Maths year 10/11: Find lengths,areas and volumes of similar shapes using scaling factors

Similar figures are identical in shape, but not in size.

For example the two triangles are similar. One is a bigger copy of the other.

The two trapeziums are similar. One is a bigger copy of the other.

If you know how much bigger a side is from one shape to another (given by a scale factor SF), you can work out lengths of sides.

**small side length × SF = large side length**

The sides on each must be corresponding ie the same side on each shape.

**Example:** Look at these two shapes. The right one is the larger copy.

Work out a scaling factor (SF) using two corresponding sides.

Use the 2cm on the smaller and 3cm on the larger shapes.

Divide the larger length by the smaller:

SF = 3 ÷ 2 = 1.5 (check: 2 × 1.5 = 3)

Now work out **y** length:

The 1 cm side and y are corresponding sides, so y = 1 × 1.5 = 1.5cm

Work out **x** length.

The corresponding side is 4.5cm.

This time we make the side smaller so **divide by the SF.**
x = 4.5 ÷ 1.5 = 3cm

**Q**: These shapes are similar.

Find the length of sides x and y

Sometimes you have to find the two similar shapes.

In this example triangles ACE and BCD are similar. Note the parallel lines.

Q1: What are the two corresponding sides you can use to make the SF?

ANSQ2: Use your SF to find the length of AE

ANSQ3: Use your SF to find the length of AB

ANSWhen two shapes are similar you can work out the area of one from the area of the other.

Do this by finding a scaling factor (SF for Area, SF_{A}).

**Example**: Look at the two trapeziums

Find the scale factor for length, SF_{L}

Then square SF_{L} to find scaling factor for area, SF_{A}.

SF_{L} = 2, (3cm → 6cm), so SF_{A} = 2^{2} = 4

Large area = SF_{L} × small area

Large area = 4cm × 9cm = 36cm^{2}

**Q**: Find the smaller area of these similar triangles.

**Q**: Find the length of the missing side of the similar trapeziums

**Q**: Find the missing area for these similar shapes

When two volume shapes are similar you can work out several scaling factors.

Scaling factor for Volume, SF_{V} = (SF_{L})^{3}

Scaling factor for Area, SF_{A} = (SF_{L})^{2}

**Example**: For two similar shapes, Large volume = small volume × SF_{V}

or SF_{V} = large volume ÷ small volume = 16 ÷ 2 = 8

Scaling factor for Length, SF_{L} = ³√ SF_{V} = 2

Scaling factor for Area, SF_{A} = (SF_{L})^{2} = 4

**Q**: For these similar shapes, find the larger volume

**Q**: For these similar shapes find the missing length

**Q:** Two solid shapes are mathematically similar.

The base of the larger is a circle diameter 10 cm.

The base of the smaller is a circle diameter 6 cm.

The surface area of the smaller is 72 cm^{2}

Work out the surface area of the larger

ANSThe volume of the larger is 500 cm^{3}

Work out the volume of the smaller

**Q:** Two cones, A and B, are mathematically similar.

The total surface area of A is 23 cm^{2}

The total surface area of cone B is 92 cm^{2}

The height of cone A is 7 cm.

Work out the height of cone B.

ANSThe volume of cone B is 120cm^{3}

Work out the volume of cone A

ANS