The question
Question: find the volume of a pyramid with a square base 6 by 6 and height 4.
1 Area of the base
The base is a square, 6 by 6.
Its area is side times side: A = 6 × 6.
A = 6 × 6
6 times 6 is 36.
So the base is A = 36 square units.
A = 36
2 A third of the box
Draw a box around the pyramid, with the same base and the same height.
Fill the pyramid with sand 3 times.
That fills the box exactly.
So the pyramid is one third of the box.
The box is A times height.
So the pyramid is V = ⅓ × A × height.
V = 13 × A × height
Now put the numbers in: A = 36 and height = 4.
V = 13 × 36 × 4
Keep the numbers small: divide 36 by 3 first.
36 divided by 3 is 12.
363 = 12
So V = 12 × 4.
V = 12 × 4
12 times 4 is 48.
So the volume is 48 cubic units.
V = 48
Let's look back.
First: area of the base.
Last: a third of the box.
▶ Watch it solved
The question
Question: find the surface area of a square pyramid with base 6 and slant height 5.
1 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.
2 The base
The base is a square, 6 by 6.
Its area is side times side: 6 × 6.
base = 6 × 6
6 times 6 is 36.
So the base is 36 square units.
base = 36
3 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.
triangle = 12 × 6 × 5
6 times 5 is 30.
triangle = 12 × 30
Half of 30 means 30 divided by 2.
That is 15.
302 = 15
So one triangle is 15 square units.
triangle = 15
There are 4 of them: 4 times 15 is 60.
4 triangles = 4 × 15 = 60
4 Add all the faces
Add the base and the triangles: 36 + 60.
S = 36 + 60
36 plus 60 is 96.
So the surface area is 96 square units.
S = 96
Let's look back.
First: count the faces.
Then: the base.
Then: the four triangles.
Last: add all the faces.
▶ Watch it solved
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 + 42 = l2
3² means 3 times 3.
That is 9.
And 4² is 16.
32 = 3 × 3 = 9
42 = 4 × 4 = 16
Write the squares: 9 + 16 = l².
9 + 16 = l2
9 plus 16 is 25.
25 = l2
l² is 25.
So l is the square root of 25.
l = √25
The square root of 25 is the positive number that times itself makes 25.
5 times 5 is 25.
5 × 5 = 25
So the slant height is l = 5.
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.
base = 6 × 6
6 times 6 is 36.
So the base is 36 square units.
base = 36
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.
triangle = 12 × 6 × 5
6 times 5 is 30.
triangle = 12 × 30
Half of 30 means 30 divided by 2.
That is 15.
302 = 15
So one triangle is 15 square units.
triangle = 15
There are 4 of them: 4 times 15 is 60.
4 triangles = 4 × 15 = 60
5 Add all the faces
Add the base and the triangles: 36 + 60.
S = 36 + 60
36 plus 60 is 96.
So the surface area is 96 square units.
S = 96
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.
▶ Watch it solved
