What this calculator does
The area of a triangle is most often taught as half the base times the height, but that is not always the information you actually have to hand. This calculator covers the three common cases: base and height, three known sides, or two sides with the angle between them, so you can find the area of a triangle whichever measurements you start with.
Switching method changes which formula runs underneath, from the simple A = ½bh through to Heron’s formula and the trigonometric ½ab·sin C, but the area is always shown as the same single headline result, so the working stays out of the way unless you want to check it.
The formula
With base and height known, area of triangle is simply ½ × base × height. With all three sides known, Heron’s formula first finds the half-perimeter s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). With two sides and the included angle, area = ½ × a × b × sin(C), where C is the angle between sides a and b.
| Term | Meaning |
|---|---|
| ½bh | Base-and-height form: half the base multiplied by the perpendicular height. |
| s | Half-perimeter used in Heron’s formula: (a + b + c) / 2. |
| ½ab·sin C | Two-sides-and-angle form: half the product of two sides times the sine of the angle between them. |
The inputs explained
| Field | What to enter |
|---|---|
| Known values | Choose which measurements you know: base and height, three sides, or two sides with the angle between them. |
| Base, or side a | The base (for base and height), or side a (for the other two methods). |
| Height, or side b | The height (for base and height), or side b (for the other two methods). |
| Side c (SSS only), or the included angle in degrees (SAS only) | Side c, used only for the three-sides method, or the included angle in degrees, used only for the two-sides method. |
When to use it
A straightforward base and height
When a triangle’s base and its perpendicular height are already known or easily measured, such as a triangular gable end or a set-square shape, ½ × base × height is the fastest route to the area of triangle.
Only the three side lengths are known
A plot of land, a triangular panel or a survey result is sometimes given as three side lengths with no height marked; Heron’s formula finds the area of triangle from those sides alone.
Two sides and the angle between them
In trigonometry problems, or when a bearing and two distances are known, the ½ab·sin C form gives the area without needing a separate height measurement at all.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How to find the area of a triangle from base and height
A fixed height of 6, across a range of base lengths, using the base-and-height method.
Area of triangle formula: three methods, same sides
The same three figures (5, 6 and 7) run through all three methods, to show how the same inputs are read differently depending on what they represent.
| Method | Area |
|---|---|
| Base and height | 15.000 |
| Three sides (Heron’s) | 14.697 |
| Two sides and angle | 1.828 |
Questions
How do you find the area of a triangle?
It depends what you know. With a base and its perpendicular height, use ½ × base × height. With three sides, use Heron’s formula. With two sides and the angle between them, use ½ab·sin C.
What is the area of triangle formula when you only have three sides?
Heron’s formula: work out the half-perimeter s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). It needs no height measurement at all, only the three side lengths.
What is the area of a triangle formula using an angle instead of a height?
Area = ½ × a × b × sin(C), where a and b are two sides and C is the angle between them in degrees. This avoids needing to drop a perpendicular height onto the base.
Do all three methods give the same answer for the same triangle?
Yes, provided the inputs genuinely describe the same triangle. Entering the same numbers under a different method usually describes a different triangle entirely, which is why the results in the comparison table above differ.
For angles, perimeter and the inradius or circumradius from three known sides, see the full triangle from three sides calculator. For a right-angled triangle specifically, see the right triangle / Pythagoras calculator.