Similarity and Ratios of Length Area and Volume for the TMUA

Definition of Similarity

If two shapes are mathematically similar, one shape is considered an enlargement of the other. This means they are the same shape and have the same angles in the same order. Crucially, all corresponding sides must be in the same ratio. For example, if every length in triangle A is doubled to create triangle B, the two triangles are similar, and the linear scale factor is 2.

Area and Volume Ratios from a Linear Scale Factor

When two shapes, A and B, are mathematically similar and the lengths of the sides of B are $x$ times the lengths of the corresponding sides of A, the following rules apply:

1. The area of the surfaces of B are $x^2$ times the areas of the corresponding surfaces of A.
2. The volume of B is $x^3$ times the volume of A.

Worked Example: Using a Linear Scale Factor

X is a cube of side $x$ cm. Y is a cube of side $3x$ cm. If the volume of X is 100 cm$^3$, what is the volume of Y?

First, identify the linear scale factor. Each length of X is multiplied by 3 to get the length of Y (since $3x \div x = 3$). Because volume scales by the cube of the linear factor, the volume of Y is $3^3$ times the volume of X. Therefore, the volume of Y is $27 \times 100 = 2700$ cm$^3$.

Area and Volume Ratios from a Linear Ratio

If the ratio of corresponding lengths on shape A to those on shape B is $x:y$, then:

1. The ratio of corresponding areas is $x^2:y^2$.
2. The ratio of the volumes is $x^3:y^3$.

Worked Example: Ratios of Spheres

The ratio of the diameters of two spheres A and B is $2:5$. If the surface area of A is 250 cm$^2$, find the surface area of B.

Diameter is a linear measure. The ratio of lengths is $2:5$, which means the ratio of areas is $2^2:5^2$, or $4:25$. To find the area of B, we use the factor $\frac{25}{4}$. The area of B is $250 \times \frac{25}{4} = 1562.5$ cm$^2$.

Calculating Linear Ratios from Area or Volume Ratios

You can work backwards from area or volume information to find linear measurements:

1. From Area: If the ratio of the area of B to the area of A is $x:y$, then the ratio of the lengths is $\sqrt{x}:\sqrt{y}$.
2. From Volume: If the ratio of the volume of B to the volume of A is $x:y$, then the ratio of the lengths is $\sqrt[3]{x}:\sqrt[3]{y}$.

Worked Example: Length from Area Ratio

Shapes A and B are mathematically similar. The ratio of the area of shape A to the area of shape B is $9:25$. If the length of shape A is 21 cm, what is the length of shape B?

If the ratio of areas is $9:25$, the ratio of lengths is $\sqrt{9}:\sqrt{25}$, which is $3:5$. Since the length of A is 21 cm, the length of B is $21 \div 3 \times 5 = 35$ cm.

Worked Example: Area from Volume Ratio

The volume of a cylinder, P, is 64 times the volume of a mathematically similar cylinder, Q. The surface area of Q is 300 cm$^2$. What is the surface area of P?

To move from a volume ratio to an area ratio, you must find the linear ratio first. The ratio of volumes is $Q:P = 1:64$. The ratio of corresponding lengths is $\sqrt[3]{1}:\sqrt[3]{64}$, which is $1:4$. Now, we use this linear ratio to find the area ratio: $1^2:4^2 = 1:16$. If the area of Q is 300 cm$^2$, the area of P is $300 \times 16 = 4800$ cm$^2$.

Similarity, Trigonometric Ratios and Scale Factors

If a right-angled triangle ABC is enlarged by a scale factor $p$ to form triangle DEF, the angles and trigonometric ratios are preserved. They do not depend on the scale factor because the ratio of two sides remains the same.

In the diagrams above, $\tan C = \frac{x}{z}$ and $\tan F = \frac{px}{pz} = \frac{x}{z}$. Since $\tan C = \tan F$, angle C must equal angle F. This applies to all trigonometric ratios including $\sin$ and $\cos$.

Worked Example: Trigonometry in Similar Triangles

Triangle DEF is an enlargement of triangle ABC with scale factor 4. The sides of triangle ABC have lengths $x, y$ and $z$, with $y < x < z$. In triangle DEF, $EF < ED < DF$. What is the sine of angle DFE in terms of $x, y$ and $z$?

Because triangle DEF is an enlargement of triangle ABC, the two triangles are similar and their corresponding angles are equal. Angle A corresponds to angle D (both opposite the shortest sides), meaning angle C corresponds to angle F. Thus, $\sin F = \sin C = \frac{x}{z}$.