Click to show tips
Pick a scenario to see how the calculator works, then adjust the values
A triangular field with fence posts 80 m, 100 m, and 60 m apart.
Key values: a = 80 m · b = 100 m · c = 60 m
A sailboat mainsail with two sides of 6 m and 4 m and an included angle of 70°.
Key values: a = 6 m · b = 4 m · C = 70°
Select your known values (SSS, SAS, ASA, AAS, or SSA) and enter measurements to solve the triangle and compute area, perimeter, inradius, and more.
Depending on what measurements you have, one formula will be more convenient than the others. This calculator supports all four major methods:
The simplest formula. The height (altitude) must be measured perpendicular to the chosen base. This works for all triangles, but you need to know (or be able to measure) the perpendicular height.
Common mistake: Using a slanted side instead of the perpendicular height. The height must form a 90° angle with the base.
When you know all three sides but not the height, Heron's formula computes the area directly:
where is the semi-perimeter.
Example: Triangle with sides 7, 8, 9:
Validity check: For three sides to form a valid triangle, they must satisfy the triangle inequality: the sum of any two sides must exceed the third. If , , or is negative, the triangle is impossible.
When you know two sides and the angle between them, this formula is the most direct. It comes from the fact that the height relative to side is .
Example: Sides 10 and 14, included angle 35°:
Given vertices at coordinates , , :
This is a special case of the shoelace formula for polygon area. The absolute value ensures a positive result regardless of vertex ordering.
| You know... | Best method |
|---|---|
| Base + perpendicular height | |
| All three sides | Heron's formula |
| Two sides + included angle | SAS formula |
| Vertex coordinates | Shoelace formula |
The simplest formula is , where is the base and is the perpendicular height. The height must form a 90-degree angle with the chosen base. This works for all triangles but requires knowing the perpendicular height.
Heron's formula calculates area from three side lengths: , where is the semi-perimeter. For example, a triangle with sides 7, 8, 9 has and area .
Use the SAS formula when you know two sides and the angle between them. It is the most direct method for this case, since the perpendicular height equals .
The shoelace formula computes area from vertex coordinates: . The absolute value ensures a positive result regardless of vertex ordering. It is ideal when you have coordinate geometry data.
Three sides form a valid triangle if and only if the sum of any two sides exceeds the third (the triangle inequality). Equivalently, when computing Heron's formula, if any of , , or is negative or zero, the triangle is impossible.