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Measurements Involving Three-Dimensional Figures

Objective

Now we’ll turn our attention to measuring various characteristics of three-dimensional figures. Specifically, we’ll review surface areas and volumes of prisms and other common solids. At the conclusion of the lesson, we should be able to calculate surface areas and volumes for almost all regular solids.

Previously Covered

  • A polyhedron is a three-dimensional solid whose faces are polygons.
  • A prism is a polyhedron that has two parallel, congruent faces called bases. The other faces are parallelograms.
  • A pyramid is a polyhedron whose base is a polygon and whose faces are triangles with a common vertex.

Measuring Polyhedra

Surface Area

The surface area of a three-dimensional figure is the sum of the areas of its faces.

Since the faces of a polyhedron are polygons, the surface area of a polyhedron is the sum of the areas of its polygonal faces.

Let’s look at a prism first. Each of the faces of the rectangular prism below is a rectangle. What is its surface area?

Rectangular prism

  • The prism has two rectangular faces with dimensions 3 in. x 8 in. Each has area (3 in)(8 in) = 24 in2
  • Two faces have dimensions 8 in. x 12 in. Each has area (8 in)(12 in) = 96 in2
  • Two faces have dimensions 3 in. x 12 in. Each has area (3 in)(12 in) = 36 in2

The surface area of the prism is:

Surface area of a prism

Question

What is the surface area of a square pyramid with sides of 4 cm and the faces measuring 5.5 cm in height? Round your answer to the nearest square centimeter.

  1. 44 cm2
  2. 52 cm2
  3. 60 cm2
  4. 104 cm2

Reveal Answer

Choice C is the correct answer. The surface area of the pyramid is the sum of the areas of its polygonal faces. The pyramid has a square base with sides 4 cm long, so its area is A = bh = (4 cm)(4 cm) = 16 cm2. The pyramid has four triangular faces, each with base 4 cm and height 5.5 cm. Each triangular face has area Solution formula, so the sum of the areas of the four triangular faces is 4(11 cm2) = 44 cm2. The surface area of the pyramid is 16 cm2 + 44 cm2, or 60 cm2.

Volume

A general formula for the volume of a prism is V = Bh, where B is the area of the base and h is the height.

Since volume is three-dimensional, it is expressed in cubic units, like cubic inches (in3), cubic centimeters (cm3), and cubic feet (ft3).

Let’s look back at the prism from the previous section. What is its volume?

Rectangular prism
The bases are any pair of congruent faces; this prism has three pairs of congruent faces. We’ll let the faces with dimensions 8 in. x 12 in. be the bases, since they are on the top and bottom (and that’s where we like our bases). Now compute the formula for volume:

Volume of a prism
The volume of a pyramid is given by the formula V = 1/3 Bh, where B is the area of the base and h is the height.

In other words, the volume of a pyramid is one-third of the volume of a prism with the same base and height. Remember, base and height must be perpendicular.

Question

What is the volume of a pyramid with a base that measures 4 cm by 2 cm and a height of 5 cm? Round your answer to the nearest tenth of a cubic centimeter.

  1. 29.3 cm3
  2. 13.33 cm3
  3. 93.6 cm3
  4. 128.7 cm3

Reveal Answer

Choice B is correct. The volume of the pyramid is given by the formula V = 1/3 Bh, where B is the area of the base. So, first multiply 2 by 4 to get the area of the base (8). Then, multiply that value by the height (5) to get 40. Divide this by 3 to get 13.33. If you answered C, you may have forgotten to multiply the product of base and height by one-third.

Review

  • The surface area of a three-dimensional figure is the sum of the areas of its faces.
  • The volume of a prism is given by the formula V = Bh, where B is the area of the base and h is the height.
  • The volume of a pyramid is given by the formula V = 1/3 Bh, where B is the area of the base and h is the height.

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