Question

This image shows a square pyramid. What is the surface area of this square pyramid?

A. 32 ft²

B. 64 ft²

C. 128 ft²

D. 136 ft²

Answer 1

`A = 128` or answer C

Explanation:

Surface area is the area of all the surfaces that comprise a single form. In a square pyramid we have five faces, 4 triangles and 1 square. So to determine surface area we need to take the areas of the triangles and square and add them all up.

The square's area is relatively simple:

`A = s^2`
`A = 8^2 = 64`

But now let's observe the triangles, a triangle's area is:

`A = 1/2bh`

So we need to determine the base and height of every triangle in the picture. The base is always 8 ft, as it is the length of the bottom of each triangle.

The height though, is a bit different. We don't know the height, but we can draw a right triangle in each triangle face, such that it splits the base in half.

In this case we have a 45/45/90 right triangle with the opposite side being the height, and the adjacent side being half the base. We need to relate adjacent to opposite with an angle, so we use tangent, as:

`tan(theta) = "opposite"/"adjacent"`

So after plugging things in, calling opposite h we get the following:

`tan(45) = h/4`
`1 = h/4` Multiply both sides by 4
`color(red)(4)*1 = (color(red)(4)*h)/4`
`4 = h`

So now the area of the triangle is:
`A = 1/2(8)(4)`
`A = 16`

Multiply the triangle's area by four to account for all four triangular faces and add the squares area for your answer:

`A = 16(4) + 64`
`A = 128` or answer C