Making shapes bigger
Double every side of a square and the area does not double: it grows four times. See why, for areas and volumes.
Make every side of a square twice as long, and you might expect a square twice as big. It is not. It is four times as big. Make every edge of a cube twice as long and it holds eight times as much. This surprises almost everyone the first time. It also explains real things. In 1638 the Italian scientist Galileo used this idea to show why a giant could not simply be a person made larger: his weight would grow much faster than the strength of his bones. Once you see why it happens, these questions become very easy.
Lengths grow by the same number
When we make a shape bigger, we multiply every length by the same number. This number is called the scale factor. If every side is doubled, the scale factor is 2. If every side is made 3 times as long, the scale factor is 3.
Everything that is a length grows by exactly the scale factor. A square with side 3 has a perimeter (the distance all the way around) of 4 × 3 = 12. Double the side to 6 and the perimeter is 4 × 6 = 24. That is 2 times as much, just like the side.
Areas grow by the scale factor times itself
Area is different, because area uses two lengths: a width and a height. When you double the side of a square, the width doubles and the height doubles. So the area is multiplied by 2, and then by 2 again.
Look at a square with side 3. Its area is 3 × 3 = 9. Now double the side to 6. The area is 6 × 6 = 36. Is that double 9? No: double 9 is 18. It is 4 × 9 = 36, four times as much.
You can see it in the big square. Draw a line down the middle and a line across the middle. You get 4 squares, each with side 3. So the big square holds exactly 4 of the small ones.
Remember
Multiply every length by k, and the area is multiplied by k × k = k².
This is true for every flat shape: squares, rectangles, triangles and circles.
Example 1
Each side of a square is made 3 times as long. The area is multiplied by what number?
- The width grows 3 times, and the height grows 3 times.
- So the area grows 3 × 3 = 9 times.
- Check with side 2: the area is 4. With side 6, the area is 36, and 36 ÷ 4 = 9.
Answer: 9
Example 2
A square has an area of 16. Each side is made 3 times as long. What is the new area?
- The old side: which number times itself is 16? It is 4.
- The new side: 4 × 3 = 12.
- The new area: 12 × 12 = 144.
- Check: 16 × 9 = 144. The area grew 9 times, as the rule says.
Answer: 144
Circles too
A circle grows in the same way. Its area is π r², where r is the radius, the distance from the center to the edge. The radius is used twice, because r² means r × r. So if the radius doubles, the area grows 2 × 2 = 4 times.
Example 3
The radius of a circle is doubled, from 3 to 6. How many times bigger is the area?
- Old area: π × 3² = 9π.
- New area: π × 6² = 36π.
- 36π is 4 times 9π.
Answer: 4 times
Volumes grow by the scale factor three times
Volume, the space inside a solid, uses three lengths: length, width and height. Double every edge of a cube and all three double. So the volume is multiplied by 2 × 2 × 2 = 8.
A cube with edge 2 has volume 2 × 2 × 2 = 8. A cube with edge 4 has volume 4 × 4 × 4 = 64. And 64 ÷ 8 = 8. You could fill the big cube with 8 of the small ones: 2 along, 2 across and 2 up.
Remember
Multiply every length by k, and the volume is multiplied by k × k × k = k³.
Example 4
A box is 2 long, 3 wide and 1 tall. Every edge is made 3 times as long. What is the new volume?
- The old volume: 2 × 3 × 1 = 6.
- The new box is 6 long, 9 wide and 3 tall.
- The new volume: 6 × 9 × 3 = 162.
- Check: 6 × 27 = 162. The volume grew 3 × 3 × 3 = 27 times.
Answer: 162
One table for all of it
| Every length times | Lengths times | Areas times | Volumes times |
|---|---|---|---|
| 2 | 2 | 4 | 8 |
| 3 | 3 | 9 | 27 |
| 4 | 4 | 16 | 64 |
| 5 | 5 | 25 | 125 |
| 10 | 10 | 100 | 1000 |
Try it yourself
1.Each side of a square is made 5 times as long. The area is multiplied by what number?
Show the answer
25
The width and the height both grow 5 times, so the area grows 5 × 5 = 25 times.
2.A rectangle is 3 by 4. Each side is doubled. What is the new area?
Show the answer
48
The new rectangle is 6 by 8, and 6 × 8 = 48. That is 4 times the old area of 12.
3.Each edge of a cube is made 3 times as long. The volume is multiplied by what number?
Show the answer
27
Length, width and height all grow 3 times: 3 × 3 × 3 = 27.
4.The radius of a circle is made 3 times as long. The area is multiplied by what number?
Show the answer
9
The area is π r², and r² grows 3 × 3 = 9 times.
5.A cube has a volume of 10. Each edge is doubled. What is the new volume?
Show the answer
80
Doubling every edge multiplies the volume by 2 × 2 × 2 = 8, and 10 × 8 = 80.
6.A square has an area of 36. Each side is halved. What is the new area?
Show the answer
9
The side was 6 and is now 3, so the new area is 3 × 3 = 9. That is a quarter of 36.
Ready to test yourself?
Play Geometry: 20 seconds per question, and it gets harder as you go.