The Diagonals of a Rhombus are 7.5 cm and 12 cm, Find its Area

Find Rhombus Area Based on Diagonal

The diagonals of a rhombus are 7.5 cm and 12 cm, find its area. The answer, in a word, is 45 cm². Let's deep dive. The diagonals of a rhombus are line segment connected by two opposite corners of the rhombus.

The two diagonals of rhombus are unequal to each other. And the diagonals of rhombus bisect to one another.

Again, all the sides of a rhombus are equal in length.

Moreover, the diagonals of rhombus bisects at right angle.

First Method

Let ◇PQRS be a rhombus and its two diagonals are PR and QS.

Suppose, PR = d₁ and QS = d₂ where d₁ = 7.5 cm and d₂ = 12 cm.

The area of a rhombus is half the product of the diagonals.

∴ Area of ◇PQRS = 12 × d₁ × d₂
or, Area of ◇PQRS = 12 × 7.5 cm × 12 cm
or, Area of ◇PQRS = (12 × 7.5 × 12) cm²
or, Area of ◇PQRS = (7.5 cm × 6) cm²
or, Area of ◇PQRS = (7510 × 6)cm²
or, Area of ◇PQRS = (755 × 3) cm²
or, Area of ◇PQRS = (15 × 3) cm²
∴ Area of ◇PQRS = 45 cm²

Hence, the area of a rhombus is 45 cm² if the diagonals of a rhombus are 7.5 cm and 12 cm.

Second Method

Let ◇ABCD be a rhombus whose two diagonals are AC and BD intersect at O.

If AC = d₁ and BD = d₂,
d₁ = 7.5 cm and
d₂ = 12 cm.

AC and BD bisect at O.

∴ OA = 12 AC
or, OA = 12 d₁
or, OA = 12× 7.5 cm
or, OA = 7.52 cm
∴ OA = 3.75 cm

Again, OB = 12 BD
or, OB = 12 d₂
or, OB = 12× 12 cm
∴ OB = 6 cm

Again, the diagonals AC and BD intersect at right angle at the point O. So, ∠AOB = 90°.

Thus, △AOB is a right triangle whose hypotenuse is AB and other two sides are OA and OB.

We know if a and b are the sides of right angle of a right triangle,
Area of right triangle = 12 ab square unit.

∴ Area of △AOB = (12×OA×OB) square unit
or, Area of △AOB = (12×3.75×6) cm²
or, Area of △AOB = (3.75×3) cm²
∴ Area of △AOB = 11.75 cm²

The two diagonals divides the rhombus into four congruent right triangles.

∴ Area of ◇ABCD = 4×△AOB

or, Area of ◇ABCD = (4×11.25) cm²

∴ Area of ◇ABCD = 45 cm²

Therefore, the area of rhombus is found 45 cm² if the diagonals of a rhombus are 7.5 cm and 12 cm.