Volume Unit 2

Warm Up Josh marked off a section of the playground in the shape of a right triangle. If the hypotenuse of the triangular section measure 20 meters and one of the legs measures 6 meters, what is the area of the section of the playground? If Josh needs to build a fence around his section, how much fencing would he need? Give the answer in meters.

Objectives: Find the volume of a prism, pyramid, cone, cylinder, and sphere Review other area and volume formulas

Vocabulary A prism is a three-dimensional figure named for the shape of its bases. The two bases are congruent polygons. All of the other faces are parallelograms. A cylinder has two circular bases. A pyramid is named for the shape of its base. The base is a polygon, and all of the other faces are triangles.

More Vocabulary A cone has a circular base. The height of a pyramid or cone is measured from the highest point to the base along a perpendicular line. A sphere is the set of points in three dimensions that are a fixed distance from a given point, the center. A plane that intersects a sphere through its center divides the two halves or hemispheres. The edge of a hemisphere is a great circle.

If all six faces of a rectangular prism are squares, it is a cube. Remember! Triangular prism Rectangular prism Cylinder Height Height Height Base Base Base

Words Numbers Formula VOLUME OF PRISMS AND CYLINDERS Prism: The volume V of a prism is the area of the base B times the height h. Cylinder: The volume of a cylinder is the area of the base B times the height h. B = 2(5) = 10 units2 V = Bh V = 10(3) = 30 units3 B = p(22) V = Bh = 4p units2 = (pr2)h V = (4p)(6) = 24p  75.4 units3

VOLUME OF PYRAMIDS AND CONES (22)

Volume Formula Summary Pyramid: Prism: Cylinder: or Cone: or Sphere:

Try This: Example 1A Find the volume of the figure to the nearest tenth. A. A rectangular prism with base 5 mm by 9 mm and height 6 mm. B = 5 • 9 = 45 mm2 Area of base V = Bh Volume of prism = 45 • 6 = 270 mm3

Try This: Example 1B Find the volume of the figure to the nearest tenth. B = p(82) Area of base B. 8 cm = 64p cm2 V = Bh Volume of a cylinder 15 cm = (64p)(15) = 960p  3,014.4 cm3

Try This: Example 1C Find the volume of the figure to the nearest tenth. C. B = • 12 • 10 1 2 Area of base 10 ft = 60 ft2 V = Bh Volume of a prism 14 ft = 60(14) = 840 ft3 12 ft

Additional Example 1B: Finding the Volume of Pyramids and Cones Find the volume of the figure. B. B = (32) = 9 in2 1 3 V = • 9 • 10 V = Bh 1 3 V = 30  94.2 in3 Use 3.14 for .

Try This: Example 1B Find the volume of the figure. B. B = (32) = 9 m2 7 m 1 3 V = • 9 • 7 V = Bh 1 3 3 m V = 21  65.9 m3 Use 3.14 for .

Additional Example 1: Finding the Volume of a Sphere Find the volume of a sphere with radius 9 cm, both in terms of p and to the nearest tenth of a unit. 4 3 V = pr3 Volume of a sphere = p(9)3 4 3 Substitute 9 for r. = 972p cm3  3,052.1 cm3

Lesson Quiz: Part 2 5. A basketball has a circumference of 29 in. To the nearest cubic inch, what is its volume? 412 in3

1. the triangular pyramid 6.3 m3 Lesson Quiz: Part 1 Find the volume of each figure to the nearest tenth. Use 3.14 for p. 1. the triangular pyramid 6.3 m3 2. the cone 78.5 in3

Additional Example 2A: Exploring the Effects of Changing Dimensions A juice box measures 3 in. by 2 in. by 4 in. Explain whether tripling the length, width, or height of the box would triple the amount of juice the box holds. The original box has a volume of 24 in3. You could triple the volume to 72 in3 by tripling any one of the dimensions. So tripling the length, width, or height would triple the amount of juice the box holds.

Additional Example 2B: Exploring the Effects of Changing Dimensions A juice can has a radius of 2 in. and a height of 5 in. Explain whether tripling the height of the can would have the same effect on the volume as tripling the radius. By tripling the height, you would triple the volume. By tripling the radius, you would increase the volume to nine times the original.