How this calculator works
With a base and its perpendicular height, the area is simply half their product. With only the three sides, Heron’s formula first finds the semi-perimeter s, then takes the square root of s times its differences with each side. The three sides must satisfy the strict triangle inequality: the two shortest must add to more than the longest, or the shape collapses onto a line.
Formula
Base-height: Area = b × h / 2. Heron’s formula: s = (a + b + c)/2, then Area = √(s(s - a)(s - b)(s - c)).
Worked example
A base of 10 with height 6 gives an area of 30. The 3-4-5 triangle has s = 6 and area √(6 × 3 × 2 × 1) = 6.
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Frequently asked questions
Does the height have to be a side of the triangle?
No. The height is any perpendicular segment from the base to the opposite vertex, even when it lands outside the triangle, as with obtuse triangles.
Why do sides 2, 3 and 5 fail?
They fail the triangle inequality: 2 + 3 equals 5 exactly, so the three segments lie flat on a line and enclose no area.
Does the order of the three sides matter?
No. The calculator sorts the sides before checking the triangle inequality, so 3, 4 and 5 and 5, 3 and 4 give the same area.