Triangle Calculator

What this calculator does

The Triangle Calculator measures a triangle in two ways. If you know all three sides, it computes the area, perimeter, semiperimeter, interior angles, and the height to each side. If you know a base and the perpendicular height, it computes the area.

Triangles are the simplest closed polygons and appear throughout engineering, construction, surveying, and design. Three side lengths determine a triangle completely, which is why the three-side path can return angles and heights rather than only an area. A base and height are enough for area but not enough to determine the other sides or angles.

How to use this calculator

1. Choose Three sides or Base and height.
2. For three sides, enter side a, side b, and side c.
3. For base and height, enter the base and the perpendicular height.
4. Click Calculate and review the result panel.

The default three-side example is 1, 1, and 1, an equilateral triangle. Its area is about 0.4330 square units, its perimeter is 3, and each angle is 60 degrees. The base-and-height example uses a base of 6 and a height of 4, giving an area of 12 square units.

Triangle formulas

Semiperimeter: s = (a + b + c) / 2
Area (Heron):  A = √(s(s − a)(s − b)(s − c))
Law of cosines: cos A = (b² + c² − a²) / (2bc)
Height to side a: hₐ = 2A / a
Base and height: A = ½ × base × height

The angles are labeled A, B, and C, opposite sides a, b, and c. The law of cosines converts the three side lengths into angles, and the last angle can also be found by subtracting the other two from 180 degrees. The calculator reports all three angles from the side lengths directly.

What the results mean

The highlighted result is the area. The supporting rows show the perimeter, semiperimeter, angle A, angle B, angle C, and the height to each side. The heights are perpendicular distances from a vertex to the opposite side or its extension.

Angles are reported in degrees. The three angles of a valid triangle always add up to 180 degrees. Small rounding differences in the displayed values may make the sum appear slightly different.

Assumptions and limitations

• All side lengths must use the same unit. No unit conversion is performed.
• Every side must be greater than zero.
• Three sides must satisfy the triangle inequality: the sum of any two sides must be greater than the third side.
• The base-and-height mode calculates area only, because one base and one height do not determine a unique triangle.
• The calculator does not accept angle-side combinations such as SAS, ASA, AAS, or SSA. Use the Right Triangle Calculator when a right angle is known.
• Results are rounded for display.

Frequently asked questions

What is the triangle inequality?

The triangle inequality states that the sum of any two sides must be greater than the third side. For example, sides 1, 1, and 3 cannot form a triangle because 1 + 1 is not greater than 3. The calculator rejects these inputs rather than returning an impossible area.

Why does the three-side mode also report angles?

Three side lengths determine the shape of a triangle completely. Once the sides are fixed, the angles and heights are also fixed. The calculator uses the law of cosines to find the angles and uses the area to find the heights.

Why can't the base-and-height mode report the other sides?

Many different triangles can share the same base and height. The apex can slide along a line parallel to the base without changing the area. Additional information, such as another side or an angle, is needed to determine the remaining measurements.

Source

Reference: Calculator.net — Triangle Calculator.

This page is for general educational and estimation purposes. Confirm critical dimensions independently before construction or fabrication work.