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

Explanatory Answer

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

Formula 1: Area of an equilateral triangle if its side is known

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).