Presentation on theme: "EXPLORING VOLUME The volume of a solid is the number of cubic units"— Presentation transcript:
1 EXPLORING VOLUMEThe volume of a solid is the number of cubic unitscontained in its interior. Volume is measured in cubicunits, such as cubic meters (m3).
2 EXPLORING VOLUME VOLUME POSTULATES s The volume of a cube is the cube Postulate 27 Volume of a CubesThe volume of a cube is the cubeof the length of its side, or V = s3.Postulate 28 Volume Congruence PostulateIf two polyhedra are congruent, then they have thesame volume.Postulate 29 Volume Addition PostulateThe volume of a solid is the sum of thevolumes of all its nonoverlapping parts.
3 The box shown is 5 units long, 3 units wide, and 4 units high. How Finding the Volume of a Rectangular Prismunit cube1The box shown is 5 units long,3 units wide, and 4 units high. Howmany unit cubes will fit in the box?What is the volume of the box?4 units3 units5 units
4 The base of the box is 5 units by 3 units. Finding the Volume of a Rectangular PrismSOLUTION1unit cube1The base of the box is 5 units by 3 units.This means 5 • 3, or 15 unit cubes, willcover the base.14 unitsThree more layers of 15 cubes each canbe placed on top of the lower layer to fillthe box. Because the box contains 4 layerswith 15 cubes in each layer, the boxcontains a total of 4 • 15, or 60 unit cubes.3 units5 unitsThe volume of the prism can be found by multiplying the area of thebase by the height. This method can also be used to find the volumeof a cylinder.
5 FINDING VOLUMES OF PRISMS AND CYLINDERS THEOREMTheorem Cavalieri’s PrincipleIf two solids have the same height and the samecross-sectional area at every level, then they havethe same volume.
6 FINDING VOLUMES OF PRISMS AND CYLINDERS hBBBhrVOLUME THEOREMSTheorem Volume of a PrismThe volume V of a prism is V = B h, whereB is the area of a base and h is the height.Theorem Volume of a CylinderThe volume V of a cylinder is V = B h = r 2 h, whereB is the area of a base, h is the height, andr is the radius of a base.
7 Using Volumes in Real Life 0.66 ft1.31 ft0.39 ft0.33 ftThe 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 largerectangleArea of smallrectangleB =– 2 •= (1.31) (0.66) – 2(0.33) (0.39) 0.61 ft 2Using the formula for the volume of a prism, the volume isV = Bh 0.61(0.66) 0.4 ft 3.
8 Using Volumes in Real Life To find the weight of the block, multiplythe pounds per cubic foot, 145 lb/ft 3, bythe number of cubic feet, 0.40 ft 3.114 lb1 ft 3• 0.4 ft 3Weight = 58 lb