Lesson 15

Putting All the Solids Together

  • Let’s calculate volumes of prisms, cylinders, cones, and pyramids.

15.1: Math Talk: Volumes

Right triangular prism. For triangular bases, base = 8, height = 3. Prism height = 10.
Right cone. Base radius = 5. Cone height= 6.
Cylinder. Base radius = 5. Height of cylinder = 8.
Rectangular pyramid. Base side lengths = 3 and 4. Height = 6.

Evaluate the volume of each solid mentally.

15.2: Missing Measurements

  1. Answer the questions for each of the two solids shown.

    A

    Cone. Height = 7, slant length =8, radius = r.

    B

    Rectangular prism. Base, side lengths = 7 point 5 and 4 point 5 meters. Prism height = a leg of a right triangle. Other leg = 3 meters. Opposite angle = 65 degrees.
    1. Which measurement that you need to calculate the volume isn’t given?
    2. How can you find the value of the missing measurement?
    3. What volume formula applies?
    4. Calculate the volume of the solid, rounding to the nearest tenth if necessary.
  2. Calculate the volume of each solid, rounding to the nearest tenth if necessary.

    A

    Right Triangular prism. Base, hypotenuse = 73 centimeters, leg = 48 centimeters. Prism height = 31 centimeters.

    B

    Cone, height 10 inches. The side of the cone makes a 30 degree angle with the base of the cone.

 



For a sphere with radius \(r\), its volume is \(\frac43 \pi r^3\) and its surface area is \(4 \pi r^2\). Here is a half-sphere bowl pressed out of a piece of sheet metal with area 1 square foot. What is the volume of the bowl?

Half of a sphere, radius r.

15.3: Spinning into Three Dimensions

Suppose this two-dimensional figure is rotated 360 degrees using 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 and a triangular part.
  1. Draw the solid that would be traced out. Label the dimensions of the solid.
  2. Find the volume of the solid. Round your answer to the nearest tenth if needed.

Summary

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.

Right cone, base has radius 6 centimeters, cone has height 16 centimeters.

To calculate the cylinder toy’s volume, use the expression \(Bh\). The radius measures 9 cm, so the area of the base, \(B\), is \(81\pi\) cm2. The volume is \(202.5\pi\), or approximately 636 cm3, because \(81\pi \boldcdot 2.5 = 202.5\pi\).

For the cone, the height is unknown. A right triangle is formed by the radius 6 cm and the height \(h\), with hypotenuse 16 cm. By the Pythagorean Theorem, \(6^2+h^2=16^2\). Solving, we get \(h=\sqrt{220}\).

Since this is a cone, use the expression \(\frac13 Bh\). The area of the base, \(B\), is \(36\pi\) cm2. The volume is approximately 559 cm3 because \(\frac13 \boldcdot 36 \pi \boldcdot \sqrt{220} \approx 559\). The cylinder-shaped toy uses more material.

Glossary Entries

  • apex

    The single point on a cone or pyramid that is the furthest from the base. For a pyramid, the apex is where all the triangular faces meet.