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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?

Matthew
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  • Show that the altitude of an equilateral triangle is $ \ \frac{\sqrt{3}}{2} \ $ times the length of one of its sides. Then apply the formula for the area of a triangle. – colormegone Apr 02 '14 at 03:12

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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.

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

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

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

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

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

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

$$s \approx 9.118028228$$

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

Brad
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  • -1. The OP asked for an explanation, not a string of equations, meaningless to someone unfamiliar with the context of this question. After the second line, you fail to answer the main question ('how') rather than 'what'. – beep-boop Jul 02 '14 at 22:44
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    @alexqwx: to be fair, the question does not give much context as to what can or cannot be used. The formula $A=\frac{\sqrt3}4s^2$ is fairly common. More basic than that, is showing that the apothem of an equilateral triangle is $\frac{s}{2\sqrt3}$. – robjohn Jul 06 '14 at 22:25