Step by step
The question
Question: find the surface area of a square pyramid with base 8 cm and height 3 cm.
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 8 divided by 2.
That is 4.
The height, half the base and the slant height make a right triangle.
So the Pythagorean theorem works: 4² + 3² = l².
4² means 4 times 4.
That is 16.
And 3² is 9.
Write the squares: 16 + 9 = l².
16 plus 9 is 25.
l² is 25.
So l is the square root of 25.
The square root of 25 is the positive number that times itself makes 25.
5 times 5 is 25.
So the slant height is l = 5 cm.
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, 8 by 8.
Its area is side times side: 8 × 8.
8 times 8 is 64.
So the base is 64 cm².
4 The four triangles
Each triangle has base 8 cm.
Its height is the slant height, 5 cm: 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 × 8 × 5.
8 times 5 is 40.
Half of 40 means 40 divided by 2.
That is 20.
So one triangle is 20 cm².
There are 4 of them: 4 times 20 is 80.
5 Add all the faces
Add the base and the triangles: 64 + 80.
64 plus 80 is 144.
So the surface area is 144 cm².
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.