Similar areas and volumes
Similar areas
We already know that if two shapes are similar their corresponding sides are in the same ratio and their corresponding angles are equal.
When calculating a missing area, we need to calculate the Area Scale Factor.
Area Scale Factor (ASF) = (Linear Scale Factor)2
Example
The figures below are similar. Calculate the missing area.
\(LS{F_{enlargement}} = \frac{{Big}}{{Small}} = \frac{{16}}{8} = \frac{2}{1} = 2\)
\(ASF = {2^2} = 4\)
\(Area = 4 \times 22 = 88c{m^2}\)
Similar volumes
When calculating a missing volume, we need to calculate the Volume Scale Factor.
Volume Scale Factor (VSF) = (Linear Scale Factor)3
Example
The vases below are similar. Calculate the missing volume.
\(LS{F_{{\mathop{\rm re}\nolimits} duction}} = \frac{{Small}}{{Big}} = \frac{6}{8} = \frac{3}{4}\)
\(VSF = {(\frac{3}{4})^3} = \frac{{27}}{{64}}\)
\(Volume = \frac{{27}}{{64}} \times 640 = 270ml\,(to\,nearest\,1ml)\)
Now try the example questions below.
Question
Two similar figures have corresponding sides 3cm and 5cm. The area of the smaller one is 12 cm2. What is the area of the larger one? Give your answer to 1 decimal place.
Enlargement
Linear scale factor \(=\frac{5}{3}\)
Area scale factor \(=(\frac{5}{3})^{2}=\frac{25}{9}\)
Area \(=\frac{25}{9}\times 12\)
\(=33.3cm^{2}\) (1 decimal place)
Question
Two similar objects have corresponding sides 4cm and 7cm. The volume of the larger one is 230 cm3. What is the volume of the smaller one? Give your answer to 1 decimal place.
Reduction
Linear scale factor \(=\frac{4}{7}\)
Volume scale factor \(=(\frac{4}{7})^{3}=\frac{64}{343}\)
Volume \(=\frac{64}{343}\times 230\)
\(=42.9cm^{3}\) (1 decimal place)
Question
These two shapes are similar. What is the length of \(x\)?
(This time we start with the Area Scale Factor and have to get back to the Linear Scale Factor).
\(AS{F_{enlargement}} = \frac{{Big}}{{Small}} = \frac{{36}}{{25}}\)
\(LSF = \frac{{\sqrt {36} }}{{\sqrt {25} }}\)
\(=\frac{\sqrt{36}}{\sqrt{25}}\)
\(=\frac{6}{5}\)
\(x = \frac{6}{5} \times 2 = 2.4cm\)
More guides on this topic
- Determine the gradient of a straight line
- Circle geometry
- Calculating the volume of a standard solid
- Applying Pythagoras Theorem
- Applying the properties of shapes to determine an angle
- Working with two-dimensional vectors
- Working with three-dimensional coordinates
- Using vector components
- Calculating the magnitude of a vector
- An Approximate History of Co-ordinates