Cubes, boxes and cylinders
Faces, edges and corners, the volume of a cube, a box and a cylinder, and the total area of a cube’s six faces.
A solid is a shape that is not flat: it has length, width and height. A dice is a cube, a shoebox is a box, and a tin can is a cylinder. The most useful question about a solid is how much it holds. That is its volume, the amount of space inside it. It tells you how much water fills a tank or how many small cubes fit in a crate. Builders, cooks and shipping companies use these rules every day.
Faces, edges and corners
A cube is a box whose length, width and height are all the same. Every one of its sides is a square. Three words describe the parts of a cube or a box:
- A face is one flat side. A cube has a top, a bottom, a front, a back, a left and a right: 6 faces.
- An edge is a straight line where two faces meet. There are 4 edges around the top, 4 around the bottom and 4 going up the sides: 12 edges.
- A corner (also called a vertex, and in the plural vertices) is a point where edges meet. There are 4 corners on the top and 4 on the bottom: 8 corners.
| Part | How to count it | Cube or box |
|---|---|---|
| Faces | top, bottom and 4 around the sides | 6 |
| Edges | 4 on top, 4 on the bottom, 4 going up | 12 |
| Corners | 4 on top and 4 on the bottom | 8 |
The volume of a box
Volume is measured in small cubes of side 1. Think of a box that is 5 long, 3 wide and 2 tall. On the floor of the box you can lay a row of 5 small cubes, and 3 such rows: 5 × 3 = 15 cubes. That is one layer. The box is 2 tall, so there are 2 layers: 15 × 2 = 30 cubes. The volume is 30.
Remember
Volume of a box = length × width × height
Multiply all three lengths. The order does not matter.
Example 1
What is the volume of this box?
- One layer on the floor: 7 × 4 = 28.
- There are 3 layers: 28 × 3 = 84.
Answer: 84
The volume of a cube
A cube is a box with all three lengths equal. So you multiply the edge by itself, and then by itself again. This is why "a number cubed", like 4³, means 4 × 4 × 4.
Remember
Volume of a cube = s × s × s = s³
s is the length of one edge.
Example 2
What is the volume of this cube?
- 6 × 6 = 36. That is one layer.
- 36 × 6 = 216.
Answer: 216
| Edge | 2 | 3 | 4 | 5 | 6 | 10 |
|---|---|---|---|---|---|---|
| Volume | 8 | 27 | 64 | 125 | 216 | 1000 |
The area of the six faces
Sometimes the question is not how much a cube holds, but how much paint covers it. Then you need the total area of its six faces. Every face is a square with the same side, so find one face and multiply by 6.
Remember
Total area of a cube = 6 × s × s
Example 3
What is the total area of all six faces of a cube with edge 4?
- One face: 4 × 4 = 16.
- Six faces: 6 × 16 = 96.
Answer: 96
The volume of a cylinder
A cylinder is the shape of a tin can: a circle at the bottom, the same circle at the top, and a curved wall that goes straight up between them. Think of it as a pile of coins. Each coin is a thin circle, and the pile is as tall as the cylinder. So the volume is the area of the bottom circle times the height, just as a box is the area of its floor times its height.
The area of a circle is π r², where r is the radius (the distance from the center to the edge). The answers are usually written with π in them, like 45π. That means 45 times π, where π is about 3.14.
Remember
Volume of a cylinder = π r² h
Square the radius, multiply by the height h, and keep the π.
Example 4
What is the volume of this cylinder?
- The area of the bottom circle: π × 2² = 4π.
- Times the height: 4π × 7 = 28π.
- As a normal number, that is about 88.
Answer: 28π
Example 5
A cylinder has a radius of 6 and a height of 3. What is its volume?
- The area of the bottom circle: π × 6² = 36π.
- Times the height: 36π × 3 = 108π.
Answer: 108π
Try it yourself
1.How many edges meet at each corner of a cube?
Show the answer
3
At every corner, one edge goes along the length, one along the width and one up or down.
2.What is the volume of a box that is 8 long, 5 wide and 3 tall?
Show the answer
120
One layer is 8 × 5 = 40, and 3 layers make 40 × 3 = 120.
3.What is the volume of a cube with edge 7?
Show the answer
343
7 × 7 = 49, and 49 × 7 = 343.
4.What is the total area of all six faces of a cube with edge 6?
Show the answer
216
One face is 6 × 6 = 36, and six faces make 6 × 36 = 216.
5.What is the volume of a cylinder with radius 3 and height 10?
Show the answer
90π
The bottom circle is π × 3² = 9π, and 9π × 10 = 90π.
6.A cube has a volume of 64. What is the total area of its six faces?
Show the answer
96
4 × 4 × 4 = 64, so the edge is 4. One face is 16, and six faces make 96.
Ready to test yourself?
Play Geometry: 20 seconds per question, and it gets harder as you go.