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Unit F3 Section 2
Similarity

Similar shapes have the same shape but may be different sizes. The two rectangles shown below are similar – they have the same shape but one is smaller than the other.

They are similar because they are both rectangles and the sides of the larger rectangle are three times longer than the sides of the smaller rectangle.

It is interesting to compare the area of the two rectangles. The area of the smaller rectangle is 6 cm2 and the area of the larger rectangle is 54 cm2 , which is nine times (32) greater.

Note

In general, if the lengths of the sides of a shape are increased by a factor k, then the area is increased by a factor k2.

These two triangles are not similar.
The sides lengths of the triangles are not in the same ratio and so the triangles are not similar.

For two triangles to be similar, they must have the same internal angles, as shown in the similar shapes below.

The diagrams below show 3 similar cubes.

Length
of side		Area of
one face		Surface
area		Volume
1 cm 1 cm2 6 cm2 1 cm3
2 cm 4 cm2 24 cm2 8 cm3
3 cm 9 cm2 54 cm2 27 cm3

The table gives the lengths of sides, area of one face, total surface area and volume.

Comparing the larger cube with the 1 cm cube we can note that:

For the 2 cm cube
The lengths are 2 times greater.
The areas are 4 = 22 times greater.
The volume is 8 = 23 times greater.
For the 3 cm cube
The lengths are 3 times greater.
The areas are 9 = 32 times greater.
The volume is 27 = 33 times greater.

Note

If the lengths of a solid are increased by a factor, k, its surface area will increase by a factor k2 and its volume will increase by a factor k3 .

Worked Examples

1
(a)

Which of the triangles, A, B, C, D, shown below are similar?

First compare triangles A and B.

Here all the lengths of the sides are twice the length of the sides of triangle A, so the two triangles are similar.

Then compare triangles A and C.

Here all the angles are the same in both triangles, so the triangles must be similar.

Finally, compare triangles A and D.

Note that 4 = × 8 and 4.52 = × 9.04, but 3.5 ≠ × 6.13.

So these triangles are not similar.
(b)

How do the areas of the triangles which are similar compare?

The lengths of the sides of triangle B are 2 times greater than the lengths of the sides of triangle A, so the area will be 22 = 4 times greater.

The side lengths of triangle C are 3 4 of the side lengths of triangle A.

So the area will be ⎛
⎝⎞
⎠2 = of the area of triangle A.

The ratio of the areas of triangles C : A : B can be written as:

: 1 : 4

or

9 : 16 : 64
2
(a)

Explain why triangles ABE and ACD are similar.

As the lines BE and CD are parallel,

∠ABE = ∠ACD

and

∠AEB = ∠ADC

As the vertex A, is common to both triangles,

∠DAC = ∠EAB

So the three angles are the same in both triangles and therefore they are similar.
(b)

Find the lengths of x and y.

Comparing the sides BE and CD, the lengths in the larger triangle are 1.5 times the lengths in the smaller triangle. Alternatively, it can be stated that the ratio of the lengths is 2 : 3.

So the length AC will be 1.5 times the length AB.

AC = 1.5 × 6
= 9
So y = 3

In the same way,

AD = 1.5 × AE,
so 4 + x = 1.5x
4 = 0.5x
x = 8
(c)

Find the ratio of the area of ABE to BCDE.

As the lengths are increased by a factor of 1.5 or for the larger triangle, the areas will be increased by a factor of 1.52 or ⎛
⎝⎞
⎠2. We can say that the ratio of the areas of the triangles is 1 : 2.25   or   1 :   or   4 : 9.

If the area of triangle ABE is 4k, then the area of triangle ACD is 9k and hence the area of the quadrilateral BCDE is 5k.

So the ratio of the area of ABE to BCDE is 4 : 5.
3

The diagrams show two similar triangles.

If the area of triangle DEF is 26.46 cm2, find the lengths of its sides.

If the lengths of the sides of triangle DEF are a factor k greater than the lengths of the sides of triangle ABC, then its area will be a factor k2 greater than the area of ABC.

Area of ABC = × 4 × 3
= 6 cm2 .
So 6 × k2 = 26.46
k2 = 4.41
k =
= 2.1

So the lengths of the sides of triangle DEF will be 2.1 times greater than the lengths of the sides of triangle ABC.

DE = 2.1 × 5
= 10.5 cm
DF = 2.1 × 3
= 6.3 cm
EF = 2.1 × 4
= 8.2 cm
4

A can has a height of 10 cm and has a volume of 200 cm3. A can with a similar shape has a height of 12 cm.

(a)

Find the volume of the larger can.

The lengths are increased by a factor of 1.2, so the volume will be increased by a factor of 1.23.

Volume = 200 × 1.23
= 345.6 cm3
(b)

Find the height of a similar can with a volume of 675 cm3 .

If the lengths are increased by a factor of k, then the volume will be increased by k3.
675 = 200 × k3
k3 = 3.375
k =
= 1.5

So the height must be increased by a factor of 1.5, to give

height = 1.5 × 10
= 15 cm

Exercises

Which of the triangles below are similar? Diagrams are not drawn to scale.

and

and

and

The diagram shows two regular hexagons and AB = BC.

(a)

What is the ratio of the area of the smaller hexagon to the area of the larger hexagon?

:
(b)

What is the ratio of the area of the smaller hexagon to that of the shaded area?

:
(a)

Find the lengths of:

(i)
AD cm
(ii)
DE cm
(iii)
BC cm
(b)

Find the ratio of the areas of:

(i)

ABE to ACD

:
(ii)

ABE to BCDE.

:

A box has a volume of 50 cm3 and a width of 6 cm. A similar box has a width of 12 cm.

(a)

Find the volume of the larger box.

cm3
(b)

How many times bigger is the surface area of the larger box?

times

A packet has the dimensions shown in the diagram.

All the dimensions are increased by 20%.

(a)

Find the percentage increase in:

(i)
surface area %
(ii)
volume. %
(b)

Find the percentage increase needed in the dimensions of the packet to increase the volume by 50%.

%

Two similar cans have volumes of 400 cm3 and 1350 cm3 .

(a)

Find the ratio of the heights of the cans.

:
(b)

Find the ratio of the surface areas of the cans.

:

(a)

Calculate the length of OY.

cm
(b)

Calculate the size of angle XOY.

° (to 1 d.p.)

I stood 420 m away from the tallest building in Singapore. I held a piece of wood 40 cm long at arms length, 60 cm away from my eye. The piece of wood, held vertically, just blocked the building from my view.

Use similar triangles to calculate the height, h metres, of the building.

h = m

AD = 4 cm, BC = 6 cm, angle BCD = 35°. BD is perpendicular to AC.

(a)

Calculate BD.

cm (to 2 d.p.)
(b)

Calculate angle BAC.

° (to 1 d.p.)
(c)

Triangle A′B′C′ is similar to triangle ABC. The area of triangle A′B′C′ is nine times the area of triangle ABC.

(i)

What is the size of angle A′B′C′?

° (to 1 d.p.)
(ii)

Work out the length of B′C′ .

cm

Two wine bottles have similar shapes. The standard bottle has a height of 30 cm. The small bottle has a height of 23.5 cm.

(a)

Calculate the ratio of the areas of the bases of the two bottles.
Give your answer in the form n : 1.

: 1 (to 2 d.p.)
(b)

What is the ratio of the volumes of the two bottles?
Give your answer in the form n : 1.

: 1 (to 2 d.p.)
(c)

Is it a fair description to call the small bottle a 'half bottle'?

(a)

Two bottles of perfume are similar to each other. The heights of the bottles are 4 cm and 6 cm. The smaller bottle has a volume of 24 cm3.
Calculate the volume of the larger bottle.

cm3
(b)

Two bottles of aftershave are similar to each other. The areas of the bases of these bottles are 4.8 cm2 and 10.8 cm2. The height of the smaller bottle is 3 cm. Calculate the height of the larger bottle.

cm