Calculate the area of a triangle without the height using three sides, two sides and the included angle, or the three vertex coordinates.
Area of a Triangle Without Height Formula
You can find a triangle's area without the perpendicular height as long as you know enough other measurements. The right formula depends on what you are given.
When you know all three side lengths, use Heron's formula:
Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2
When you know two sides and the angle between them, use the sine formula:
Area = 1/2 * a * b * sin(C)
When you know the three vertex coordinates, use the shoelace formula:
Area = 1/2 * |xA(yB-yC) + xB(yC-yA) + xC(yA-yB)|
- a, b, c: the three side lengths of the triangle.
- s: the semiperimeter, equal to half the perimeter (a + b + c) / 2.
- C: the angle included between sides a and b.
- xA, yA, xB, yB, xC, yC: the x and y coordinates of the three vertices A, B, and C.
Heron's formula takes the three sides, builds the semiperimeter, and returns the area directly, so no height is needed. The sine formula uses two sides and their included angle, which fixes the shape of the triangle without measuring its height. The shoelace formula works straight from coordinates and gives the absolute area regardless of how the points are ordered. For a right triangle the two legs act as base and height, so the area is simply half their product, and for an equilateral triangle the area follows from a single side.
Which Method to Use Based on What You Know
Pick the row that matches the information you already have.
| What you know | Method | Formula |
|---|---|---|
| All three sides | Heron's formula | sqrt(s(s-a)(s-b)(s-c)) |
| Two sides and the included angle | Sine (SAS) | 1/2 * a * b * sin(C) |
| Three vertex coordinates | Shoelace | 1/2 * |xA(yB-yC)+xB(yC-yA)+xC(yA-yB)| |
| Two legs of a right triangle | Leg product | 1/2 * leg a * leg b |
| One side, equilateral triangle | Equilateral | (sqrt(3)/4) * side^2 |
Example Problems
Example 1: three known sides. A triangle has sides a = 13, b = 14, and c = 15. The semiperimeter is s = (13 + 14 + 15) / 2 = 21. Heron's formula gives Area = sqrt(21(21-13)(21-14)(21-15)) = sqrt(21 * 8 * 7 * 6) = sqrt(7056) = 84 square units.
Example 2: two sides and the included angle. A triangle has sides a = 8 and b = 10 with a 45 degree angle between them. Area = 1/2 * 8 * 10 * sin(45 degrees) = 40 * 0.7071 = 28.28 square units.
Frequently Asked Questions
Can you find the area of a triangle without the height? Yes. If you know all three sides, two sides and the included angle, or the coordinates of the three vertices, you can find the area without ever measuring the perpendicular height. Each of these inputs fully determines the triangle, so the area is fixed.
Which formula should I use when I only know the three sides? Use Heron's formula. Add the three sides and divide by two to get the semiperimeter s, then take the square root of s(s-a)(s-b)(s-c). It works for any triangle whose sides satisfy the triangle inequality, meaning each side is shorter than the sum of the other two.
Why does my coordinate calculation return zero? A zero area means the three points are collinear, so they sit on a single straight line and do not enclose any space. Check that the coordinates you entered are correct and that the three points are not in a line.
