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How many cubic inches of candy will fit in a cone-shaped piñata that is 18 inches long and 12 inches across its base? Round the answer to the nearest hundredth.

678.24 cu. in.

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What is the volume of a cone-shaped party hat that is 10 inches tall and 7 inches across at the base? Round the answer to the nearest hundredth.

128.2 cu. in.

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Summary of geometry formulas

The following charts summarize all of the formulas covered in this chapter.

A table is shown that summarizes all of the formulas in the chapter. The first cell is for Supplementary and Complementary Angles, and says that the measure of angle A plus the measure of angle B equals 180 degrees for supplementary angles A and B and the measure of angle C plus the measure of angle D equals 90 degrees for complementary angles C and D. There is an image of two angles A and B that together form a straight line and two angles C and D that together form a right angle. The next cell says Rectangular Solid and shows the formulas Volume equals LWH and Surface Area equals 2LH plus 2LW plus 2WH. An image of a rectangular solid with sides L, W, and H is shown. The next cell says Triangle. An image of a triangle is shown with sides a, b, and c, vertices A, B, and C, and height h. It says, “For triangle ABC, angle measures measure of angle A plus measure of angle B plus measure of angle C equal 180 degrees. Below this is Perimeter, P equals a plus b plus c. Below this is Area, A equals one-half bh. The next cell says Cube and shows an image of a cube with sides s. It says Volume V equals s cubed and Surface Area S equals 6 times s squared. The next cell says Similar Triangles. It shows two similar triangles ABC and XYZ. It says if triangle ABC is similar to triangle XYZ, then measure of angle A equals measure of angle X, measure of angle B equals measure of angle Y, and measure of angle C equals measure of angle Z. It then says a over x equals b over y equal c over z. The next cell says Sphere and shows an image of a sphere with radius r. It says volume V equals four-thirds times pi times r and Surface Area S equals 4 times pi times r squared. The next cell says Circle. There is an image with two radii labeled r and the diameter labeled d. It says Circumference C equals 2 pi times r and C equals pi times d. It says Area equals pi times r squared. The next cell says Cylinder and shows an image of a cylinder with height h and radius of the base r. It says Volume V equals pi times r squared times h. Below this is V equals Bh. Below that is Surface Area S equals 2 times pi times r squared plus 2 times pi times rh. The next cell says Rectangle and shows an image of a rectangle with sides W and L. It says Perimeter P equals 2L plus 2W, then Area A equals LW. The next cell says Cone and shows an image of a cone with height h and radius of the base r. It says Volume V equals one-third times pi times r squared times h. The last cell says Trapezoid and shows an image of a trapezoid with bases little b and capital B, and height h. It says Area A equals one-half times h times parentheses little b plus capital B. This image shows a row with three columns. The first column says Rectangular solid with the formula below that says volume: V equals LWH. Under this, it says Surface Area: S equals 2LH plus 2LW plus 2WH. An image shows an image of a rectangular solid with the sides labeled L , W and H. The middle column says Rectangle. Under this it says Perimeter P equals 2L plus 2W, then Area A equals LW.  An image of a rectangle with sides W and L. The right column says Cube. Under this it says “Volume: V equals s to the third power.” Under this is says “Surface area: S equals 6 times s squared. Below it is an image of a cube with three sides labeled “s”.

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Key concepts

  • Volume and Surface Area of a Rectangular Solid
    • V = L W H
    • S = 2 L H + 2 L W + 2 W H
  • Volume and Surface Area of a Cube
    • V = s 3
    • S = 6 s 2
  • Volume and Surface Area of a Sphere
    • V = 4 3 π r 3
    • S = 4 π r 2
  • Volume and Surface Area of a Cylinder
    • V = π r 2 h
    • S = 2 π r 2 + 2 π r h
  • Volume of a Cone
    • For a cone with radius r and height h :
      Volume: V = 1 3 π r 2 h

Practice makes perfect

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find the volume and the surface area of the rectangular solid with the given dimensions.

length 2 meters, width 1.5 meters, height 3 meters

  1. 9 cu. m
  2. 27 sq. m

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length 5 feet, width 8 feet, height 2.5 feet

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length 3.5 yards, width 2.1 yards, height 2.4 yards

  1. 17.64 cu. yd.
  2. 41.58 sq. yd.

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length 8.8 centimeters, width 6.5 centimeters, height 4.2 centimeters

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In the following exercises, solve.

Moving van A rectangular moving van has length 16 feet, width 8 feet, and height 8 feet. Find its volume and surface area.

  1. 1,024 cu. ft
  2. 640 sq. ft

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Gift box A rectangular gift box has length 26 inches, width 16 inches, and height 4 inches. Find its volume and surface area.

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Carton A rectangular carton has length 21.3 cm, width 24.2 cm, and height 6.5 cm. Find its volume and surface area.

  1. 3,350.49 cu. cm
  2. 1,622.42 sq. cm

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Shipping container A rectangular shipping container has length 22.8 feet, width 8.5 feet, and height 8.2 feet. Find its volume and surface area.

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In the following exercises, find the volume and the surface area of the cube with the given side length.

5 centimeters

  1. 125 cu. cm
  2. 150 sq. cm

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10.4 feet

  1. 1124.864 cu. ft.
  2. 648.96 sq. ft

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In the following exercises, solve.

Science center Each side of the cube at the Discovery Science Center in Santa Ana is 64 feet long. Find its volume and surface area.

  1. 262,144 cu. ft
  2. 24,576 sq. ft

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Museum A cube-shaped museum has sides 45 meters long. Find its volume and surface area.

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Base of statue The base of a statue is a cube with sides 2.8 meters long. Find its volume and surface area.

  1. 21.952 cu. m
  2. 47.04 sq. m

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Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its volume and surface area.

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Find the Volume and Surface Area of Spheres In the following exercises, find the volume and the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

3 centimeters

  1. 113.04 cu. cm
  2. 113.04 sq. cm

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7.5 feet

  1. 1,766.25 cu. ft
  2. 706.5 sq. ft

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In the following exercises, solve. Round answers to the nearest hundredth.

Exercise ball An exercise ball has a radius of 15 inches. Find its volume and surface area.

  1. 14,130 cu. in.
  2. 2,826 sq. in.

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Balloon ride The Great Park Balloon is a big orange sphere with a radius of 36 feet . Find its volume and surface area.

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Practice Key Terms 3

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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