Triangle Calculator

About Triangle Calculator

Calculate all properties of a triangle including sides, angles, area, perimeter, heights, and radii. Enter any 3 known values to solve the complete triangle.

Triangle Solution Methods

SSS (Side-Side-Side)

When all three sides are known, use the Law of Cosines to find angles.

SAS (Side-Angle-Side)

Two sides and the included angle are known.

ASA (Angle-Side-Angle)

Two angles and the included side are known.

AAS (Angle-Angle-Side)

Two angles and a non-included side are known.

SSA (Side-Side-Angle)

Two sides and a non-included angle - ambiguous case, may have 0, 1, or 2 solutions.

Triangle Formulas

Area Formulas

• Heron's Formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
• Base × Height: Area = (1/2) × base × height
• Two Sides and Angle: Area = (1/2) × a × b × sin(C)

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C) = 2R (where R is circumradius)

Law of Cosines

• a² = b² + c² - 2bc·cos(A)
• b² = a² + c² - 2ac·cos(B)
• c² = a² + b² - 2ab·cos(C)

Other Properties

• Perimeter: P = a + b + c
• Semiperimeter: s = (a + b + c)/2
• Height: h = 2·Area/base
• Inradius: r = Area/s
• Circumradius: R = abc/(4·Area)

Triangle Types

By Angles

Type	Condition	Description
Acute	All angles < 90°	All angles are acute
Right	One angle = 90°	Has one right angle
Obtuse	One angle > 90°	Has one obtuse angle

By Sides

Type	Condition	Description
Equilateral	a = b = c	All sides equal, all angles 60°
Isosceles	Two sides equal	Two equal sides, two equal angles
Scalene	All sides different	No equal sides or angles

Special Right Triangles

30-60-90 Triangle

Side ratios: 1 : √3 : 2

45-45-90 Triangle

Side ratios: 1 : 1 : √2

Pythagorean Theorem

For right triangles only: a² + b² = c² (where c is the hypotenuse)

Common Triangle Examples

3-4-5 Right Triangle

Classic Pythagorean triple. Sides: 3, 4, 5. Angles: 90°, 53.13°, 36.87°

5-12-13 Right Triangle

Another Pythagorean triple. Sides: 5, 12, 13. Right angle opposite the longest side.

Equilateral Triangle

All sides equal, all angles 60°. Area = (√3/4) × side²

Triangle Inequality Theorem

The sum of any two sides must be greater than the third side:

• a + b > c
• a + c > b
• b + c > a

Tips for Using the Calculator

• Enter exactly 3 known values (can be any combination of sides and angles)
• Angles must be entered in degrees
• The sum of all angles in a triangle must equal 180°
• For SSA (ambiguous case), the calculator shows one possible solution
• Check that your input satisfies the triangle inequality
• Results are rounded to 4 decimal places for accuracy

Common Use Cases

• Construction: Calculate roof angles and dimensions
• Navigation: Determine distances using triangulation
• Engineering: Structural calculations and design
• Surveying: Land measurement and mapping
• Education: Geometry homework and learning
• Art & Design: Perspective and composition

Frequently Asked Questions

Can I solve a triangle with only angles?

No, you need at least one side length. Angles alone only determine the shape, not the size.

What is the ambiguous case?

SSA (two sides and a non-included angle) can have 0, 1, or 2 valid solutions depending on the values.

Why do my angles not add up to 180°?

Check your inputs. The sum of angles in any triangle must equal exactly 180°. Small rounding differences may occur in calculations.

What's the difference between inradius and circumradius?

Inradius is the radius of the inscribed circle (inside the triangle). Circumradius is the radius of the circumscribed circle (passing through all vertices).

How accurate are the results?

Results are calculated using standard trigonometric formulas and displayed to 4 decimal places for practical precision.