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9.6 Solve geometry applications: volume and surface area  (Page 7/11)

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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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6 inches

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10.4 feet

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

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12.5 meters

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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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9 inches

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7.5 feet

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

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2.1 yards

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