Source: Pyramid Arena, Exothermic, Flickr

Recall from the lesson introduction that the Pyramid Arena is a regular rectangular pyramid in Memphis, Tennessee.

Suppose that Marshall has been contracted to resurface the exterior of the Pyramid Arena using solar electricity panels. Before he can start, he needs to know exactly how much material it will take for him to resurface this regular pyramid.

Before beginning, letâ€™s examine the problem situation more closely.

• Will Marshall need to calculate the total surface area of the pyramid or the lateral surface area? How do you know?

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Which faces of the pyramid will Marshall be resurfacing with solar electricity panels?

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Marshall will need to calculate the lateral surface area because he will only be resurfacing the lateral faces of the pyramid.
• Which area formulas will Marshall need to use in order to make his calculations?

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What shape(s) are the lateral faces of the pyramid?

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Since the lateral faces are all triangles, Marshall will need to use the area formula for a triangle, A = 1 over 2 1 2 bh.
• What dimensions of each triangular face will Marshall need to know in order to calculate the area of each face?

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What do the variables b and h stand for in the triangle area formula?

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Marshall will need to know the base length, b, and the height, h, of each triangular face of the pyramid.
• To which dimension of the pyramid does the height of the triangular lateral face compare?

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A pyramid has an altitude height, which is the vertical distance from the apex to the base of the pyramid. It also has a slant height, which is the perpendicular distance from the apex to the base edge along the triangular lateral face.

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The slant height of the pyramid is the same as the height of the triangular lateral face.

Marshall is provided the height of the Pyramid Arena, 321 feet, and the length of each base edge of the pyramid, 590 feet, as shown in the diagram below.

In most problem situations, the slant height of a pyramid will be given. However, sometimes, as in the situation with Marshall and the Pyramid Arena, you will have to find the slant height using the information given in the problem.

Watch the video below to see how the slant height can be calculated when you know the height of the pyramid and the base edge length.

The images below summarize how the slant height was calculated in the video.

Using the side view of the pyramid, you see that the height of the pyramid is perpendicular with the base.

Therefore, a right triangle is created. Its hypotenuse is the slant height of the pyramid. Using the Pythagorean Theorem with the height and half of the length of the base as the legs of the right triangle, you can calculate the slant height.

Using the Pythagorean Theorem and rounding to the nearest whole number value, the slant height is calculated to be about 436 feet.

Next, let's look at the Pyramid Arena as a net with all of the measurements that you know now.

• What is the area of each triangle in the net?

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Use the triangle area formula, A = 1 over 2 1 2 bh.

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A = 1 over 2 1 2 bh
A = 1 over 2 1 2 (590 ft)(436 ft)
A = 128,620 ft2
• What is the complete area that Marshall will need to resurface?

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Which faces of the pyramid will Marshall need to resurface? Determine the sum of the areas of these faces.

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Surface Area = 4 × Area of One Triangle

Surface Area = 4 × 128,620 ft2

Surface Area = 514,480 ft2
• If solar electricity panels cost \$25 per square foot, then what will be the total cost to completely cover the surface of the Pyramid Arena with solar electricity panels?

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How much will 1 square foot cost? 2 square feet?

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Total Cost = \$25 per ft2 × Surface Area

Total Cost = 25 × 514,480 = \$12,862,000

### Pause and Reflect

How could nets help you determine the surface area of other figures?

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If you know the net of a figure and its dimensions, then you can calculate the area of the relevant faces of the figure.

### Practice

For each of the figures below, determine the total surface area.

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What shapes are the faces that are present in each net? What are the area formulas that you can use to determine the area of each of those shapes?

 1. 2. Interactive popup. Assistance may be required. Check Your Answer This net is a triangular prism, which has two congruent triangular bases and 3 rectangular lateral faces.Area of Base=1 over 2 1 2 bh = 1 over 2 1 2 (3.5 cm)(3 cm) = 5.25 cm2Area of Lateral Faces= bh + bh+ bh= (8.6 cm)(3.5 cm) + (8.6 cm)(3.5 cm) +(8.6 cm) (3.5 cm)= 30.1 cm2 + 30.1 cm2 + 30.1 cm2 = 90.3 cm2 Surface Area= Area of Bases + Area of Lateral Faces = 2(5.25 cm2) + 90.3 cm2 = 100.8 cm2 Interactive popup. Assistance may be required. Check Your Answer