Geometry · Trigonometry
Triangle
Calculator
Enter any 3 values (sides and/or angles) to solve the complete triangle. Calculates all sides, angles, area, perimeter, heights, medians, inradius, and circumradius — with SVG diagram.
—
Area
—
Perimeter
—
Type
Enter Any 3 Values
Enter at least 3 values (including at least 1 side). Leave unknown fields blank. Angles in degrees.
Quick Examples
Sides
opposite A
opposite B
opposite C
Angles (degrees)
opposite a
opposite b
opposite c
Triangle Diagram
Area
—
Perimeter
—
Inradius (r)
—
Circumradius (R)
—
📐 Complete Triangle Solution
📐 Triangle Formulas Reference
Law of Cosines
a² = b² + c² − 2bc·cos(A)
Use when: SAS or SSS
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
Use when: AAS, ASA, SSA
Area (base×height)
A = ½ × b × h
h = height to base b
Area (two sides)
A = ½ × b × c × sin(A)
Two sides + included angle
Heron's Formula
A = √(s(s-a)(s-b)(s-c))
s = (a+b+c)/2 (all 3 sides)
Inradius
r = Area / s
s = semi-perimeter
Circumradius
R = abc / (4 × Area)
Circumscribed circle
Height (ha)
ha = 2·Area / a
Height from vertex A to side a
Solving Triangles
A triangle has 6 unknowns: 3 sides (a, b, c) and 3 angles (A, B, C). To solve a triangle completely, you need any 3 values — as long as at least one is a side. Any combination of 3 angles (AAA) only gives shape, not size.
When to Use Each Law
SSS (3 sides): Law of Cosines → find angle A
SAS (2 sides + incl. ∠): Law of Cosines → find opposite side
AAS or ASA (2∠ + side): A+B+C=180° then Law of Sines
SSA (2 sides + opp. ∠): Law of Sines (watch for ambiguous case!)
SSA may have 0, 1, or 2 triangle solutions
What is the ambiguous case (SSA)?
When given two sides and an angle not between them (SSA), there may be 0, 1, or 2 valid triangles. Let h = b·sin(A) (height from B to side a). If a < h: no triangle (side too short). If a = h: exactly one right triangle. If h < a < b: two triangles (both angles for B are valid). If a ≥ b: one triangle. This calculator solves for both solutions when they exist.
What makes a triangle right, obtuse, or acute?
A right triangle has one 90° angle: a² + b² = c². An acute triangle has all angles < 90°: a² + b² > c² (for all pairs). An obtuse triangle has one angle > 90°: a² + b² < c² (for the largest side c). These follow from the Law of Cosines: cos(C) = (a²+b²-c²)/(2ab) — if negative, angle C > 90°.
How is Heron's formula derived?
Heron's formula Area = √(s(s−a)(s−b)(s−c)) where s=(a+b+c)/2 is derived from the cosine rule. Start with Area = ½ab·sin(C), substitute cos(C) = (a²+b²-c²)/(2ab), use sin²C = 1-cos²C, factor the resulting expression, and you get Heron's form. It's especially useful when you know all 3 sides but no angles.
What are the inradius and circumradius?
The inradius r is the radius of the largest circle that fits inside the triangle, touching all three sides. r = Area/s (s = semi-perimeter). The circumradius R is the radius of the circle passing through all three vertices. R = abc/(4·Area). For a right triangle, R = hypotenuse/2. The relationship r ≤ R/2 (Euler's inequality) always holds, with equality only for equilateral triangles.