If an equilateral triangle has an area of 36 units squared, what is the length of a side to the nearest tenth?

Question

I have been working with finding the area of a regular triangles, squares, and hexagons using special right triangle formulas drawn from the radii and apothems, but I cannot for the life of me work backwards. How would I find a side length given the area?

Answer 1

I will leave the formula for you to prove ($s$ is side length). Consider drawing an altitude for an equilateral triangle and using symmetry to find the angles of the two triangles you are left with.

$$A_{\text{equilateral}} = \dfrac{s^2\sqrt{3}}{4}$$

$$36 = \dfrac{s^2\sqrt{3}}{4}$$

$$144 = s^2\sqrt{3}$$

$$\frac{144\sqrt{3}}{3} = s^2$$

$$s = 12\frac{3^{1/4}}{3^{1/2}}$$

$$s = 12 \cdot 3^{-1/4}$$

$$s \approx 9.118028228$$

$$\boxed{s \approx 9.1}$$