons and Composite Figures SOL: G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; d) solve real-world problems about similar geometric objects. Objective: Find areas of regular polygons Find areas of composite figures Vocabulary: Apothem perpendicular segment from center to side of a regular polygon Irregular figure a figure that can be classified into the specific shapes studied so far. Irregular polygon a polygon that is not regular (all sides and angles are not congruent) Area of Regular Polygons Polygons Area A = 1/2 * P * a P is the perimeter (# of sides * length) a is apothem (altitude like) Regular (all sides equal!) Octagon eample: A = ½ * 8 * * a a Irregular Shapes Area: Sum of Separate Parts Area of Irregular Shapes h y y y y a y y Heagon Eample Area A = ½ * P * a Heagon: A = ½ * 6 * y * a r Eample Area: A = ½ circle + triangle + square A = ½ * πr 2 + ½ * h + * Eample 1: Find the area of the irregular shape to the right Eample 2: Find the area of the field inside the track as pictured below (measurements in yds):

8 Eample 2a-d: Name the parts: a) b) c) d) Eample 3: Find the area of a heagon, if the apothem is 4 3 Eample 4: Find the area of a heagon Eample 5: Find the area of a regular pentagon with a perimeter of 90 meters. Concept Summary: If a regular polygon has an area of A square units, a perimeter of P units, and an apothem of a units, then A = ½ Pa The area of an irregular figure is the sum of the areas of its nonoverlapping parts Homework: pg ; 3, 8-13

9 Section 11-5: Areas of Similar Figures SOL: G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; d) solve real-world problems about similar geometric objects. Objective: Find areas of similar figures by using scale factors Find scale factors or missing measures given the areas of similar figures Vocabulary: none new Theorems: Theorem 11.1: Area of Similar Polygons: If two polygons are similar, then their areas are proportional to the square of the scale factor between them. Key Concepts: Eample 1: If ABCD ~ PQRS and the area of ABCD is 48 square inches, find the area of PQRS.

10 Eample 2: The area of ΔABC is 98 square inches. The area of ΔRTS is 50 square inches. If ΔABC ~ ΔRTS, find the scale factor from ΔABC to ΔRTS and the value of. Eample 3: The area of one side of a skyscraper is 90,000 square feet. The area of one side of a scale model is 200 square inches. If the skyscraper is 720 feet tall, about how tall is the model? Concept Summary: Similar figures have their areas in squared form of the ratio of their sides Homework: pg ; 1-4, 8-11, 30

11 Lesson minute Check Find the area and the perimeter of each parallelogram. Round to the nearest tenth if necessary Find the height and base of this parallelogram, if the area is 168 square units 6. Find the area of a parallelogram if the height is 8 cm and the base length is 10.2 cm. A cm² B cm² C cm² D cm² Lesson minute Check Find the area of each figure. Round to the nearest tenth if necessary Trapezoid LMNO has an area of 55 square units. Find the height. 6. Rhombus ABCD has an area of 144 square inches. Find AC if BD = 16. A. 8 in B. 9 in C. 16 in D. 18 in

12 Lesson minute Check Find the area of each regular polygon. Round to the nearest tenth if necessary. 1. A circle with radius of 8 cm. 2. A circle with a diameter of 14 in. 3. A 120 sector of a circle with radius of 9 m. Find the area of each shaded region. Assume all polygons are regular. Round to the nearest tenth if necessary Find the area of a circle with a diameter of 8 inches. A. 4π B. 8π C. 16π D. 64π Lesson minute Check Find the area of each figure. Round to the nearest tenth if necessary A triangle with a side length of 18.6 m Find the area of the figure 6. Find the figure s area. A. 112 units² B units² C units² D units²