Triangle Area Calculator (Heron’s Formula)
Calculate the area and perimeter of any triangle from its three sides using Heron’s formula.
Step-by-step working
- s = (5 + 6 + 7) / 2 = 9
- Area = √(s(s−a)(s−b)(s−c)) = √(9·4·3·2) = 14.6969
Mathematical Formula
Overview & Explanation
Heron’s formula gives a triangle’s area from its side lengths alone, without needing a height or angle.
How It Works
- Check the triangle inequality.
- Compute the semi-perimeter s.
- Apply Heron’s formula.
Practical Applications
- Land and plot area from measured sides.
- Geometry problems.
Heron's formula
When you know all three sides a, b and c but not the height, use Heron's formula. First find the semi-perimeter s = (a + b + c) ÷ 2, then area = √(s(s − a)(s − b)(s − c)). For sides 5, 6 and 7: s = 9 and area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square units.
The three sides must satisfy the triangle inequality: each side must be shorter than the sum of the other two, otherwise no triangle exists and the formula gives the square root of a negative number.
Two sides and the included angle
If you know two sides and the angle between them, area = ½ × a × b × sin(C). Sides 8 and 10 with a 30° angle between them give ½ × 8 × 10 × 0.5 = 20 square units. This is handy in surveying, where angles are measured directly, and for irregular plots that can be split into triangles.
Example 1: 3, 4, 5
Right triangle, area 6.
Inputs
Side a: 3 · Side b: 4 · Side c: 5
Step 1
s = (3 + 4 + 5) / 2 = 6
Step 2
Area = √(s(s−a)(s−b)(s−c)) = √(6·3·2·1) = 6
Area
Result
Example 2: 5, 6, 7
Scalene triangle.
Inputs
Side a: 5 · Side b: 6 · Side c: 7
Step 1
s = (5 + 6 + 7) / 2 = 9
Step 2
Area = √(s(s−a)(s−b)(s−c)) = √(9·4·3·2) = 14.6969
Area
Result
How it's calculated & sources
s = (a + b + c) / 2; Area = √(s(s − a)(s − b)(s − c)). Valid only if each side is less than the sum of the other two.
Frequently Asked Questions
What if the sides cannot form a triangle?
Is Heron’s formula exact?
Other area formulas?
Which formula should I use?
Do units matter?
Related Calculators
Pythagorean Theorem Calculator
Find the hypotenuse or a missing leg of a right triangle using a² + b² = c².
Circle Calculator (Area, Circumference)
Enter a radius, diameter, circumference or area and get every other circle measurement.
Percentage Calculator
Calculate percentages, percentage change, and percentage of a number instantly.
Percentage Increase Calculator
Calculate the percentage growth or increase from an initial value to a final value.