Updated · Built and checked by the PlinzoTools team, formulas tested against published sources
Calculate a triangle's area from its three side lengths using Heron's formula.
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This tool calculates a triangle's area directly from its three side lengths, using Heron's formula -- no need to know the height separately.
Enter the three side lengths a, b, and c. The area appears instantly (the tool checks the sides form a valid triangle).
s = (a + b + c) / 2 (the semi-perimeter)
Area = √(s(s-a)(s-b)(s-c))
Besides the area, the tool also computes the perimeter, which is simply the sum of the three sides. All three sides must be positive numbers, and the results are displayed to four decimal places. The calculator uses whatever unit you enter, so sides in meters give an area in square meters.
Take a hypothetical triangular plot with sides 7, 8, and 9 meters. It is not a right triangle, so there is no obvious height, which is exactly the situation Heron's formula is designed for.
s = (7 + 8 + 9) / 2 = 12
Area = √(12 × 5 × 4 × 3) = √720 ≈ 26.8328 square meters
Perimeter = 24 meters
If you were fencing the plot you would need about 24 meters of fencing, and if you were covering it with turf you would need roughly 26.83 square meters before allowing for waste.
The tool needs the quantity inside the square root to be positive. That happens only when the three sides satisfy the triangle inequality: each side must be shorter than the sum of the other two. With sides 2, 3, and 10, the two short sides cannot reach across the long one, so no triangle exists and no area is shown. A "degenerate" triangle such as 3, 4, 7 is also rejected, because the sides would lie flat along a single line and enclose zero area. If you get no result, check for a typing mistake first, and then check whether your measurements really describe a closed triangle.
When the sides come from real measurements, small errors matter most for long thin triangles, where one side is nearly equal to the sum of the other two. A difference of a few millimeters in one side can change the calculated area by a surprising percentage. For land or building work, measure each side at least twice and consider using a different method, such as splitting the shape into right triangles, as a cross-check.
Once you know the area from Heron's formula, you can find the height above any side. Height = 2 × Area / base. In the example above, the height above the 9-meter side is 2 × 26.8328 / 9, about 5.963 meters. This is useful when you need a roof-truss rise, a sail height, or a garden bed layout but only measured the edges.
Heron's formula is handy for surveying odd-shaped lots divided into triangles, for craft and sewing patterns, for roofing gables measured from the ground, and for checking geometry homework. If your measurements include angles rather than only sides, a different formula may be simpler, but for three known sides this calculator gives the area directly.
Three lengths only form a valid triangle if the sum of any two sides exceeds the third (the triangle inequality) -- otherwise no triangle can be formed.
Area = ½ × base × height is simpler in that case -- Heron's formula is specifically for when you only know the three sides.
No. The formula treats all three sides symmetrically, so entering them in any order gives the same area and perimeter.
Multiply the area by 2 and divide by the length of the side you are using as the base. The result is the perpendicular height above that side.
When one side is almost as long as the other two combined, the triangle is nearly collapsed and encloses very little space. The area approaches zero as the sides approach a straight line.
Enter the same length three times. For sides of 6, the area is about 15.5885, which equals the square root of 3 divided by 4, times 36.
Reference: NIST Weights and Measures