Using Heron's Formula in Geometry

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  • 0:06 Heron's Formula
  • 1:52 The Sides of a Triangle
  • 2:18 Finding S
  • 3:02 Finding the Area
  • 4:46 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Heron's formula has been around for a couple thousand years. Watch this video lesson to see how you can use Heron's formula to find the area of a triangle while only knowing the side lengths.

Heron's Formula

Heron's formula is a formula that helps you to find the area of a triangle when all you know are the lengths of the sides. It is an ancient formula that has been around for a couple thousand years!

The formula itself is a two-step process. Let me give you an overview of the process before going into detail about it. I know it's a two-step formula, but it's not hard. I want to warn you that it may look tricky, but it's really not. Don't let your eyes get the better of you. Stay calm, and tell yourself this is a straightforward formula just like all the other formulas that you know. Are you ready? Let's go.

First, we have a triangle with three sides.

To use this formula, the sides of a triangle should be labeled.
triangle with sides labeled a, b and c

We label our three sides a, b, and c. It doesn't matter which side is which as long as we have all three labeled in some way.

Next, we need to find half of the triangle's perimeter. We call this value s.

s = (a + b + c)/2

We find s by adding up all our sides and then dividing by two. The next part of Heron's formula gives us the area.

A = sqrt (s(s - a)(s - b)(s - c))

We see that we have to subtract each side from the half perimeter and then multiply all those values together along with the half perimeter again. Then, our area is the square root of our multiplication.

What did you think? Did Heron's formula look a bit complicated? I think when you first look at it, it's a little tricky, but once you see it in real action, it becomes less complicated. So, let's go over a real example problem so you can see how all the numbers work together.

The Sides of a Triangle

We begin with a triangle. We aren't limited by the type of triangle we have. It can be any shape or type. The only things that we must know about the triangle are how long each side is.

So, the triangle we'll be working with has sides of 3, 4, and 5 inches long. To keep things simple, I will label the 3-inch side a, the 4-inch side b, and the 5-inch side c.

Example triangle
example triangle with sides 3, 4 and 5

Finding S

The s is part of Heron's formula, so we need to find this half perimeter before finding the area. Do you remember the formula for finding the half perimeter s? It is the sum of all three sides divided by 2.

s = (a + b + c)/2

Using the sides from our triangle, we find s by plugging in our labeled values and then evaluating for s.

s = (3 + 4 + 5)/2

s = (12)/2

s = 6 inches

We find our half-perimeter to equal 6 inches. We can use this number in the second part of Heron's formula to find the area now.

Finding the Area

The second part of Heron's formula gives you the area.

A = sqrt (s(s - a)(s - b)(s - c))

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