Triangle Area Calculator
Calculate the area of a triangle from three side lengths (Heron's formula) or from a base and height. Enter your known values, pick a unit, and get the area with step-by-step calculations.
Find the Area of a Triangle
Triangle Area Formula
Where:
- \(A\) — area of the triangle (in the chosen unit squared)
- \(b\) — base of the triangle (length of the side the height is measured from)
- \(h\) — height, the perpendicular distance from the base to the opposite vertex
- \(a, b, c\) — the three side lengths of the triangle
- \(s\) — semiperimeter, half of the perimeter
The first formula is the classic base-height method: the area is half the product of a side and the perpendicular height dropped onto it. The second is Heron's formula, which finds the area from the three side lengths alone, without needing any angle or height.
Triangle Area – Calculation Example
Example: a triangle has sides \(a = 3\) cm, \(b = 4\) cm and \(c = 5\) cm (a right triangle). Find its area using Heron's formula.
\[ s = \frac{3 + 4 + 5}{2} = 6 \text{ cm} \]</n
\[ A = \sqrt{6 \times (6-3) \times (6-4) \times (6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6 \text{ cm}^2 \]
Result: the area is 6 cm². As a check, the base-height formula with legs 3 and 4 gives \(\tfrac{1}{2} \times 3 \times 4 = 6\) cm² — the same answer.
How to use the calculator
- Open the Known values dropdown and choose Three side lengths (Heron's formula) or Base and height, depending on what you know about your triangle.
- If you chose three side lengths, enter Side a, Side b and Side c. If you chose base and height, enter Base and Height instead.
- Pick the Unit (cm, m, in or ft). It applies to all lengths you enter and to the result.
- Read the Triangle Area in square units. For the side-length method you also get the Semiperimeter, and the Calculation Steps panel shows exactly how the answer was found.
What is the area of a triangle?
The area of a triangle is the amount of surface enclosed by its three sides, measured in square units (cm², m², in², ft²). Two equivalent formulas cover almost every practical situation. If you can measure one side together with the perpendicular height dropped onto that side, use \(A = \tfrac{1}{2}bh\). If you only know the three side lengths — common in surveying, land measurement and geometry problems — Heron's formula gives the area directly.
Heron's formula, attributed to Heron (Hero) of Alexandria in the 1st century AD, first computes the semiperimeter \(s\), which is simply half the perimeter. The area is then the square root of the product \(s(s-a)(s-b)(s-c)\). The result is always the same no matter which side you label \(a\), \(b\) or \(c\).
Interpreting the result is straightforward: the number tells you how many unit squares would fit inside the triangle. For instance, an area of 6 cm² means six 1 cm × 1 cm squares would cover the triangle exactly. Keep in mind that the height in the base-height formula must be perpendicular to the base — a slanted side length is not a height unless the triangle is right-angled and the leg is used as the base.
Assumptions, limitations and tips
- Triangle inequality: the three sides must satisfy \(a + b > c\), \(a + c > b\) and \(b + c > a\). If any side is as long as or longer than the sum of the other two, no triangle exists and the calculator shows an error.
- Positive values only: all lengths must be greater than zero. The calculator rejects zero or negative inputs.
- Same unit for everything: enter all lengths in the same unit; the area comes out in that unit squared. For a result in m², enter sides in metres, not centimetres.
- Height must be perpendicular: when using the base-height method, the height is the shortest distance from the base to the opposite vertex, not the length of a slanted side.
- Units don't mix with exact values: because of rounding, very "thin" triangles (one side barely shorter than the sum of the other two) can give extremely small areas that are sensitive to input precision — measure carefully in such cases.
Related calculations
Once you have the sides of a triangle, you may also want the average of the three sides (which relates directly to the semiperimeter), or to scale your measurements with the multiplication calculator. If your triangle work involves the quadratic formula, our nth root calculator helps with taking roots of any degree.
Frequently Asked Questions
How do I calculate the area of a triangle by hand?
Use \(A = \tfrac{1}{2} b h\): multiply the base by the perpendicular height and divide by 2. If you only know the three sides, compute the semiperimeter \(s = \tfrac{a+b+c}{2}\) and apply Heron's formula \(A = \sqrt{s(s-a)(s-b)(s-c)}\). A quick sanity check: the result can never exceed \(\tfrac{1}{2}\) of the longest side times the longest other side.
What is the area of an equilateral triangle with side a?
An equilateral triangle with side length \(a\) has area \(A = \tfrac{\sqrt{3}}{4}a^2 \approx 0.433\,a^2\). For example, a side of 6 cm gives \(0.433 \times 36 \approx 15.59\) cm². This follows from Heron's formula and is a handy shortcut because all three sides are equal, so the height is \(h = \tfrac{\sqrt{3}}{2}a\).
Can the height of a triangle fall outside the triangle?
Yes. In an obtuse triangle, the perpendicular from the opposite vertex to the base can land outside the triangle's side — the height segment extends beyond it. The base-height formula still works: the height is simply the perpendicular distance between the base line and the vertex, no matter where the foot of the perpendicular lands. Only its length matters, not its position.
How do I convert the area between square units?
Square the length conversion factor. Since 1 m = 100 cm, you have 1 m² = 10,000 cm². Similarly 1 ft = 12 in gives 1 ft² = 144 in², and 1 ft = 30.48 cm gives 1 ft² \(\approx\) 929 cm². To convert 500 cm² to m², divide by 10,000 to get 0.05 m². A common mistake is dividing by 100 instead of 10,000.
Why does the calculator reject my three side lengths?
The sides violate the triangle inequality: each side must be strictly shorter than the sum of the other two. Lengths like 2, 3 and 6 cannot close into a triangle — the two short sides can't reach each other. Even an exact equality such as 2, 3 and 5 only produces a degenerate, flat "triangle" with zero area. Double-check your measurements; a typo in one side is the usual culprit.
Does it matter which side I label as the base?
No, the area is the same. With Heron's formula you can enter the sides in any order — the result is identical. With the base-height method you just have to be consistent: the height you enter must be the perpendicular height to that specific base. Pairing a height with the wrong side is one of the most common mistakes in triangle area calculations.
How do I find the area of a right triangle?
For a right triangle, the two legs are perpendicular to each other, so either leg can serve as the base and the other as the height: \(A = \tfrac{1}{2} \times \text{leg}_1 \times \text{leg}_2\). With legs 5 and 12, the area is \(\tfrac{1}{2} \times 5 \times 12 = 30\) square units. Do not use the hypotenuse as the height — it is not perpendicular to the legs.