Area of a Triangle with 3 Sides Calculator

What is the area of a triangle with 3 sides calculator?

This calculator finds the area of any triangle when you only know the lengths of its three sides. There is no need to measure the height, an angle, or any other parameter: enter the three sides, and the tool returns both the area and the perimeter instantly. It relies on Heron’s formula, a classic result of plane geometry that works for every triangle, whether it is acute, right, or obtuse.

Knowing all three sides is one of the most common situations in practice. Surveyors, builders, and designers frequently measure distances directly but rarely have a convenient way to measure a triangle’s height. This calculator turns those three measurements into an area in a single step.

How does the calculator work?

The calculation happens in two stages. First, the calculator computes the semi-perimeter, which is half of the perimeter. Then it substitutes the semi-perimeter and the three side lengths into Heron’s formula to obtain the area.

The semi-perimeter $s$ is found as:

$s = \frac{a + b + c}{2}$

The area $A$ then follows from Heron’s formula:

$A = \sqrt{s(s-a)(s-b)(s-c)}$

where $a$, $b$, and $c$ are the lengths of the three sides. The perimeter is simply the sum of the sides, $a + b + c$, which the tool also reports.

For the result to describe a real triangle, the three sides must satisfy the triangle inequality: the sum of any two sides must be greater than the third side. If this condition fails, the expression under the square root becomes negative and no triangle exists.

Examples

Example 1: Right triangle (3, 4, 5)

Consider a triangle with sides 3, 4, and 5.

1. Calculate the semi-perimeter:
   $s = \frac{3 + 4 + 5}{2} = 6$

2. Substitute into Heron’s formula:
   $A = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \times 3 \times 2 \times 1}$

3. Solve:
   $A = \sqrt{36} = 6$

The area is 6 square units, matching the familiar right-triangle formula $\frac{1}{2} \times 3 \times 4 = 6$.

Example 2: Scalene triangle (7, 8, 9)

Imagine a triangle with sides 7, 8, and 9.

1. Calculate the semi-perimeter:
   $s = \frac{7 + 8 + 9}{2} = 12$

2. Substitute into Heron’s formula:
   $A = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3}$

3. Solve:
   $A = \sqrt{720} \approx 26.83$

The area is approximately 26.83 square units.

Example 3: Equilateral triangle (6, 6, 6)

Consider an equilateral triangle with each side equal to 6.

1. Calculate the semi-perimeter:
   $s = \frac{6 + 6 + 6}{2} = 9$

2. Substitute into Heron’s formula:
   $A = \sqrt{9(9-6)(9-6)(9-6)} = \sqrt{9 \times 3 \times 3 \times 3}$

3. Solve:
   $A = \sqrt{243} \approx 15.59$

The area is approximately 15.59 square units.

Practical notes

• The method works for every triangle type, so you do not need to know whether the triangle is acute, right, or obtuse.
• Always confirm the triangle inequality: the sum of the two shorter sides must exceed the longest side.
• If you know parameters other than three sides, such as a base and height or two sides and an angle, use the more general triangle area calculator instead.
• This tool uses the same mathematics as the dedicated Heron’s formula calculator; choose whichever framing suits your problem.

Frequently asked questions

Can I find a triangle’s area knowing only its three sides?

Yes. Heron’s formula gives the area directly from the three side lengths, with no need to measure height or angles.

What is the semi-perimeter?

The semi-perimeter is half of the perimeter, $s = \frac{a + b + c}{2}$. It is an intermediate quantity that simplifies Heron’s formula.

Why must the sides satisfy the triangle inequality?

If the sum of any two sides is not greater than the third, the three lengths cannot form a triangle, and the value under the square root becomes negative, so no real area exists.

Does this calculator handle different units?

Yes. You can choose units for each side, and the area and perimeter are reported in the units you select, with conversions handled automatically.