Free Triangle Area Calculator
Calculate the area of any triangle in two ways: from a base and its height, or from the lengths of all three sides using Heron's formula. Results update live as you type.
Choose a method, then enter values to calculate the triangle's area. Results update live and use whatever unit your inputs share (the area is that unit squared).
Base & height: A = ½ · b · h, where h is the perpendicular height to the base. Three sides: s = (a + b + c) / 2, then A = √(s(s − a)(s − b)(s − c)). Sides must obey the triangle inequality.
Quick answer
A triangle's area equals one-half its base times its height: A = ½ · b · h. For a base of 10 and a height of 6, A = ½ · 10 · 6 = 30 square units. If you only know the three side lengths a, b, and c, use Heron's formula: compute the semi-perimeter s = (a + b + c) / 2, then A = √(s(s − a)(s − b)(s − c)).
Formula & method
Base and height
A = ½ · b · h
- A — Area of the triangle
- b — Length of the base
- h — Perpendicular height to that base
The height (altitude) must be measured perpendicular to the chosen base, not along a slanted side.
Heron's formula (three sides)
s = (a + b + c) / 2, A = √(s(s − a)(s − b)(s − c))
- s — Semi-perimeter (half the perimeter)
- a, b, c — The three side lengths
- A — Area of the triangle
Valid only when the three sides satisfy the triangle inequality: each side must be shorter than the sum of the other two.
Examples
- Input
- Mode: Base & height, base b = 10, height h = 6
- Result
- Area = 30 square units
- Why
- A = ½ · b · h = ½ · 10 · 6 = ½ · 60 = 30. The base and its perpendicular height are all you need; the triangle's exact shape does not change the area.
- Input
- Mode: Three sides, a = 3, b = 4, c = 5
- Result
- Area = 6 square units
- Why
- Semi-perimeter s = (3 + 4 + 5) / 2 = 6. Then A = √(6·(6−3)·(6−4)·(6−5)) = √(6·3·2·1) = √36 = 6. This matches ½·3·4 = 6 because 3-4-5 is a right triangle.
- Input
- Mode: Three sides, a = 13, b = 14, c = 15
- Result
- Area = 84 square units
- Why
- s = (13 + 14 + 15) / 2 = 21. A = √(21·(21−13)·(21−14)·(21−15)) = √(21·8·7·6) = √7056 = 84. Heron's formula handles triangles with no right angle and no known height.
- Input
- Mode: Three sides, a = 1, b = 2, c = 10
- Result
- No triangle exists
- Why
- The triangle inequality fails because 1 + 2 = 3 is less than 10, so these lengths cannot close into a triangle. The calculator flags this instead of returning a meaningless number from the square root of a negative value.
When to use this tool
- You know a base and its perpendicular height and want the area in one step (A = ½·b·h).
- You only have the three side lengths — from a survey, a drawing, or a word problem — and no angle or height (use Heron's formula).
- You need to verify a triangle is geometrically valid before computing area, via the built-in triangle-inequality check.
- Homework, drafting, land-area estimates, or any geometry task where you want the arithmetic shown and a copyable result.
Common mistakes
- Using a slanted side length as the height. In A = ½·b·h, h must be the perpendicular distance from the base to the opposite vertex, not the length of a leg drawn at an angle.
- Forgetting to halve in Heron's formula — using the full perimeter instead of the semi-perimeter s = (a + b + c)/2 doubles or quadruples the value under the root.
- Mixing units. If the base is in centimeters and the height is in meters, convert to one unit first; otherwise the area is wrong by a factor of 100.
- Entering side lengths that violate the triangle inequality (e.g. 1, 2, 10). No real triangle exists, so the area is undefined — the value under Heron's square root goes negative.
Frequently asked questions
Why is the triangle area formula one-half base times height?
Any triangle is exactly half of a parallelogram (or rectangle) built on the same base and height. A parallelogram's area is base × height, so a triangle covers half of that: A = ½ · b · h. This holds for every triangle regardless of its angles.
What is Heron's formula and when do I need it?
Heron's formula finds a triangle's area from its three side lengths alone. First compute the semi-perimeter s = (a + b + c)/2, then A = √(s(s − a)(s − b)(s − c)). Use it whenever you know all three sides but have no height or angle to work with.
How do I find the height if I only know the sides?
Compute the area with Heron's formula, then rearrange A = ½·b·h to get h = 2A / b for the base you care about. For example, the 13-14-15 triangle has area 84, so the height to the side of length 14 is 2·84/14 = 12.
What is the triangle inequality and why does it matter here?
The triangle inequality says each side must be shorter than the sum of the other two (a + b > c, a + c > b, b + c > a). If it fails, the lengths cannot form a closed triangle and Heron's formula would take the square root of a negative number, so this calculator reports that no triangle exists.
Does the calculator work for right, isosceles, and equilateral triangles?
Yes. Both methods apply to every triangle type. For a right triangle the two legs are a valid base and height, so A = ½·leg₁·leg₂. For equilateral or isosceles triangles where you only know the sides, Heron's formula gives the area directly.
What units does the result use?
The calculator is unit-agnostic: it returns the area in whatever squared unit your inputs use. If you enter lengths in centimeters the area is in cm²; in meters it is m². Just keep every input in the same unit.
Sources & references
External references open in a new tab. We are independent and not affiliated with these organizations.
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- ✓ Formula and method shown above
Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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