# Finding the area of a parallelogram given the length of its diagonals and their intersection angle

The diagonals of a parallelogram have lengths of $15.6 \text{cm}$ and $17.2 \text{cm}$. They intersect at an angle of $120$. Find the area of the parallelogram.

The part I find most confusing is the intersection angle. I thought diagonals intersect at a right angle.

This comes from an 8th-grade school math textbook.

• A rectangle is a parallelogram. Do its diagonals intersect at right angles? – Lord Shark the Unknown Aug 31 '17 at 11:16

## 2 Answers

The area is $\mathcal{A}=\dfrac{1}{2} 15.6\times 17.2 \,\dfrac{\sqrt{3}}{2}\approx 232.372$cm$^2$

The answer is 116 cm square. The answer would be 232 cm square if there was no halving required.

• Wow! Absolute wow! May I ask how you made the illustration? MSPaint can never be this good. – Soha Farhin Pine Aug 31 '17 at 11:30
• Geogebra. It's free (as in "free speech") and you can find it for almost any OS geogebra.org – Raffaele Aug 31 '17 at 11:52
• @amd Now it is :) Anyway the formula for the area doesn't change – Raffaele Aug 31 '17 at 18:43

Recall that the area of the parallelogram is $\frac{1}{2}d_1 d_2\sin\theta$, where $d_1$ and $d_2$ are the lengths of the diagonals, and $\theta$ is the angle between the diagonals.

As regards the intersection angle take a look here: Diagonals of parallelogram intersect at $90^\circ$ if and only if figure is rhombus