How do I calculate the area of a triangle?

Use Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter and a, b, c are the three sides. For a right triangle, Area = (1/2) * base * height. For a triangle with two sides and the included angle, Area = (1/2) * a * b * sin(C). Example: a 3-4-5 triangle has s = 6, Area = sqrt(6*3*2*1) = 6 square units.

What is the Law of Cosines?

The Law of Cosines states c^2 = a^2 + b^2 - 2ab*cos(C), where C is the angle opposite side c. It generalizes the Pythagorean theorem (when C = 90 degrees, cos(C) = 0, giving c^2 = a^2 + b^2). Use it to find a missing side when you know two sides and the included angle (SAS), or to find an angle when you know all three sides (SSS).

What is the Law of Sines?

The Law of Sines states a/sin(A) = b/sin(B) = c/sin(C), where lowercase letters are sides and uppercase are their opposite angles. Use it to solve ASA (two angles and included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle) triangles. The SSA case may have zero, one, or two solutions (the ambiguous case).

What are the different types of triangles?

By sides: Equilateral (all sides equal, all angles 60 degrees), Isosceles (two sides equal, two angles equal), Scalene (all sides different). By angles: Acute (all angles 90 degrees). A triangle can be both, e.g., an isosceles right triangle has two equal sides and a 90-degree angle.

What is the triangle inequality theorem?

The triangle inequality theorem states that the sum of any two sides must be greater than the third side. For sides a, b, c: a + b > c, a + c > b, and b + c > a. If any condition fails, the three lengths cannot form a triangle. Example: sides 3, 4, 5 form a triangle (3+4>5, 3+5>4, 4+5>3), but sides 1, 2, 5 do not (1+2 < 5).

What are the inradius and circumradius of a triangle?

The inradius (r) is the radius of the largest circle that fits inside the triangle (inscribed circle), calculated as r = Area / s where s is the semi-perimeter. The circumradius (R) is the radius of the circle that passes through all three vertices (circumscribed circle), calculated as R = abc / (4 * Area). For an equilateral triangle with side a: r = a*sqrt(3)/6, R = a*sqrt(3)/3.

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Triangle Calculator

A triangle is solved by finding all 3 sides, 3 angles, and area from any 3 known values. Methods: SSS uses Law of Cosines (c^2 = a^2 + b^2 - 2ab*cos(C)), SAS/ASA/AAS use Law of Sines (a/sin(A) = b/sin(B)), area uses Heron's formula (A = sqrt(s(s-a)(s-b)(s-c))). A 3-4-5 right triangle has area = 6, perimeter = 12, inradius = 1, circumradius = 2.5.

Solve any triangle given 3 known values. Supports SSS, SAS, ASA, AAS, and SSA modes. Calculate all sides, angles, area, perimeter, heights, inradius, and circumradius using the Law of Cosines, Law of Sines, and Heron's formula.

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Input Mode

Values

Side a

Side b

Side c

How to Use

1
Select input mode
Choose SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), AAS (2 angles + non-included side), or SSA (2 sides + non-included angle).
2
Enter known values
Type the 3 known measurements. For angles, enter values in degrees. Use presets like 3-4-5 or Equilateral for quick testing.
3
Review the solution
The calculator shows all 6 values (3 sides, 3 angles), area, perimeter, heights, inradius, circumradius, and triangle type classification.
4
Copy results
Click Copy All Results to get a formatted summary of all calculated values for your homework, project, or reference.

Frequently Asked Questions

How do I calculate the area of a triangle?: Use Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter and a, b, c are the three sides. For a right triangle, Area = (1/2) * base * height. For a triangle with two sides and the included angle, Area = (1/2) * a * b * sin(C). Example: a 3-4-5 triangle has s = 6, Area = sqrt(6*3*2*1) = 6 square units.
What is the Law of Cosines?: The Law of Cosines states c^2 = a^2 + b^2 - 2ab*cos(C), where C is the angle opposite side c. It generalizes the Pythagorean theorem (when C = 90 degrees, cos(C) = 0, giving c^2 = a^2 + b^2). Use it to find a missing side when you know two sides and the included angle (SAS), or to find an angle when you know all three sides (SSS).
What is the Law of Sines?: The Law of Sines states a/sin(A) = b/sin(B) = c/sin(C), where lowercase letters are sides and uppercase are their opposite angles. Use it to solve ASA (two angles and included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle) triangles. The SSA case may have zero, one, or two solutions (the ambiguous case).
What are the different types of triangles?: By sides: Equilateral (all sides equal, all angles 60 degrees), Isosceles (two sides equal, two angles equal), Scalene (all sides different). By angles: Acute (all angles < 90 degrees), Right (one angle = 90 degrees), Obtuse (one angle > 90 degrees). A triangle can be both, e.g., an isosceles right triangle has two equal sides and a 90-degree angle.
What is the triangle inequality theorem?: The triangle inequality theorem states that the sum of any two sides must be greater than the third side. For sides a, b, c: a + b > c, a + c > b, and b + c > a. If any condition fails, the three lengths cannot form a triangle. Example: sides 3, 4, 5 form a triangle (3+4>5, 3+5>4, 4+5>3), but sides 1, 2, 5 do not (1+2 < 5).
What are the inradius and circumradius of a triangle?: The inradius (r) is the radius of the largest circle that fits inside the triangle (inscribed circle), calculated as r = Area / s where s is the semi-perimeter. The circumradius (R) is the radius of the circle that passes through all three vertices (circumscribed circle), calculated as R = abc / (4 * Area). For an equilateral triangle with side a: r = a*sqrt(3)/6, R = a*sqrt(3)/3.

Triangle Solving Methods

Method	Known Values	Formula Used	Example
SSS	3 sides	Law of Cosines	a=3, b=4, c=5
SAS	2 sides + included angle	Law of Cosines	a=5, C=60, b=5
ASA	2 angles + included side	Law of Sines	A=45, c=10, B=60
AAS	2 angles + non-included side	Law of Sines	A=30, B=60, a=5
SSA	2 sides + non-included angle	Law of Sines (ambiguous)	a=8, b=6, A=40