Area of triangle with 3 sides formula

Let a,b,c be the lengths of the sides of a triangle. The area is given by: where p is half the perimeter, or

Try this Drag the orange dots to reshape the triangle. The formula shown will re-calculate the triangle's area using Heron's Formula

Heron was one of the great mathematicians of antiquity and came up with this formula sometime in the first century BC, although it may have been known earlier. He also extended it to the area of quadrilaterals and higher-order polygons.

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 (aaa, bbb, and ccc), 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 AAA. Here is that mathematically:

A=14[(a+b+c)(−a+b+c)(a−b+c)(a+b−c)]\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=41​​[(a+b+c)(−a+b+c)(a−b+c)(a+b−c)]​

A more compact way to write the same formula is:

A=144a2b2−(a2+b2−c2)2\footnotesize A = \frac{1}{4}\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2}A=41​4a2b2−(a2+b2−c2)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:

• Triangle area;
• Similar triangles;
• Square feet of a triangle;
• Scalene triangle area;
• Area of obtuse triangle;
• Area of oblique triangle;
• Area of triangle with coordinates; and
• Area of triangle SAS.

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.

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How do you find the area of a triangle?