Triangle Area Calculator — Three Methods

Base & height: A = ½ × b × h

Heron's formula (3 sides):
s = (a+b+c)/2
A = √(s(s−a)(s−b)(s−c))

Two sides + angle:
A = ½ × a × b × sin(C)

Use base & height when you have them. Heron's formula when you only know all three sides. The SAS formula when you know two sides and the angle between them.

Base & height (b A1, h B1)

=0.5*A1*B1

Heron's (a A1, b B1, c C1)

=SQRT((A1+B1+C1)/2*((A1+B1+C1)/2-A1)*((A1+B1+C1)/2-B1)*((A1+B1+C1)/2-C1))

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Three Ways to Find Triangle Area

The base-and-height formula (½ × b × h) is the simplest but requires knowing the perpendicular height — the distance from the base to the opposite vertex measured at a right angle. This height is sometimes called the altitude, and it is not the same as the length of a slanted side.

Heron's formula is elegant because it only needs the three side lengths — no angles, no heights. Given sides a, b, and c, compute the semi-perimeter s = (a+b+c)/2, then area = √(s(s-a)(s-b)(s-c)). It works for any triangle and is particularly useful in surveying and construction where side lengths are easier to measure than heights.

The SAS (Side-Angle-Side) formula uses two sides and the included angle: A = ½ × a × b × sin(C). This is powerful in navigation, physics, and engineering where angles are measured directly. All three formulas give identical results for the same triangle — choose based on what information you have.

Triangle Inequality: Can These Sides Form a Triangle?

Not every combination of three lengths can form a triangle. The triangle inequality requires that each side must be strictly less than the sum of the other two sides. Sides 3, 4, 8 cannot form a triangle because 3 + 4 = 7 < 8. If you enter three sides that violate this rule, Heron's formula will attempt to take the square root of a negative number — which signals an invalid triangle.

This rule has a useful geometric interpretation: if you try to construct a triangle where one side is longer than the combined length of the other two, the shorter sides simply cannot reach each other to close the triangle.

Frequently Asked Questions

Base & height: simplest if you have the perpendicular height. Heron's: when you only know the three side lengths. SAS: when you know two sides and the angle between them. All give the same result for the same triangle — the choice depends on what information is available.

s = (a+b+c)/2 — half the triangle's perimeter. It appears in Heron's formula as a convenient computational shorthand. The formula can be derived geometrically and is one of several elegant results attributed to the ancient Greek mathematician Heron of Alexandria.

Any three positive angles that sum to 180° form a valid triangle (though you'd need at least one side length to determine the actual size). Any three positive lengths satisfying the triangle inequality form a valid triangle.

For a right triangle with legs a and b (the two sides that meet at the right angle), area = ½ × a × b. The base and height are simply the two legs — the right angle guarantees they are perpendicular to each other. No special formula needed beyond the basic ½ × base × height.