Relationship of Length, Area and Volume

## Relationship of Length, Area and Volume

A line is drawn that is 5cm long. Applying a scale factor of 2, the line becomes 2 x 5cm = 10cm long.

A rectangle is drawn that is 5cm by 10 cm.

The area of the rectangle is 5 xx 10 = 50 text(cm)^2.

Applying a linear scale factor of 2, the lengths of each side of the rectangle are doubled. The area of the rectangle becomes 10 xx 20 = 200 text(cm)^2, which is four times larger.

If a linear scale factor is applied to an area, then the area is increased by the square of the scale factor.

A cuboid is drawn that is 2 text(cm) xx 3 text(cm) xx 4 text(cm), which is a volume of 24cm3. Applying a linear scale factor of 2, the sides become 4 text(cm) xx 6 text(cm) xx 8 text(cm), giving a volume of 192cm3, which is 8 times larger.

If a linear scale factor is applied to a volume, then the volume is increased by the cube of the scale factor.

## Example 1

A cuboid has a volume of 22 cm3. A similar cuboid has a length that is 1.5 times the size of the original. What is the volume of the similar cuboid?

Give the answer to 1 decimal place.

The volume of a shape increases by a cube of the scale factor.

The volume of the enlarged cuboid is 22 xx 1.5^2 = 74.25

To 1 dp this is 74.3cm3

## Example 2

A rectangle, A, has an area of 100cm2. A similar shape, B, has an area of 200cm2. What is the scale factor to map A onto B? Give the answer correct to 2 decimal places.

A scale factor is squared when relating area

Rectangle A x scale factor2 = Rectangle B

Let f be the scale factor

 Write the relationship A x f^2 = B Substitute: 100 x f^2 = 200 Divide both sides by 100 f^2 = 2 Square root both sides f = 1.412 To 2dp f = 1.41