Triangle Area (Heron's Formula)
Find the area of a triangle from its three side lengths with Heron's formula, including the semiperimeter, validity checks and a 5-6-7 worked example.
Results
Area
- Semiperimeter
- 9
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Formula / how it works
s = (a + b + c) ÷ 2. Area = √( s (s − a)(s − b)(s − c) ). The sides must satisfy the triangle inequality: each side is less than the sum of the other two.
Detailed guide
What it is, how to use it, how to read the numbers, common mistakes, a worked example, formulas, and FAQs.
What this calculator is
Heron's formula gives a triangle's area when you know all three sides but not the height. First find the semiperimeter s, half the perimeter. Then the area is the square root of s times s minus each side. It works for any triangle, whether acute, obtuse or right, so you do not need an angle or an altitude. The calculator returns the area and the semiperimeter. If the three lengths cannot form a triangle, the value under the square root is zero or negative, so there is no valid area. Use consistent units; the area comes out in squares of those units.
When to use / who it helps
- You measured the three sides of a plot, sail or panel and need its area.
- You do not know the height or any angles.
- You are checking geometry homework.
- You want to test whether three lengths can form a triangle.
- You are estimating material for a triangular surface.
How to use
Steps
- Enter side a.
- Enter side b.
- Enter side c.
- Use the same unit for all three sides.
- Read the area and the semiperimeter.
- Check that each side is shorter than the sum of the others.
How to interpret results
- Area is in square units of the length unit used.
- The semiperimeter is half the perimeter.
- A zero area means the three points are collinear.
- An invalid triangle produces no real area.
- Doubling every side multiplies the area by four.
Common mistakes & gotchas
- Entering sides that violate the triangle inequality.
- Mixing units, such as feet and inches.
- Using the perimeter instead of the semiperimeter.
- Rounding the semiperimeter too early.
- Reporting area in linear units.
Worked example
Worked example — sides 5, 6 and 7
- Semiperimeter: s = (5 + 6 + 7) ÷ 2 = 9.
- Differences: s − a = 4, s − b = 3, s − c = 2.
- Product: 9 × 4 × 3 × 2 = 216.
- Area = √216 ≈ 14.697. Check with a 3-4-5 triangle: s = 6, √(6 × 3 × 2 × 1) = 6, which matches ½ × 3 × 4.
Result: A 5-6-7 triangle has an area of about 14.697 square units.
To load the form's own demo values, tap Try example above the Calculate button.
How to calculate / formula
s = (a + b + c) ÷ 2. Area = √( s (s − a)(s − b)(s − c) ). The sides must satisfy the triangle inequality: each side is less than the sum of the other two.
Frequently asked questions
What is Heron's formula?
A formula for the area of a triangle from its three side lengths.
What is the semiperimeter?
Half of the triangle's perimeter.
Why do I get no result for some inputs?
The three lengths do not form a triangle.
Does it work for right triangles?
Yes, it works for every triangle.
Can I use any unit?
Yes, but use the same unit for all sides; area is in that unit squared.
Is Heron's formula exact?
Yes, though floating-point rounding can matter for very thin triangles.