Worked examples
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.
base = 18 × 10
18 times 10 is 180.
So the base is 180 square units.
base = 180
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.
front = 12 × 18 × 13
18 times 13 is 234.
front = 12 × 234
Half of 234 means 234 divided by 2.
That is 117.
2342 = 117
So the front is 117 square units.
front = 117
The back is the same.
2 times 117 is 234.
front and back = 2 × 117 = 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.
Careful: not 4 × 117 ✗
One side triangle is half of base times height: ½ × 10 × 15.
side = 12 × 10 × 15
10 times 15 is 150.
side = 12 × 150
Half of 150 means 150 divided by 2.
That is 75.
1502 = 75
So one side is 75 square units.
side = 75
The other side is the same.
2 times 75 is 150.
left and right = 2 × 75 = 150
5 Add all the faces
Add the base and the two pairs: 180 + 234 + 150.
S = 180 + 234 + 150
180 plus 234 plus 150 is 564.
So the surface area is 564 square units.
S = 564
Let's look back.
First: count the faces.
Then: the base.
Then: front and back.
Then: left and right.
Last: add all the faces.
▶ Watch it solved
The question
Question: find the surface area of a rectangular pyramid with base 18 cm by 10 cm, and slant heights 13 cm and 15 cm.
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.
base = 18 × 10
18 times 10 is 180.
So the base is 180 cm².
base = 180 cm2
3 Front and back
The front triangle has base 18 cm.
Its height is the slant height up its middle: 13 cm.
One triangle is half of base times height: ½ × 18 × 13.
front = 12 × 18 × 13
18 times 13 is 234.
front = 12 × 234
Half of 234 means 234 divided by 2.
That is 117.
2342 = 117
So the front is 117 cm².
front = 117 cm2
The back is the same.
2 times 117 is 234.
front and back = 2 × 117 = 234 cm2
4 Left and right
The side triangle is different.
Its base is 10 cm, and its slant height is 15 cm.
Careful!
The side triangles are not the same as the front and back.
They have their own base and slant height.
Careful: not 4 × 117 ✗
One side triangle is half of base times height: ½ × 10 × 15.
side = 12 × 10 × 15
10 times 15 is 150.
side = 12 × 150
Half of 150 means 150 divided by 2.
That is 75.
1502 = 75
So one side is 75 cm².
side = 75 cm2
The other side is the same.
2 times 75 is 150.
left and right = 2 × 75 = 150 cm2
5 Add all the faces
Add the base and the two pairs: 180 + 234 + 150.
S = 180 + 234 + 150
180 plus 234 plus 150 is 564.
So the surface area is 564 cm².
S = 564 cm2
Let's look back.
First: count the faces.
Then: the base.
Then: front and back.
Then: left and right.
Last: add all the faces.
▶ Watch it solved
The question
Question: find the surface area of a rectangular pyramid with base 15 by 6, and slant heights 5 and 8.5.
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, 15 by 6.
Its area is length times width: 15 × 6.
base = 15 × 6
15 times 6 is 90.
So the base is 90 square units.
base = 90
3 Front and back
The front triangle has base 15.
Its height is the slant height up its middle: 5.
One triangle is half of base times height: ½ × 15 × 5.
front = 12 × 15 × 5
15 times 5 is 75.
front = 12 × 75
Half of 75 means 75 divided by 2.
That is 37.5.
752 = 37.5
So the front is 37.5 square units.
front = 37.5
The back is the same.
2 times 37.5 is 75.
front and back = 2 × 37.5 = 75
4 Left and right
The side triangle is different.
Its base is 6, and its slant height is 8.5.
Careful!
The side triangles are not the same as the front and back.
They have their own base and slant height.
Careful: not 4 × 37.5 ✗
One side triangle is half of base times height: ½ × 6 × 8.5.
side = 12 × 6 × 8.5
Forget the points: 6 times 85 is 510.
The numbers have 1 digit after the point.
So the answer has 1 too: 51.0, which is 51.
6 × 85 = 510 → 51.0
side = 12 × 51
Half of 51 means 51 divided by 2.
That is 25.5.
512 = 25.5
So one side is 25.5 square units.
side = 25.5
The other side is the same.
2 times 25.5 is 51.
left and right = 2 × 25.5 = 51
5 Add all the faces
Add the base and the two pairs: 90 + 75 + 51.
S = 90 + 75 + 51
90 plus 75 plus 51 is 216.
So the surface area is 216 square units.
S = 216
Let's look back.
First: count the faces.
Then: the base.
Then: front and back.
Then: left and right.
Last: add all the faces.
▶ Watch it solved
