Introduction to Surface Area Word Problems
Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.
—BERTRAND RUSSELL (1872–1970)
This chapter will include details on common geometric solids and the use of the formulas needed to find the surface area of each figure. Strategies on solving word problems with surface area will also be modeled and explained.
Geometric Solids
Geometric solids are threedimensional figures that are defined by the number of faces, edges, and vertices that they have.
Tip:
Here are some important terms to know about geometric solids.
Face: the twodimensional "sides" of a figure
Edge: the line segment formed where two faces meet
Vertices: the points where two or more edges meet

The surface area of a threedimensional figure is the number of square units it takes to cover each face of the figure. You can think of it as the outer cover of the object.
Rectangular Prisms
A rectangular prism is a geometric solid that has a rectangle for each of the faces. There are three dimensions in this solid: a length, a width, and a height. This is shown in the following figure.
In a rectangular prism, each of the opposite faces is congruent. To find the surface area, find the area of three of the rectangular faces and multiply each by two to represent the side opposite. Then, add these values together. The formula is SA = 2lw + 2lh + 2wh where w is the width, l is the length, and h is the height of the prism.
Example
What is the volume of a rectangular prism with a length of 5 m, a width of 6 m, and a height of 10 m?
Read and understand the question. This question is looking for the surface area of a rectangular prism. Each of the three dimensions is known.
Make a plan. Use the formula SA = 2lw + 2lh + 2wh, and substitute the given values for the length, width, and height.
Carry out the plan. The formula becomes SA = 2(5)(6) + 2(5)(10) + 2(6)(10). Multiply to get SA = 60 + 100 + 120. Add to get a surface area of 280 m^{2}.
Check your answer. Substitute the values into the formula again to doublecheck your solution. The formula SA = 2(5)(6) + 2(5)(10) + 2(6)(10) simplifies to 2(30) + 2(50) + 2(60) = 60 + 100 + 120 = 280. This answer is checking.
Tip:
Surface area is a measure of area, so it is always calculated in square units.

A Special Prism: Cubes
A cube is a special type of rectangular prism. The faces of a cube are six congruent squares, as shown in the following figure.
The area of each face can be found by multiplying the length of each edge of the cube by itself. To find the surface area of a cube, find the area of one face and multiply this area by six to represent all of the faces. Therefore, the formula is SA = 6e^{2}, where e is the length of an edge of the cube.
Example 1
What is the surface area of a cube with an edge that measures 7 in.?
Read and understand the question. This question is looking for the surface area of a cube when the measure of the edge of the cube is given.
Make a plan. Use the formula SA = 6e^{2}and substitute the value of e. Remember that the surface area will be represented in square units.
Carry out the plan. Substitute into the formula to get SA = 6(7)^{2}. Evaluate the exponent first to get SA = 6(49). Multiply to find the surface area: SA = 294 in.^{2}.
Check your answer. To check your answer, divide the total surface area by 6: 294 ÷ 6 = 49. Then, take the positive square root of 49 to find the measure of the edge of the cube. The positive square root of 49 is 7, so this result is checking.
Example 2
What is the measure of the edge of a cube with a total surface area of 54 ft.^{2}?
Read and understand the question. This question is looking for the measure of the edge of a cube when the total surface area is known.
Make a plan. Use the formula for the surface area of a cube, and work backward to solve for e.
Carry out the plan. Substitute the values into the formula to get 54 = 6e^{2}. Divide each side of the equation by 6 to get 9 = e^{2}. Take the positive square root of each side of the equation to find the value of e: e = 3 ft.
Check your answer. To check this answer, substitute the value of e into the surface area formula. The formula becomes SA = 6(3)^{2}. Evaluate the exponent to get SA = 6(9). Multiply to get a surface area of 54 ft.^{2}. This result is checking.
Solids With Curved Surfaces
Cylinders
A cylinder is a solid with two circles as the bases. A right cylinder has the height perpendicular to the bases; in other words, the cylinder is upright and not tilted at all.
To find the surface area of a cylinder, imagine a label on a soup can. If the label is peeled off and flattened, it is in the shape of a rectangle. The height of this rectangle is the height of the cylinder, and the base of this rectangle is the circumference of the base of the cylinder. (See the following figure.) The area of this rectangle can be expressed as πdh. The area of each of the circular bases can be found by the expression πr^{2}, the formula for the area of a circle.
Thus, the formula for the surface area of a cylinder is SA = 2πr^{2}+ πdh, where r is the radius of the base, d is the diameter of the base, and h is the height of the cylinder.
Example
What is the surface area of a cylinder with a height of 8 m and a base with a radius of 6 m? (Leave your answer in terms of π.)
Read and understand the question. This question asks for the surface area of a cylinder when the dimensions are given.
Make a plan. Use the formula for the surface area of a cylinder, and substitute the known values. Evaluate to find the surface area.
Carry out the plan. Because the radius is 6 m, the diameter is 2 × 6, or 12 m. The formula SA = 2πr^{2}+ πdh becomes SA = 2π(6)^{2}+ π(12)(8). Evaluate the exponents and multiply to get SA = 72π + 96π. Combine like terms to get SA = 168πm^{2}.
Check your answer. Substitute again into the formula to doublecheck your solution. The formula is
SA = 2π(6)^{2}+ π(12)(8) = 2π(36) + 96π = 72π + 96π = 168π
This answer is checking.
Spheres
A sphere is a round threedimensional figure in the shape of a ball. An example of a sphere is shown in the following figure.
Think about having to wrap a baseball in paper. The amount of paper needed would be approximately equal to four times the area of a circle with the same radius. Therefore, the formula for the surface area of a sphere is SA = 4πr^{2}, where r is the length of the radius of the sphere.
Example
What is the surface area of a sphere with a radius of 3 m? (Leave your answer in terms of π.)
Read and understand the question. This question is looking for the surface area of a sphere when the radius of the sphere is known.
Make a plan. Use the formula SA =4πr^{2}, and substitute r = 3.
Carry out the plan. The formula becomes SA = 4π(3)^{2}. Evaluate the exponent to get SA = 4π(9). Multiply to get the total surface area: SA = 36π m^{2}.
Check your answer. To check this solution, work backward, and divide the total surface area by 4π: 36π divided by 4π is equal to 9. Take the positive square root of 9 to get r = 3. This answer is checking.
Find practice problems and solutions for these concepts at Surface Area Word Problems Practice Questions.
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