Step by step
The question
Question: find the surface area of a square pyramid with base 10 m and height 12 m.
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 10 divided by 2.
That is 5.
The height, half the base and the slant height make a right triangle.
So the Pythagorean theorem works: 5² + 12² = l².
5² means 5 times 5.
That is 25.
And 12² is 144.
Write the squares: 25 + 144 = l².
25 plus 144 is 169.
l² is 169.
So l is the square root of 169.
The square root of 169 is the positive number that times itself makes 169.
13 times 13 is 169.
So the slant height is l = 13 m.
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, 10 by 10.
Its area is side times side: 10 × 10.
10 times 10 is 100.
So the base is 100 m².
4 The four triangles
Each triangle has base 10 m.
Its height is the slant height, 13 m: 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 × 10 × 13.
10 times 13 is 130.
Half of 130 means 130 divided by 2.
That is 65.
So one triangle is 65 m².
There are 4 of them: 4 times 65 is 260.
5 Add all the faces
Add the base and the triangles: 100 + 260.
100 plus 260 is 360.
So the surface area is 360 m².
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.