GMAT Math : Heron’s formula

Heron’s formula is an amazing tool to solve GMAT Quant questions involving areas of triangles

Let’s start with an example :

What is the area of a triangle with sides 25 cm, 60 cm, 65 cm?

A. 750 cm2

B. 520 cm2

C. 800 cm2

D. 790 cm2

E. 950 cm2

How do we get to the area of an arbitrary triangle with just the sides given?

Enter Heron’s formula

Area = √(sx(s-a)x(s-b)x(s-c))

Where s = (a+b+c)/2, and a, b, c are sides of the triangle

This formula works with ANY triangle – right, isosceles, equilateral – you name it. It is quite a nifty tool.

Now, if you wanted to solve this question EVEN faster – here’s an observation.

3, 4, 5

5, 12, 13

8, 15, 17

7, 24, 25

Are Pythagorean triplets – essentially they represent sides of a right triangle (the shortest side and second longest side are at 900 to each other) and satisfy:

Longest side2 = Second longest side2 + Shortest side2

If you multiply a Pythagorean triplet with a constant, the resultant series remains a Pythagorean triplet.

So, (3, 4, 5) x 2 = (6, 8, 10) is also a Pythagorean triplet and satisifes 102 = 82 + 62

What this also means is the area of triangle with sides 6, 8, 10 is straightforward to calculate.

Area = 0.5 x Height x Base (where Height and Base are at right angles to each other)

Coming to our question, notice 25, 60 and 65 is also a Pythagorean triplet

(5, 12, 13) x 5 = (25, 60, 65)

So area of the triangle = 0.5 x 25 x 60 = 750 cm2

Hence the correct option is (A)

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