Step by step
The question
Question: find the surface area of a rectangular pyramid with base 12 by 30, and slant heights 17 and 10.
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, 12 by 30.
Its area is length times width: 12 × 30.
12 times 30 is 360.
So the base is 360 square units.
3 Front and back
The front triangle has base 12.
Its height is the slant height up its middle: 17.
One triangle is half of base times height: ½ × 12 × 17.
12 times 17 is 204.
Half of 204 means 204 divided by 2.
That is 102.
So the front is 102 square units.
The back is the same.
2 times 102 is 204.
4 Left and right
The side triangle is different.
Its base is 30, and its slant height is 10.
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 × 30 × 10.
30 times 10 is 300.
Half of 300 means 300 divided by 2.
That is 150.
So one side is 150 square units.
The other side is the same.
2 times 150 is 300.
5 Add all the faces
Add the base and the two pairs: 360 + 204 + 300.
360 plus 204 plus 300 is 864.
So the surface area is 864 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.