3 Sides Triangle Area Calculator

How to Calculate Triangle Area from 3 Sides (Heron's Formula)

When you know the lengths of all three sides of a triangle ($a, b, c$) but do not know its height or angles, you can calculate the exact surface area using Heron's formula (also known as Hero's formula), established by Heron of Alexandria in 60 AD.

Heron's Formula and Semi-Perimeter

First, compute the triangle's semi-perimeter ($s$), which is half of the total perimeter:

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

Next, apply Heron's formula to determine the total area ($A$):

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

The Triangle Inequality Theorem

For any three positive lengths to form a valid geometric triangle, they must satisfy the Triangle Inequality Theorem: the sum of the lengths of any two sides must always be strictly greater than the length of the remaining side:

$a + b > c$
$a + c > b$
$b + c > a$

If $a + b = c$, the three vertices fall on a single straight line (a degenerate triangle with zero area). If $a + b < c$, the sides cannot meet to form a closed polygon.

Related Geometric Properties

1. Inradius (Incircle Radius)

The radius ($r$) of the largest circle that fits completely inside the triangle:

$$r = \frac{\text{Area}}{s}$$

2. Circumradius (Circumcircle Radius)

The radius ($R$) of the circle that passes through all three vertices of the triangle:

$$R = \frac{a \cdot b \cdot c}{4 \cdot \text{Area}}$$

3. Internal Angles (Law of Cosines)

Each internal angle can be computed from the three side lengths:

$$\cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos(\beta) = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab}$$

Step-by-Step Example

Calculate the area of a triangle with sides $a = 5\text{ cm}$, $b = 6\text{ cm}$, and $c = 7\text{ cm}$:

Perimeter: $P = 5 + 6 + 7 = 18\text{ cm}$.
Semi-perimeter: $s = \frac{18}{2} = 9\text{ cm}$.
Differences: $s - a = 4$, $s - b = 3$, $s - c = 2$.
Product under square root: $9 \times 4 \times 3 \times 2 = 216$.
Area: $\sqrt{216} \approx 14.6969\text{ cm}^2$.

Explore related tools: check out our Half Square Triangle Calculator, calculate with the AAA Triangle Calculator, or use the AAS Triangle Calculator.

Frequently Asked Questions

What happens if three side lengths do not satisfy the triangle inequality?

If the sum of two sides is less than or equal to the third side, the side lengths cannot connect in Euclidean space. The calculator detects this condition and warns you of an invalid triangle configuration.

Does Heron's formula work for right-angled triangles?

Yes. For a right triangle with legs $a, b$ and hypotenuse $c = \sqrt{a^2 + b^2}$, Heron's formula simplifies algebraically to $\frac{1}{2}ab$, matching the standard base-times-height formula.

Can Heron's formula calculate the area of an equilateral triangle?

Yes. For an equilateral triangle with equal sides $a$, Heron's formula simplifies to $\frac{\sqrt{3}}{4}a^2$.

How are triangle altitudes (heights) determined from 3 sides?

Because $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$, the altitude to any side $a$ is $h_a = \frac{2 \times \text{Area}}{a}$.

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