Step by step
The question
Question: find the surface area of a square pyramid with base 6 and height 4.
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 6 divided by 2.
That is 3.
The height, half the base and the slant height make a right triangle.
So the Pythagorean theorem works: 3² + 4² = l².
3² means 3 times 3.
That is 9.
And 4² is 16.
Write the squares: 9 + 16 = l².
9 plus 16 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.
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, 6 by 6.
Its area is side times side: 6 × 6.
6 times 6 is 36.
So the base is 36 square units.
4 The four triangles
Each triangle has base 6.
Its height is the slant height, 5: 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 × 6 × 5.
6 times 5 is 30.
Half of 30 means 30 divided by 2.
That is 15.
So one triangle is 15 square units.
There are 4 of them: 4 times 15 is 60.
5 Add all the faces
Add the base and the triangles: 36 + 60.
36 plus 60 is 96.
So the surface area is 96 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.