3 Sides Triangle Area Calculator

Created by Steven Wooding
Reviewed by Aleksandra Zając, MD
Last updated: Oct 20, 2022

The 3 sides triangle area calculator is here to the rescue if you need to calculate the area of a triangle but only know the three side lengths. A good example is trying to figure out the area of a triangular-shaped room – with this calculator, you'll learn how to find the square footage of a triangle room.

Read on to learn about:

  • Calculating the area of a triangle with 3 sides.
  • Heron's formula.
  • How to find the third side of a triangle without angles.

Calculating the area of a triangle with 3 sides – Heron's formula

To calculate the area of a triangle using the three side lengths is surprisingly tricky. If we know the height, then the area is found by simply multiplying the height by the base length and dividing by two. What we have built is a triangle area calculator – 3 sides and without height.

The area can be found using Heron's formula, first published by Heron (or Hero) of Alexandria in around 60 AD. It's believed Archimedes knew the formula 200 years earlier, but it was never published at the time, as far as we know.

Heron's formula can be expressed in many ways. The longest form is to take the three sides (aa, bb, and cc), sum them together, then multiply by another three sums, but each time one of the sides is subtracted. Then the square root is taken, and we divide it by four to get the area AA. Here is that mathematically:

A=14[(a+b+c)(a+b+c)(ab+c)(a+bc)]\footnotesize \begin{align*} A = \frac{1}{4}\sqrt{}[(a + b + c)(-a + b + c)\\[0.5em] (a - b + c)(a + b - c)] \end{align*}

A more compact way to write the same formula is:

A=144a2b2(a2+b2c2)2\footnotesize A = \frac{1}{4}\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2}

All rather complicated. So why not just use our 3 side triangle area calculator to make life easy.

How to use this 3 side triangle area calculator

It's super easy to use the calculator. Simply enter the three side lengths, and the calculator will show the area result instantly.

You can even try using the calculator to find a missing side length if you know the area and the other two side lengths:

  1. Enter a value for the area of the triangle.
  2. Input the other two known side lengths of the triangle.
  3. The third, missing side length will then be calculated for you.

And that is how to find the third side of a triangle without angles.

Other triangle area calculators

Here are some other triangle area calculators you can explore here on the Omni Calculator website:

FAQ

Can any 3 sides make a triangle?

No, it is not the case that any 3 sides can make a triangle. If one side of the triangle is longer than the sum of the other two sides, then you cannot form a triangle.

How do I find the square footage of a triangle?

To find the square footage area of a triangle, follow these steps:

  1. Measure each side of the triangle in feet and label them a, b, and c.

  2. Input them into Heron's formula, shown below:

    A = √[4a²b² - (a² + b² - c²)²]/4

  3. The result is the area of your triangle in square feet.

What is the area of a triangle with side lengths of 9, 6 and 5 inches?

14.1 square inches. This result comes from using Heron's formula:

A = √[4a²b² - (a² + b² - c²)²]/4

Substituting in the values, we get:

A = √[4×9²×6² - (9² + 6² - 5²)²]/4
A = √[11664 - 8464]/4
A = 56.57/4 = 14.1 inch²

Steven Wooding
Triangle area: triangle with three sides





a
in
b
in
c
in
Area
in²
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