Step by step
The question
Question: find the surface area of a rectangular pyramid with base 18 by 10, and slant heights 13 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 10.
Its area is length times width: 18 × 10.
18 times 10 is 180.
So the base is 180 square units.
3 Front and back
The front triangle has base 18.
Its height is the slant height up its middle: 13.
One triangle is half of base times height: ½ × 18 × 13.
18 times 13 is 234.
Half of 234 means 234 divided by 2.
That is 117.
So the front is 117 square units.
The back is the same.
2 times 117 is 234.
4 Left and right
The side triangle is different.
Its base is 10, 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 × 10 × 15.
10 times 15 is 150.
Half of 150 means 150 divided by 2.
That is 75.
So one side is 75 square units.
The other side is the same.
2 times 75 is 150.
5 Add all the faces
Add the base and the two pairs: 180 + 234 + 150.
180 plus 234 plus 150 is 564.
So the surface area is 564 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.