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 cm ^{2}**

**B. 520 cm ^{2}**

**C. 800 cm ^{2}**

**D. 790 cm ^{2}**

**E. 950 cm ^{2}**

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 90^{0} to each other) and satisfy:

**Longest side ^{2} = Second longest side^{2} + Shortest side^{2}**

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 10^{2} = 8^{2} + 6^{2}

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 cm^{2}

Hence the **correct option is (A)**

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