# The Area of a Rhombus is Equal to Half the Product of its Diagonals

Here we will prove that the area of a rhombus is equal to half the product of its diagonals.

Solution:

Given:

PQRS is a rhombus whose diagonals are PR and QS. The diagonals intersect at O.

To prove: ar(rhombus PQRS) = $$\frac{1}{2}$$ ×PR × QS.

 Statement Reason 1. ar(∆RSQ) = $$\frac{1}{2}$$ ×Base × Altitude                  = $$\frac{1}{2}$$ ×QS × RO. 1. QS ⊥ PR, because diagonals of a rhombus are perpendicular to each other. 2. ar(∆PQS) = $$\frac{1}{2}$$ ×Base × Altitude                  = $$\frac{1}{2}$$ ×QS × PO. 2. As in reason 1. 3. ar(∆RSQ) + ar(∆PQS)                   = $$\frac{1}{2}$$ ×QS × (RO + PO). 3. By addition from statements 1 and 2. 4. ar(rhombus PQRS) = $$\frac{1}{2}$$ ×PR × QS. 4. By addition axiom for area.