Triangle Area Calculator
Calculate the area of any triangle using five different methods: base and height, three sides (Heron's formula), two sides and an included angle (SAS), two angles and an included side (ASA), or the equilateral triangle shortcut. Pick the method that matches the measurements you have. Every calculation shows its step-by-step working.
Formula
Worked example
Base 10 cm, height 6 cm: A = 1/2 x 10 x 6 = 30 sq cm. Sides 6, 8, 10 cm: s = 12, A = sqrt(12 x 6 x 4 x 2) = 24 sq cm. SAS sides 8 and 10 cm, angle 30 deg: A = 1/2 x 8 x 10 x sin(30) = 20 sq cm. Equilateral side 8 cm: A = sqrt(3)/4 x 64 = 27.71 sq cm.
Five ways to find a triangle's area
Different problems give you different measurements. If you have the base and perpendicular height, the classic formula A = 1/2 x b x h is all you need. If you only have the three side lengths, Heron's formula uses the semi-perimeter to get the area without needing any angle or height. When two sides and the enclosed angle are known (SAS), the formula A = 1/2 x a x b x sin(C) gives the area directly. When two angles and the side between them are known (ASA), the Law of Sines lets you express the area as c squared times sin(A) times sin(B), divided by twice sin(C). For an equilateral triangle, all sides are equal, so the area simplifies to sqrt(3)/4 times the side squared. This calculator covers all five methods and shows the step-by-step working for each one.
Why height means perpendicular height
The single most common mistake is using the length of a slanted side as the "height." In the 1/2 x base x height formula, the height is the straight-line distance from the chosen base to the opposite vertex, measured at a right angle to the base. For an obtuse triangle this perpendicular can even fall outside the triangle's footprint. If you only have the three side lengths and no perpendicular height, use Heron's formula instead, which never needs a height at all.
Heron's formula and the triangle inequality
Heron's formula computes the semi-perimeter s = (a + b + c)/2, then takes the square root of s(s - a)(s - b)(s - c). It only works for lengths that can actually close into a triangle: each side must be strictly shorter than the sum of the other two. If that triangle inequality fails, the expression under the root turns negative and no real triangle exists. This calculator flags that case instead of returning a number.
SAS and ASA formulas explained
When two sides and the included angle are known (SAS), the area is half the product of those sides and the sine of the angle between them: A = 1/2 x a x b x sin(C). The angle must be the one enclosed between the two known sides. When two angles and the side between them are known (ASA), the third angle is found from the fact that angles in a triangle sum to 180 degrees, and the area formula becomes A = c squared x sin(A) x sin(B) divided by (2 x sin(C)). Both formulas are derived from the basic 1/2 x base x height formula using trigonometry.
Equilateral triangle as a special case
An equilateral triangle has all three sides equal and all three angles equal to 60 degrees. Because of that symmetry, the area simplifies from Heron's formula to A = sqrt(3)/4 x s squared, where s is the side length. The height is sqrt(3)/2 x s. These are worth memorizing for quick estimates: a side-10 equilateral triangle has an area of about 43.3 square units.
Area formulas for every common triangle case
| What you know | Formula | Notes |
|---|---|---|
| Base and perpendicular height | A = 1/2 x b x h | Simplest and most common case |
| Three side lengths (SSS) | A = sqrt(s(s-a)(s-b)(s-c)), s=(a+b+c)/2 | Heron's formula, no angles needed |
| Two sides + included angle (SAS) | A = 1/2 x a x b x sin(C) | C is between sides a and b |
| Two angles + included side (ASA) | A = c^2 x sin(A) x sin(B) / (2 x sin(C)) | C = 180 - A - B |
| Equilateral triangle | A = sqrt(3)/4 x s^2 | All sides equal, all angles 60 deg |
| Right triangle (two legs) | A = 1/2 x leg1 x leg2 | Legs are perpendicular to each other |
Match the formula to the measurements you have at hand.
Frequently asked questions
Do all three sides have to be the same length?
No. All five methods work for any triangle: scalene, isosceles, equilateral, right, acute, or obtuse. The only constraint is the triangle inequality for Heron's formula: each side must be shorter than the sum of the other two.
What units does the area come out in?
The area is in the square of whatever unit you enter. If your sides are in centimetres, the area is in square centimetres. Keep every length in the same unit before calculating. Switch the unit system selector to toggle metric (cm) or imperial (in) labels.
Why do I get "cannot form a triangle" with Heron's formula?
Because one side is at least as long as the other two combined. The triangle inequality says each side must be strictly shorter than the sum of the other two; otherwise the shape cannot close and has no area.
What is the SAS formula and when do I use it?
SAS stands for Side-Angle-Side. Use A = 1/2 x a x b x sin(C) when you know two sides and the angle between them. This is common in surveying and construction where you measure two lengths from a corner and the angle at that corner.
How do I find the area from two angles and one side (ASA)?
With two angles A and B and the side c between them, find the third angle C = 180 - A - B, then apply A = c squared x sin(A) x sin(B) / (2 x sin(C)). Enter the values in the ASA method and this calculator does the trigonometry automatically.
Can I enter angles in radians?
Yes. Change the Angle unit selector from Degrees to Radians. All angle inputs for the SAS and ASA methods will then be interpreted as radians. Make sure to switch the selector before entering your values.