๐ Heron Area Calculator: Triangle Area From 3 Sides (SSS)
By Shihab Mia ยท Updated 2026-07-26
Enter the three side lengths of a triangle (a, b and c) and this heron area calculator returns the area instantly using Heron's formula: Area = sqrt(s(s - a)(s - b)(s - c)), where s = (a + b + c) / 2 is the semi-perimeter. No height or angle is needed, only the three edge lengths. The calculator also checks the triangle inequality first, reports the perimeter and semi-perimeter, and shows the full step-by-step working so you can verify the answer by hand.
What is the Triangle Area (Heron) Calculator?
Heron's formula gives the area of any triangle from its three side lengths alone. First compute the semi-perimeter s = (a + b + c) / 2, which is just half the perimeter. Then the area is the square root of s(s - a)(s - b)(s - c). The big advantage over other area formulas is that you never need the height or an angle, which is exactly what you want for a survey plot, a triangular roof panel, or any shape measured only by its edges (the SSS, or side-side-side, case).
The formula is named after Hero of Alexandria, a first-century Greek engineer and mathematician whose book Metrica contains a proof of it. Historians of mathematics, including Thomas Heath, have noted that the 11th-century scholar al-Biruni attributed the result to Archimedes, who lived roughly two centuries before Heron, so the formula may have been known earlier and simply recorded by Heron. Either way it has been in continuous use for close to two thousand years, which says something about how directly useful it is.
The method only works for lengths that can form a real triangle. The triangle inequality says each side must be shorter than the sum of the other two: a + b must exceed c, a + c must exceed b, and b + c must exceed a. If any of these fails, the sides cannot close into a triangle, and the product s(s - a)(s - b)(s - c) turns out to be zero or negative, so its square root is not a real area. This calculator flags that case rather than returning a meaningless number. A borderline case where a + b equals c exactly is called degenerate: the three points fall on a straight line and the area is zero.
Heron's formula is one of several ways to find a triangle's area, and which one you reach for depends on what you actually know. If you have a base and the perpendicular height, 0.5 x base x height is faster and needs no square root. If you know two sides and the included angle, the SAS formula 0.5 x a x b x sin(C) applies directly. If you know the coordinates of the three vertices, the shoelace formula is quicker. Heron's formula wins specifically in the SSS case, when the only data you have is three edge lengths, because it works for every triangle shape, acute, right-angled or obtuse, without any extra measurement.
One subtlety worth knowing: the textbook version of Heron's formula loses accuracy on very thin, needle-shaped triangles, where one side is tiny compared to the other two. When that happens s is nearly equal to two of the sides, so (s - a) and (s - b) become the difference of two close floating-point numbers, and small rounding errors get amplified. Computer scientist William Kahan published a numerically stable rearrangement of the same formula in 1986 specifically to fix this for calculators and software. For everyday triangles (roof panels, land parcels, garden beds) the plain formula used here is accurate to many decimal places; the stability issue only bites on extreme, sliver-thin shapes.
When to use it
- Finding the area of a triangular plot of land when only the three boundary lengths were measured with a tape or GPS.
- Working out material needed for a triangular roof gable, sail, garden bed or solar panel when you know the edge lengths but not the height.
- Checking homework, exam answers or a geometry proof for the SSS (side-side-side) case.
- Computing the area of an irregular polygon or GIS parcel by splitting it into triangles and measuring each set of sides.
- Estimating the surface area of a triangular face in a 3D model or mesh when only vertex-to-vertex distances are available.
- Verifying a contractor or surveyor's area figure for a triangular room, deck or lot before signing off on a quote.
How to use the Triangle Area (Heron) Calculator
- Enter the length of the first side in the Side a box.
- Enter the lengths of the other two sides as Side b and Side c, using any consistent unit.
- Optionally type a unit label such as cm, m or ft so the result is labelled.
- Press Calculate area to see the area, perimeter, semi-perimeter and the step-by-step working.
- If you see a triangle inequality error, double-check your measurements: one side may be too long relative to the other two.
Formula & method
Worked examples
A right triangle with sides 3, 4 and 5.
- s = (3 + 4 + 5) / 2 = 12 / 2 = 6
- s - a = 6 - 3 = 3, s - b = 6 - 4 = 2, s - c = 6 - 5 = 1
- Area = sqrt(6 x 3 x 2 x 1) = sqrt(36)
- Area = 6
Result: Area = 6 square units, perimeter = 12
A scalene triangle with sides 5, 6 and 7.
- s = (5 + 6 + 7) / 2 = 18 / 2 = 9
- s - a = 9 - 5 = 4, s - b = 9 - 6 = 3, s - c = 9 - 7 = 2
- Area = sqrt(9 x 4 x 3 x 2) = sqrt(216)
- Area โ 14.6969
Result: Area โ 14.6969 square units, perimeter = 18
A classic textbook triangle with sides 13, 14 and 15 (a Heronian triangle: every value works out to a whole number).
- s = (13 + 14 + 15) / 2 = 42 / 2 = 21
- s - a = 21 - 13 = 8, s - b = 21 - 14 = 7, s - c = 21 - 15 = 6
- Area = sqrt(21 x 8 x 7 x 6) = sqrt(7056)
- Area = 84
Result: Area = 84 square units, perimeter = 42
Worked areas for sample triangles (Heron's formula)
| Sides (a, b, c) | Semi-perimeter s | Perimeter | Area |
|---|---|---|---|
| 3, 4, 5 | 6 | 12 | 6 |
| 5, 6, 7 | 9 | 18 | 14.6969 |
| 6, 6, 6 | 9 | 18 | 15.5885 |
| 7, 8, 9 | 12 | 24 | 26.8328 |
| 8, 15, 17 | 20 | 40 | 60 |
| 13, 14, 15 | 21 | 42 | 84 |
Common Heronian triangles (integer sides, integer area)
| Sides (a, b, c) | Area | Type |
|---|---|---|
| 3, 4, 5 | 6 | Right triangle |
| 5, 5, 6 | 12 | Isosceles |
| 5, 5, 8 | 12 | Isosceles |
| 6, 8, 10 | 24 | Right triangle |
| 9, 10, 17 | 36 | Scalene |
| 13, 14, 15 | 84 | Scalene |
Which triangle area formula to use
| What you know | Formula | When to use it |
|---|---|---|
| Three sides (SSS) | Area = sqrt(s(s-a)(s-b)(s-c)) | Heron's formula, no height or angle needed |
| Base and height | Area = 0.5 x base x height | Fastest option when height is already known |
| Two sides and included angle (SAS) | Area = 0.5 x a x b x sin(C) | Common in trigonometry and surveying |
| Coordinates of three vertices | Shoelace formula | Best for GIS, CAD or graphics work |
Common mistakes to avoid
- Entering sides that cannot form a triangle. If one side is equal to or longer than the sum of the other two (for example 2, 3 and 10), no triangle exists. The formula then gives a zero or negative value under the root, so the calculator reports an error instead of a number.
- Mixing different units. All three sides must be in the same unit. Mixing centimetres with metres, or inches with feet, gives a nonsense area. Convert every side to one unit before you calculate.
- Confusing the semi-perimeter with the perimeter. Heron's formula uses s, the semi-perimeter, which is half of a + b + c. Plugging the full perimeter into the formula instead of half of it produces a far too large answer.
- Forgetting the area is in square units. Side lengths are linear, but the area is in square units. If your sides are in metres the area is in square metres, not metres, so label the result accordingly.
- Rounding side lengths too early. Rounding a, b or c before you calculate compounds error through the square root. Enter the full measured value and let the calculator round only the final answer.
- Trusting the result on an extremely thin, sliver-shaped triangle. When one side is tiny compared to the other two, s nearly equals two of the sides and the subtraction inside the square root becomes very sensitive to rounding. For everyday triangles this is not an issue, but on needle-like shapes a numerically stable rearrangement of the formula (Kahan, 1986) gives a more reliable answer.
Glossary
- Heron's formula
- A formula that gives a triangle's area from its three side lengths, with no need for height or angles.
- Semi-perimeter (s)
- Half of the perimeter, s = (a + b + c) / 2. It is the key intermediate value in Heron's formula.
- Perimeter
- The total distance around the triangle, the sum of all three side lengths a + b + c.
- Triangle inequality
- The rule that each side of a triangle must be shorter than the sum of the other two sides.
- SSS (side-side-side)
- The case where all three side lengths are known, which is exactly what Heron's formula needs.
- Heronian triangle
- A triangle whose three sides and area are all whole numbers, such as the 3-4-5 or 13-14-15 triangle.
- Degenerate triangle
- A borderline case where one side exactly equals the sum of the other two, so the three points lie on a straight line and the area is zero.
- Circumradius and inradius
- The circumradius R = abc / 4A is the radius of the circle through all three vertices; the inradius r = A / s is the radius of the circle inscribed inside the triangle, where A is the area.
Frequently asked questions
What is Heron's formula?
Heron's formula finds the area of a triangle from its three side lengths. You compute the semi-perimeter s = (a + b + c) / 2, then the area equals the square root of s(s - a)(s - b)(s - c). It needs no height and no angles.
How do I find the area of a triangle with only the three sides?
Use Heron's formula. Add the sides and halve them to get s, then take the square root of s(s - a)(s - b)(s - c). This calculator does it for you and shows each step, including the perimeter.
What is the SSS case?
SSS stands for side-side-side: the situation where you know all three side lengths of a triangle but no angles or height. Heron's formula is the standard way to get the area in the SSS case.
Why does the calculator say my sides cannot form a triangle?
A real triangle must satisfy the triangle inequality: each side must be shorter than the sum of the other two. If, say, one side is 10 while the others are 2 and 3, the sides cannot close into a triangle, so no area exists.
Does Heron's formula work for all triangles?
Yes. It works for acute, right-angled and obtuse triangles, and for equilateral, isosceles and scalene shapes, as long as the three lengths satisfy the triangle inequality.
What units does the area come out in?
The area is in square units of whatever length unit you enter. If the sides are in centimetres the area is in square centimetres. Add a unit label to have the result labelled automatically.
What is a Heronian triangle?
A Heronian triangle has whole-number side lengths and a whole-number area, such as 3-4-5 (area 6) or 13-14-15 (area 84). They are popular in textbooks because Heron's formula produces a clean, exact answer.
Who invented Heron's formula?
It is named for Hero of Alexandria, a first-century Greek engineer who proved it in his book Metrica. Some historians believe Archimedes, who lived about two centuries earlier, may have known the result first, based on a later attribution by the 11th-century scholar al-Biruni.
Can Heron's formula be used for a right triangle?
Yes. Heron's formula works for right triangles the same way it works for any other triangle; you do not need to treat the right angle specially. For a right triangle you could also use 0.5 x leg1 x leg2, and both methods give the same area.
How is Heron's formula different from 1/2 base times height?
0.5 x base x height needs the perpendicular height, which you often do not have. Heron's formula needs only the three side lengths and works out the equivalent area internally through the semi-perimeter, so it is the better choice whenever height is unknown or hard to measure.