# Area of equilateral triangle

This aptitude question helps recall 3 important formulae to compute area of a triangle if we know the in radius, circum radius and radius of the ex circle (ex radius) of the triangle.

## Question

What is the ratio measures of the in-radius, circum-radius and one of the ex-radius of an equilateral triangle?

1. 1 : 2 : 5
2. 1 : 3 : 5
3. 1 : 2 : 3
4. 1 : 1.4142 : 2

Correct Answer      Choice (C). 1 : 2 : 3

In an equilateral triangle all three sides are of the same length and let the length of each side be 'a' units.

A = $$frac{\sqrt{3}}{4}$a2 ##### Formula 2: Area of a triangle if its inradius, r is known Area A = r $\times$ s, where r is the in radius and 's' is the semi perimeter. The semi perimeter, s = $\frac{3a}{2}$ In-radius, 'r' for any triangle = $\frac{A}{s}$ ∴ for an equilateral triangle its in-radius, 'r' = $\frac{A}{s}$ = $\frac{a}{{2\sqrt{3}}}$ ##### Formula 3: Area of a triangle if its circumradius, R is known Area, A = $\frac{abc}{4R}$, where R is the circumradius. Circumradius, R for any triangle = $\frac{abc}{4A}$ ∴ for an equilateral triangle its circum-radius, R = $\frac{abc}{4A}$ = $\frac{a}{\sqrt{3}}$ ##### Formula 4: Area of an equilateral triangle if its exradius is known Let one of the ex-radii be r1. For an equilateral triangle, all 3 ex radii will be equal. Area = r1 *$s-a), where 's' is the semi perimeter and 'a' is the side of the equilateral triangle.
∴ ex-radius of the equilateral triangle, r1 = $$frac{A}{s-a}$ = $\frac{{\sqrt{3}}a}{2}$ Therefore, the ratio of these radii is $\frac{a}{{2\sqrt{3}}}$ : $\frac{a}{\sqrt{3}}$ : $\frac{{\sqrt{3}}a}{2}$ Or the ratio is 1 : 2 : 3 The correct answer is Choice$C).

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