Triangle Area Calculator

What this calculator does

In real problems you rarely have a triangle handed to you as a neat base and height. More often you know the three side lengths, or two sides and an angle, or a couple of angles and a side. This tool takes any of those standard combinations and works out the area.

You pick the method that matches what you already know, and it does the rest.

Using the calculator

1. Choose a method from the dropdown. The input boxes change to match it.
2. Type your known values. Angles go in degrees.
3. Press Calculate.

There are no unit menus here, so work in any one unit you like and read the area in that unit squared. The result is labelled "square units" and rounded to two decimal places.

The six methods, and when to use each

The dropdown holds six ways in, named by which pieces of the triangle you know. S stands for a side, A for an angle.

• Base and Height: you have a side and the perpendicular height to it. The simplest case.
• SSS, three sides: all three side lengths, no angles. Solved with Heron's formula.
• SAS, two sides and the angle between them: the angle sits in the corner where the two sides meet.
• SSA, two sides and an angle not between them: the tool uses the sine rule to find the rest.
• ASA, two angles and the side between them.
• AAS, two angles and a side not between them.

One rule runs through all of them: you always need at least one side. Angles on their own cannot give you the area, because triangles with the same three angles come in every size, from tiny to huge.

The formulas behind them

For a side and its perpendicular height, it is the plain triangle formula:

area = ½ × base × height

For three sides, it is Heron's formula. Take half the perimeter, s = (a + b + c) ÷ 2, then:

area = √(s × (s - a) × (s - b) × (s - c))

For two sides with the angle between them, it is the sine formula:

area = ½ × a × b × sin(angle)

The sine is quietly turning one side into the effective height of the other. For the angle-heavy cases, SSA, ASA and AAS, the tool first finds a missing angle (the three angles of a triangle always add to 180 degrees) and uses the law of sines to find a missing side, then finishes with one of the formulas above.

Worked examples across three methods

The same idea, three different starting points.

• Base and Height, base 8 and height 5: ½ × 8 × 5 = 20.00.
• SSS, sides 7, 8 and 9: s = (7 + 8 + 9) ÷ 2 = 12, so area = √(12 × 5 × 4 × 3) = √720 = 26.83.
• SAS, sides 8 and 5 with a 60 degree angle between them: ½ × 8 × 5 × sin(60°) = 20 × 0.866 = 17.32.

What makes a set of numbers a real triangle

Not every set of numbers describes a triangle, and the tool checks a few things before it answers.

For three sides, each pair has to add up to more than the third. That is the triangle inequality: sides of 2, 3 and 9 cannot close into a triangle, since 2 and 3 fall short of 9, so the tool stops and tells you. Any angle you type has to be under 180 degrees, since a single corner cannot be a straight line or more.

SSA is the one to treat with a little care. Two sides and an angle that is not between them can sometimes fit two different triangles, or none at all. This is the classic ambiguous case, and the tool returns one solution, so if your setup is borderline it is worth a sketch to check which triangle you meant.

Questions people ask

Which method should I use?

Pick the one that matches what you know. Base and height if you have them, SSS for three sides, SAS for two sides and the angle between, and ASA, AAS or SSA when you have angles and a side.

What is Heron's formula?

A way to get the area from the three side lengths alone, with no angle or height needed. Take half the perimeter s, then area = √(s(s - a)(s - b)(s - c)).

Can I find the area from the angles alone?

No. Triangles with the same angles can be any size, so the angles fix the shape but not the area. You need at least one side length.

Do the different methods give the same answer?

Yes. For one triangle, every method lands on the same area, give or take rounding. They are just different routes in, depending on what you measured.

What if my three sides do not form a triangle?

The tool flags it. If any two sides do not add up to more than the third, no triangle exists, so there is no area to give.

References

A note on where these come from. The three-sides method is Heron's formula, first written down by Hero of Alexandria in his work Metrica around 60 AD. The angle-based methods lean on the law of sines, the standard result of plane trigonometry that ties each side of a triangle to the sine of the angle opposite it. For further reading, see Triangle.

1. Hero (Heron) of Alexandria, Metrica (c. 60 AD), the first known statement of Heron's formula for the area of a triangle from its three sides.
2. The law of sines, relating each side of a triangle to the sine of its opposite angle, used to solve the angle-and-side cases.