🔺Triangle Calculator
Solve any triangle from 3 known values (at least one side). Find all sides, angles, area, perimeter, heights, medians, inradius, circumradius, and vertex coordinates.
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Provide 3 values including at least one side. Leave the rest blank.
Sides
Angles
Triangle Type
Equilateral Triangle
Sides
Angles
Area & Perimeter
Heights (Altitudes)
Medians
Circles
Coordinates
Area
0
Equilateral Triangle. a=1, b=1, c=1, A=60°, B=60°, C=60°. Area=0.43301, Perimeter=3.
Triangle Summary
0.43301
3.00000
0.28868
0.57735
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Triangle Calculator: Solve Any Triangle with Sides and Angles
This triangle calculator solves any triangle from three known values, as long as at least one value is a side length. Enter any combination of sides (a, b, c) and angles (A, B, C) — the calculator uses the Law of Sines, Law of Cosines, and the angle sum property to find every missing measurement automatically.
How to Solve a Triangle
A triangle has six measurements: three sides and three angles. Given any three of these values (with at least one being a side), the remaining three can always be found. The five classic cases are:
- SSS (three sides): Use the Law of Cosines to find all three angles.
- SAS (two sides + included angle): Use the Law of Cosines to find the third side, then Law of Sines for remaining angles.
- ASA or AAS (two angles + a side): The third angle = 180° minus the other two. Use the Law of Sines for remaining sides.
- SSA (two sides + non-included angle): Use the Law of Sines — may have 0, 1, or 2 solutions (the ambiguous case).
Law of Sines
The Law of Sines states that for any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This ratio equals 2R, where R is the circumradius. It is most useful when you know an angle-side pair and want to find another angle or side across the triangle.
Law of Cosines
The Law of Cosines generalizes the Pythagorean theorem to all triangles: c² = a² + b² − 2ab·cos(C). When C = 90°, cos(C) = 0 and the equation reduces to the Pythagorean theorem. It is used to find a side when two sides and the included angle are known, or to find angles when all three sides are known.
Triangle Properties Explained
The height (altitude) from vertex A to side a is h_a = 2·Area / a. There are three altitudes, one from each vertex. The median from vertex A to the midpoint of side a has length m_a = ½√(2b² + 2c² − a²). The inradius r is the radius of the inscribed circle: r = Area / s, where s is the semiperimeter. The circumradius R is the radius of the circumscribed circle: R = abc / (4·Area).
Triangle Classification
Triangles are classified by side length: equilateral (all sides equal), isosceles (two sides equal), or scalene (no sides equal). By angles: acute (all angles under 90°), right (one angle exactly 90°), or obtuse (one angle over 90°). An equilateral triangle is always acute with all angles equal to 60°.
Frequently Asked Questions
What values do I need to solve a triangle?
You need exactly three values, including at least one side length. Valid combinations are: three sides (SSS), two sides and an included angle (SAS), two angles and any side (ASA or AAS), or two sides and a non-included angle (SSA). Three angles alone (AAA) cannot determine a unique triangle because the scale is unknown.
What is the Law of Sines?
The Law of Sines states that in any triangle, the ratio of each side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This common ratio equals 2R, where R is the circumradius. It is useful for finding unknown sides or angles when at least one side-angle pair is known.
What is the Law of Cosines?
The Law of Cosines relates the three sides and one angle of a triangle: c² = a² + b² − 2ab·cos(C). It generalizes the Pythagorean theorem. Use it to find the third side when two sides and the included angle are known, or to find angles when all three sides are known.
What are heights, medians, and the inradius of a triangle?
A height (altitude) is the perpendicular distance from a vertex to the opposite side, calculated as h = 2·Area / side. A median connects a vertex to the midpoint of the opposite side. The inradius r is the radius of the largest circle that fits inside the triangle, equal to Area / semiperimeter. The circumradius R is the radius of the circle passing through all three vertices, equal to abc / (4·Area).