Step by step
The question
Question: find the surface area of a square pyramid with base 16 and height 6.
1 Find the slant height
To find the triangles, we need the slant height: the height of a triangle face.
We call it l.
Half of the base is 16 divided by 2.
That is 8.
The height, half the base and the slant height make a right triangle.
So the Pythagorean theorem works: 8² + 6² = l².
8² means 8 times 8.
That is 64.
And 6² is 36.
Write the squares: 64 + 36 = l².
64 plus 36 is 100.
l² is 100.
So l is the square root of 100.
The square root of 100 is the positive number that times itself makes 100.
10 times 10 is 100.
So the slant height is l = 10.
2 Count the faces
The surface area is the area of all the outside faces.
This pyramid has a square base and 4 triangles, all the same.
3 The base
The base is a square, 16 by 16.
Its area is side times side: 16 × 16.
16 times 16 is 256.
So the base is 256 square units.
4 The four triangles
Each triangle has base 16.
Its height is the slant height, 10: straight up the middle of the face.
Careful: we need the slant height, up the face.
Not the height inside the pyramid.
One triangle is half of base times height: ½ × 16 × 10.
16 times 10 is 160.
Half of 160 means 160 divided by 2.
That is 80.
So one triangle is 80 square units.
There are 4 of them: 4 times 80 is 320.
5 Add all the faces
Add the base and the triangles: 256 + 320.
256 plus 320 is 576.
So the surface area is 576 square units.
Let's look back.
First: find the slant height.
Then: count the faces.
Then: the base.
Then: the four triangles.
Last: add all the faces.