Volume of Cylinders As with prisms, the area of the base of a cylinder tells the number of cubic units in one layer. The height tells how many layers there are in the cylinder. The volume V of a cylinder with radius r is the area of the base B times the height h. V = Bh, where B = πr 2, or V = πr 2 h Find the volume of the cylinder. Round to the nearest tenth. V πr 2 h Volume of a cylinder V π(2) 2 (5) Replace r with 2 and h with 5. V 62.8318 Use a calculator The volume is about 62.8 cubic inches. Volume of Cones A cone is a three-dimensional shape with one circular base. The volume V of a cone with radius r is one third the area of the base B times the height h. V = 1 3 Bh or V = 1 3 πr2 h Find the volume of the cone. Round to the nearest tenth. V = 1 3 πr2 h Volume of a cone V = 1 3 (π 62 12) r = 6 and h = 12 V 452.4 The volume is about 452.4 cubic feet.
Volume of Spheres A sphere is a set of all points in space that are a given distance from a given point. The volume V of a sphere with radius r is four thirds the product of π and the cube of the radius r. V = 4 3 πr3. Find the volume of the sphere. Round to the nearest tenth. V = 4 3 πr3 Volume of a sphere V = 4 3 (π 43 ) r = 4 V 268.1 The volume is about 268.1 cubic feet. Use a calculator. Surface Area of Cylinders The surface area S.A. of a cylinder with height h and radius r is the sum of the area of the curved surface and the area of the circular bases. S.A. = 2πrh + 2πr 2 Find the total surface area of the cylinder. Round to the nearest tenth. S.A. = 2πrh + 2πr 2 Surface area of a cylinder S.A. = 2π(6)(8) + 2π(6) 2 Replace r with 6 and h with 8. S.A. 527.7875 The surface area of the cylinder is about 527.8 square meters.
Surface Area of Cones The lateral area L.A. of a cone is π times the radius times the slant height, or L.A. = πrl. The total surface area of a cone with slant height l and radius r is the lateral area plus the area of the base, or S.A. = L.A. + πr 2 or S.A. = πrl + πr 2. Find the lateral and total surface areas of the cone. Round to the nearest tenth. Lateral Surface Area Total Surface Area L.A. = πrl S. A. = L. A. + πr 2 L.A. = π 3 5 r = 3, l = 5 S. A. = 47.1 + π 3 2 L.A. 47.1 S. A. 75.4 The lateral and total surface areas of the cone are about 47.1 and 75.4 square centimeters. Changes in Dimensions A scale factor is how much larger or smaller one solid is than another. Similar solids have the same shape and their corresponding linear measures are proportional. For similar solids A and B: surface area of B = (surface area of A) (scale factor) 2 1 The surface area of a rectangular prism is 144 square centimeters. Find the surface area of a similar prism that is larger by a scale factor of 4. S.A. = 144 4 2 S.A. = 144 16 Square 4. S.A. = 2,304 cm 2 Multiply by the square of the scale factor. For similar solids A and B: Volume of B = (volume of A) (scale factor) 3 2 The volume of a rectangular prism is 120 cubic feet. Find the volume of a similar prism that is larger by a scale factor of 2.Circumference
cente The circumference, C, is the distance around a circle. The diameter, d, is the distance across a circle through its center. The radius, r, is the distance from the center to any point on a circle. The diameter of a circle is twice its radius. The radius is half the diameter. The circumference of a circle is equal to π times its diameter or π times twice its radius. d = 2r r = d 2 C = πd C = 2πr 1 The radius of a circle is 7 meters. Find the diameter. d = 2r d = 2 7 Replace r with 7. d = 14 The diameter is 14 meters. Multiply. 2 Find the circumference of a circle with a radius that is 13 inches. Use 3.14 for π. Round to the nearest tenth. C = 2πr Write the formula. C 2 3.14 13 Replace r with 13 and π with 3.14. C 81.64 Multiply. Rounded to the nearest tenth, the circumference is about 81.6 inches. V = 120 2 3 Multiply by the cube of the scale factor. V = 120 8 Cube 2. V = 960 ft 3
Area of Circles The area A of a circle equals the product of pi (π) and the square of its radius r. A = πr 2 1 Find the area of the circle. Use 3.14 for π. A = πr 2 Area of circle A 3.14 5 2 Replace π with 3.14 and r with 5. A 3.14 25 5 2 = 5 5 = 25 A 78.5 The area of the circle is approximately 78.5 square centimeters. The formula for the area of a semicircle or half a circle is A = 1 2 πr2. 2 Find the area of a semicircle that has a diameter of 9.4 millimeters. Use 3.14 for π. Round to the nearest tenth. A = 1 2 πr2 Area of semicircle A = 1 2 3.14 4.72 Replace π with 3.14 and r with 9.4 2 or 4.7. A 34.7 Multiply. The area of the semicircle is approximately 34.7 square millimeters. Volume of Pyramids A pyramid is a three-dimensional shape with one base and triangular lateral faces. The volume V of a pyramid is one third the area of the base B times the height h. V = 1 3 Bh Find the volume of the pyramid. Round to the nearest tenth. V = 1 3 Bh V = 1 3 (lw)h Volume of a pyramid The base is a rectangle, so B = lw V = 1 (4.3 3.2) 11 l = 4.3, w = 3.2, h = 11 3 V 50.5 The volume is about 50.5 cubic meters.
Volume of Prisms The volume of a three-dimensional shape is the measure of space occupied by it. It is measured in cubic units such as cubic centimeters (cm 3 ) or cubic inches (in 3 ). The volume of the shape at the right can be shown using cubes. A rectangular prism is a three-dimensional shape that has two parallel and congruent sides, or bases, that are rectangles. To find the volume of a rectangular prism, multiply the area of the base times the height, or find the product of the length l, the width w, and the height h. V = Bh or V = lwh Find the volume of the rectangular prism. V = lwh Volume of a rectangular prism V = 5 6 8 Replace l with 5, w with 6, and h with 8. V = 240 Multiply. 8 in. The volume is 240 cubic inches. Surface Area of Prisms The sum of the areas of all the surfaces, or faces, of a three-dimensional shape is the surface area. The surface area S.A. of a rectangular prism with length l, width w, and height h is the sum of the areas of its faces. S.A. = 2lw + 2lh + 2wh Find the surface area of the rectangular prism. Faces Area top and bottom 2 (4 3) = 24 front and back 2 (4 2) = 16 two sides 2 (2 3) = 12 sum of the areas 24 + 16 + 12 = 52 Alternatively, replace l with 4, w with 3, and h with 2 in the formula for surface area. S. A. = 2lw + 2lh + 2wh = 2 (4 3) + 2 (4 2) + 2 (3 2) = 24 + 16 + 12 = 52 So, the surface area of the rectangular prism is 52 square meters.
Surface Area of Pyramids The total surface area S.A. of a regular pyramid is the lateral area L.A. plus the area of the base. S.A. = B + L.A. or S.A. = B + 1 2 Pl 1 Find the total surface area of the pyramid. S.A. = B + 1 2 Pl Surface area of a pyramid S.A. = 49 + 1 (28 6) P = 4(7) or 28, l = 6, B = 7 7 or 49 2 S.A. = 133 Simplify The surface area of the pyramid is 133 square inches. 2 Find the total surface area of the pyramid. S.A. = B + 1 2 Pl Surface area of a pyramid S.A. = 15.6 + 1 (18 5) P = 3(6) or 18, l = 5, B = 15.6 2 S.A. = 60.6 The surface area of the pyramid is 60.6 square meters.