Unit 1, Lesson 17
Putting All the Solids Together
Integrated Math 3
17.1
Warm-up
Evaluate the volume of each solid mentally.
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Suppose this two-dimensional figure is rotated 360 degrees around the vertical axis shown. Each small square on the grid represents 1 square inch.
Grid with vertical axis drawn. Touching axis, on the right is a shaded figure. Figure includes a rectangular part, base = 2, height =5 and a triangular part above rectangular part with base = 2, height = 3.
Before computing volume, it’s important to select the right formula and find all the dimensions represented in the formula. For example, consider a company that makes two chew toys for dogs. One toy is in the shape of a cylinder with radius 9 cm and height 2.5 cm. The other looks like the cone in the image. The company wants to know which toy uses more material. The toys are solid, not hollow.
To calculate the cylinder toy’s volume, use the expression . The radius measures 9 cm, so the area of the base, , is cm2. The volume is , or approximately 636 cm3, because .
For the cone, the height is unknown. A right triangle is formed by the radius 6 cm and the height , with hypotenuse 16 cm. By the Pythagorean Theorem, . Solving, we get .
Since this is a cone, use the expression . The area of the base, , is cm2. The volume is approximately 559 cm3 because . The cylinder-shaped toy uses more material.