Triangle Calculator

Solve any triangle by entering combinations of sides and angles (SSS, SAS, ASA, AAS) with complete solutions for area, perimeter, altitudes, medians, inradius, and circumradius.

Triangle Inputs

Live geometry
Classic Triangles:

The sum of all three interior angles in Euclidean plane geometry must always equal exactly 180°.

Triangle Area
14.697sq units
ClassificationScalene Acute
Perimeter: 18 unitsSemi-perimeter s: 9
Side a5
Side b6
Side c7
Angle A44.4°
Angle B57.1°
Angle C78.5°

Altitudes (Perpendicular Heights)

To side a (h_a): 5.879
To side b (h_b): 4.899
To side c (h_c): 4.199

Incircle & Circumcircle Radii

Inradius (r): 1.633
Circumradius (R): 3.572
Median to a (m_a): 6.021

Fundamental Triangle Formulas and Laws

Any triangle is uniquely determined by three independent geometric measurements (with at least one side). The primary trigonometric laws used to solve triangles are the Law of Cosines and the Law of Sines.

1. Law of Cosines (SSS and SAS)
c²=a² + b² − 2ab · cos(C)
2. Heron's Area Formula (From 3 Sides)
Area=√[ s(s − a)(s − b)(s − c) ]wheres = (a + b + c) ÷ 2
Step-by-Step Calculation Breakdown (Example: SSS with a = 3, b = 4, c = 5)
Step 1: Compute Semi-Perimeter (s)
• Perimeter P = 3 + 4 + 5 = 12
• Semi-perimeter s = 12 ÷ 2 = 6.000
Step 2: Apply Heron's Formula for Area
• Area = √[ 6 × (6 − 3) × (6 − 4) × (6 − 5) ] = √[ 6 × 3 × 2 × 1 ] = √36 = 6.000 sq units
Step 3: Solve Angles via Law of Cosines
• Angle C = arccos((3² + 4² − 5²) ÷ (2 × 3 × 4)) = arccos(0) = 90.00° (Right Angle)
• Angle A = arcsin(3/5) = 36.87°, Angle B = 53.13°

Triangle Classifications

  • Equilateral: 3 equal sides and 3 equal 60° angles.
  • Isosceles: 2 equal sides and 2 equal base angles.
  • Scalene: All 3 sides and all 3 angles are unequal.
  • Right: Contains one 90° angle (satisfies a² + b² = c²).
  • Obtuse: Contains one angle greater than 90°.
  • Acute: All three angles are strictly less than 90°.

Frequently Asked Questions

What is the triangle inequality theorem?
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side (a + b > c, a + c > b, b + c > a). If this condition fails, the three segments cannot connect to enclose a 2D triangle.
What do SSS, SAS, ASA, and AAS mean in geometry?
These are standard triangle congruence and solving postulates: SSS (Side-Side-Side) uses 3 sides; SAS (Side-Angle-Side) uses 2 sides and their included angle; ASA (Angle-Side-Angle) uses 2 angles and their connecting side; AAS (Angle-Angle-Side) uses 2 angles and a non-included side.
How does Heron's formula calculate triangle area?
Heron's formula computes area directly from side lengths without knowing the perpendicular height: Area = √[ s(s − a)(s − b)(s − c) ], where s = (a + b + c) / 2 is the semi-perimeter.
What is the difference between inradius and circumradius?
The inradius (r) is the radius of the incircle (the largest circle tangent to all three internal edges). The circumradius (R) is the radius of the circumcircle (the circle passing through all three vertices).
What are triangle medians and altitudes?
An altitude is a perpendicular line segment dropped from a vertex to the opposite base line. A median is a line segment connecting a vertex to the exact midpoint of the opposite side.

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