Step by step
The question
Question: find the surface area of a rectangular pyramid with base 18 by 32, and slant heights 20 and 15.
1 Count the faces
The surface area is the area of all the outside faces.
This pyramid has a rectangle base and 4 triangles.
The front and back are the same.
The left and right are the same.
2 The base
The base is a rectangle, 18 by 32.
Its area is length times width: 18 × 32.
18 times 32 is 576.
So the base is 576 square units.
3 Front and back
The front triangle has base 18.
Its height is the slant height up its middle: 20.
One triangle is half of base times height: ½ × 18 × 20.
18 times 20 is 360.
Half of 360 means 360 divided by 2.
That is 180.
So the front is 180 square units.
The back is the same.
2 times 180 is 360.
4 Left and right
The side triangle is different.
Its base is 32, and its slant height is 15.
Careful!
The side triangles are not the same as the front and back.
They have their own base and slant height.
One side triangle is half of base times height: ½ × 32 × 15.
32 times 15 is 480.
Half of 480 means 480 divided by 2.
That is 240.
So one side is 240 square units.
The other side is the same.
2 times 240 is 480.
5 Add all the faces
Add the base and the two pairs: 576 + 360 + 480.
576 plus 360 plus 480 is 1416.
So the surface area is 1416 square units.
Let's look back.
First: count the faces.
Then: the base.
Then: front and back.
Then: left and right.
Last: add all the faces.