If the corresponding sides of similar figures are in the ratio k, then :

Area of image = k^{2} x area of object.

For example, for the similar triangles alongside:

**Example 1 :**

Triangle DEC has area 4.2 cm^{2}.

a) Find the area of triangle ABC.

b) Find the area of quadrilateral ABED.

**Solution :**

In the triangle given above, ΔABC and ΔDEC.

<BCA = <ECD (A)

ABC = <EDC (A)

ΔABC and ΔDEC are similar. Bases of triangle ABC and EDC are BC and DC respectively.

(i) Area of Δ ABC/Area of Δ DEC = (BC/DC)^{2}

Area of Δ ABC/4.2 = (6/4)^{2}

Area of Δ ABC = (36/16) x 4.2

= 9.45 cm^{2}

(ii) Area of quadrilateral AEBD = Area of triangle ABC - Area of triangle DEC

= 9.45-4.2

= 5.25 cm^{2}

So, area of quadrilateral is AEBD is 5.25 cm^{2}.

**Example 2 :**

If triangle ABC has area 15 cm^{2}:

a) find the area of ΔCDE

b) find the area of PQED.

**Solution :**

(a) In triangles ABC and CDE

AB is parallel to DE

Area of ABC/Area of CDE = (AB/DE)^{2}

15/Area of CDE = (5/8)^{2}

15/Area of CDE = 25/64

Area of CDE = 15(64)/25

= 38.4 cm^{2}

In triangles CPQ and CDE

PQ is parallel to DE

Area of ΔCPQ/area of ΔCDE = (CQ/CE)^{2}

CQ = x and CE = 2x

Area of ΔCPQ/38.4 = (x/2x)^{2}

Area of ΔCPQ = 38.4/4

Area of ΔCPQ = 9.6

(b) Area of quadrilateral DPQE

= Area of CDE - Area of CPQ

= 38.4-9.6

= 28.8 cm^{2}

If the corresponding sides of similar solids are in the ratio k, then :

Volume of image = k^{3} x volume of object.

**Example 3 :**

What will happen to the volume of:

a) a sphere if the radius is doubled

b) a sphere if the radius is increased by 20%

**Solution :**

Volume of sphere = (4/3)πr^{3}

(a) Radius of the sphere = r

Radius of new sphere = 2r

Volume of new sphere = (4/3)π(2r)^{3}

= (4/3)π(8r^{3})

= 8 x (4/3)πr^{3}

So, volume of the new sphere will be 8 times volume of old sphere.

(b) Radius of sphere is increased by 20%.

Radius of new sphere = 1.20r

Volume of new sphere = (4/3)π(1.20r)^{3}

= (4/3)π(1.728r^{3})

= 1.728 x (4/3)πr^{3}

So, volume of the new sphere will be 1.728 times volume of old sphere.

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