10-6 Volume of Prisms & Cylinders Geometry

EXPLORING VOLUME The volume of a solid is the number of cubic units contained in its interior. Volume is measured in cubic units, such as cubic meters (m3).

EXPLORING VOLUME VOLUME POSTULATES s The volume of a cube is the cube of the length of its side, or V = s3. Volume Congruence Postulate If two polyhedra are congruent, then they have the same volume. Volume Addition Postulate The volume of a solid is the sum of the volumes of all its nonoverlapping parts.

Example 1 Find the volume of the Cube with edge length 15 in. #3 on pg. 746

FINDING VOLUMES OF PRISMS AND CYLINDERS h B B B h r VOLUME THEOREMS Volume of a Prism The volume V of a prism is V = B h, where B is the area of a base and h is the height. Volume of a Cylinder The volume V of a cylinder is V = B h = r 2 h, where B is the area of a base, h is the height, and r is the radius of a base.

Ex. 2 Find the volume of the right rectangular prism 5 cm 3 cm 13 cm

Ex. 3 Find the volume of the right hexagonal prism

Example 4 Find the volume of the oblique cylinder. Give your answers in terms of & rounded to the nearest tenth. 14 in. 18 in EX. Ex. 2

Ex. 5 Find the volume of the cylinder with the base area 121 cm² & a height equal to twice the radius. Give your answers in terms of & rounded to the nearest tenth.

Assignment

Example 3 Use the measurements given to solve for x. a) Rectangular prism, V=396 b) Right cylinder, V=277 Diameter = x ft, height = 5ft

Using Volumes in Real Life 0.66 ft 1.31 ft 0.39 ft 0.33 ft The area of the base can be found as follows: CONSTRUCTION Concrete weighs 145 pounds per cubic foot. To find the weight of the concrete block shown, you need to find its volume. Area of large rectangle Area of small rectangle B = – 2 • = (1.31) (0.66) – 2(0.33) (0.39)  0.61 ft 2 Using the formula for the volume of a prism, the volume is V = Bh  0.61(0.66)  0.4 ft 3.

Using Volumes in Real Life To find the weight of the block, multiply the pounds per cubic foot, 145 lb/ft 3, by the number of cubic feet, 0.40 ft 3. 114 lb 1 ft 3 • 0.4 ft 3 Weight =  58 lb

Example 4 Find the volume of the entire solid. The prisms and cylinders are right. #37 Pg. 748

FINDING VOLUMES OF PRISMS AND CYLINDERS THEOREM Theorem 12.6 Cavalieri’s Principle If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.