Is there an equation to find the angle of the diagonal in a rectangle?

Question

If we have a rectangle of length 5 and height 5 the angle of the diagonal would be 45°.

We know this is true but how can we arrive at this conclusion mathematically?

Answer 1

The diagonal of a rectangle splits a right angle into two complementary angles:

• $\alpha$
• $\beta = 90^\circ - \alpha$

where $$\tan \alpha = \frac {h}{\mathcal l}\implies \alpha = \arctan\left(\frac h{\mathcal l}\right)$$

It follows that $$\tan \beta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\mathcal l}{h} \implies \beta = \arctan\left(\frac {\mathcal l}{h}\right), \;\text{ and } \alpha + \beta = \frac{\pi}2 = 90^\circ$$

The example you give of a $5\times 5$ "rectangle" is a square, in which case (for squares of any dimension) using the notation I use here, $\alpha = \beta = 45^\circ$.