GED Mathematical Reasoning: Surface Area

Cube

To find the surface area of this figure, we can find the area of each side — using the “length times width” area formula — and then add those areas up. Or, we can use the following surface area formula, which is based on that idea:  Surface Area (SA) equals 6 times the side measure times the side measure, which simplifies to SA = 6s^2

meas73

SA = 6 \times side \times side
SA = 6s^2

Rectangular prism

If you need to find the surface area of a rectangular prism that’s not a cube, the process is very similar. We can find the area of each individual rectangular side using the “length times width” area formula and then add them up, or use a surface area formula based on that idea.

meas74The surface area formula for a rectangular prism is:  SA equals p times h plus 2 times B

SA = ph + 2B

p refers to the perimeter of the base, which we can find using the perimeter formula:

p = (W + W + L + L)

h stands for the height, and capitol B represents the area of the base:

B = (L \times W)

 

Pyramid

Recall that a pyramid is a three dimensional figure with four triangular faces that come together at a common point, called the vertex. The base of a pyramid can be any figure, but in this course the focus will be on pyramids having a rectangular or square base.

meas77To find the surface area of a pyramid, one strategy involves finding the area of each side and then adding them up.

To do that you’ll need to use the appropriate area formula to find the area of each triangular side and the rectangular or square base.

The other strategy involves a surface area formula based on that idea, which is:

SA = \frac{1}{2}ps + B

Where “p” is the perimeter of the base, “s” is the slant height, which is the height of one triangular side as labeled on the figure shown, and “B” is the area of the base.

 

Cylinder

The surface area formula for a cylinder, which is a three dimensional figure with two circular bases connected by a curved surface, is based on the idea of adding the areas of the two circular ends and the rectangle that forms its side.

meas78

meas79

SA = 2\pi r h + 2\pi r^2

\pi \approx 3.14

 

Cone

To review, a cone is a three dimensional figure with a circular base and a curved side that slants inward so that it meets at a point or vertex.

meas80

To find the surface area of a cone, use the formula:

SA = \pi r s + \pi r^2

 

Sphere

Finally, the surface area of a sphere, which is a three dimensional circular figure – like a ball – can be found using the formula:

SA = 4\pi r^2

meas81

 

Example 1

Dale is a stage hand for the local theater company. He needs to paint a cube that will be used as a prop in the upcoming production. He has one pint of paint which will cover 25 square feet. If the cube has a side measure of 2 feet, will Dale have enough paint for the job?

 

In example one, paint is to be applied on each square surface of a cube having a side measure of 2 feet. Let’s first determine the surface area of the cube and then we can state whether one pint will be enough paint for the job. It may help to first draw a sketch of the cube.

meas72

To find the surface area of this figure, we can find the area of each side — using the “length times width” area formula — and then add those areas up. Or, we can use the following surface area formula, SA = 6s^2

SA = 6 \times side \times side
SA = 6s^2

If we want to use the first strategy mentioned of adding up the areas of each side, we must find the area of one side. The side measure of this cube is 2 feet so to find the area of one side we’ll multiply the length times the width, 2 times 2, which is 4 square feet.

Area of one side: A = L \times W = 2 \times 2 = 4ft^2

In a cube, all sides are the same. That means that each side has an area of 4 square feet. Since there are 6 sides, the total surface area can be found by:

Surface area of cube: 4ft^2 + 4ft^2 + 4ft^2 + 4ft^2 + 4ft^2 + 4ft^2 = 24ft^2

Giving us a total surface area of 24 square feet.

The other option we have to find the surface area of this figure is to use the surface area formula for a cube:  SA = 6s^2. In this instance, s is equal to 2 so we’ll substitute 2 for s in the formula and then follow the order of operations to simplify.

SA = 6s^2 = 6{(2)}^2 = 6(4) = 24ft^2

If one pint of paint covers 25 square feet, then yes – Dale will have enough paint for this project.

 

Example 2

Find the surface area of the figure.

meas75

 

We have two options for finding the surface area of the rectangular prism in example 2.

We can find the area of each individual side using the “length times width” area formula and then add them up, or use the surface area formula for a rectangular prism.

If we find the area of each individual side…

meas76

We have 2 sides measuring 9 by 10 inches having an area of 90 square inches, 2 sides measuring 3 by 9 inches having an area of 27 square inches and 2 sides measuring 3 by 10 inches having an area of 30 square inches.

SA = 90in^2 + 90in^2 + 27in^2 + 27in^2 + 30in^2 + 30in^2 = 294in^2

Our other option is to use the surface area formula for a rectangular prism:

SA = ph + 2B

The perimeter of the base is 38, since the base is a rectangle measuring 9 inches by 10 inches:

p = W + W + L + L = 9 + 9 + 10 + 10 = 38inches

The height is 3 inches. And the area of the base is 90 square inches, since the base is a rectangle measuring 9 inches by 10 inches:

B = L \times W = 9 \times 10 = 90in^2

Using the formula and substituting 38 for p, 3 for h and 90 for B gives us:

SA = (38)(3) + 2(90)

SA = 114 + 180

SA = 294in^2

So using the surface area formula gives us the same surface area of 294 square inches.

 

Example 3

Find the surface area of the figure.

meas82

 

In example 3, the figure is a cylinder which uses the surface area formula:

SA = 2\pi r h + 2 \pi r^2

The information needed to find the cylinder’s surface area is the radius and height, which are given to be 2 centimeters and 5 centimeters, respectively. So to find the surface area of this cylinder, we’ll use the formula, replacing pi with 3.14, r with 2 and h with 5.

SA = 2(3.14)(2)(5) + 2(3.14){(2)}^2
SA = (62.8) + (25.12)
SA = 87.92cm^2

 

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