Triangle Calculator

How to Use This Triangle Calculator

This comprehensive triangle calculator solves any triangle when you provide enough information. Select a solving method based on what measurements you know, then enter your values:

Understanding the Methods (S = Side, A = Angle):

• SSS (Side-Side-Side): You know all three side lengths. Enter sides a, b, and c. The calculator will find all three angles, area, perimeter, and heights.
• SAS (Side-Angle-Side): You know two sides and the angle between them. Enter sides a and b, plus angle C (the angle between them). Best for situations where you can measure an angle directly.
• ASA (Angle-Side-Angle): You know two angles and the side between them. Enter angles A and B, plus side c (between those vertices). Since angles sum to 180 degrees, the third angle is calculated automatically.
• AAS (Angle-Angle-Side): You know two angles and a side not between them. Enter angles A and B, plus side a (opposite to angle A). Common in surveying when one side and two angles are measurable.
• SSA (Side-Side-Angle): You know two sides and an angle opposite one of them. Enter sides a and b, plus angle A (opposite to side a). Note: This case can have zero, one, or two solutions depending on the values (the ambiguous case).

After entering your values, click Calculate Triangle. The calculator displays all sides, angles, area, perimeter, and heights. Click any result box to copy that value to your clipboard.

What is a Triangle?

A triangle is a polygon formed by three straight sides connecting three vertices (corner points). It is the simplest polygon and the only one that is always rigid when its side lengths are fixed. This rigidity is why triangles appear throughout architecture, engineering, and structural design.

Every triangle has several fundamental properties. The sum of interior angles always equals exactly 180 degrees. No side can be longer than the sum of the other two sides (the triangle inequality). The longest side is always opposite the largest angle, and the shortest side is opposite the smallest angle.

Classification by Sides:

• Equilateral: All three sides are equal. All three angles are 60 degrees.
• Isosceles: Two sides are equal. The angles opposite the equal sides are also equal.
• Scalene: No sides are equal. All three angles are different.

Classification by Angles:

• Acute: All three angles are less than 90 degrees.
• Right: One angle is exactly 90 degrees. The side opposite the right angle is the hypotenuse.
• Obtuse: One angle is greater than 90 degrees.

Triangles are fundamental to trigonometry (which literally means "triangle measurement"), surveying, navigation, computer graphics, physics, and countless engineering applications.

Triangle Formulas and Laws

Law of Cosines:

c^2 = a^2 + b^2 - 2ab * cos(C)

This formula generalizes the Pythagorean theorem to any triangle. It relates one side to the other two sides and the included angle. Use it when you know two sides and the included angle (SAS) or all three sides (SSS).

Law of Sines:

a/sin(A) = b/sin(B) = c/sin(C)

This powerful relationship states that the ratio of any side to the sine of its opposite angle is constant for all three side-angle pairs in a triangle. Use it for ASA, AAS, and SSA problems.

Area Formulas:

Area = 0.5 * base * height (basic formula)

Area = 0.5 * a * b * sin(C) (when you know two sides and included angle)

Area = sqrt(s * (s-a) * (s-b) * (s-c)) (Heron's formula, where s = (a+b+c)/2)

Height Formula:

h_a = 2 * Area / a

The height (altitude) to any side can be calculated once you know the area. The height to side a equals twice the area divided by the length of side a.

Triangle Inequality Theorem: The sum of any two sides must be greater than the third side. If a + b is less than or equal to c, no triangle can be formed with those measurements.