Triangle Calculator

A triangle is a fundamental three-sided polygon defined by three vertices connected by straight line segments (edges). Denoted as Δabc, triangles serve as the building blocks of modern geometry, trigonometry, and structural engineering. Whether you are solving complex architectural drafts, completing physics homework, or calculating navigation vectors, understanding the mathematical relationships between a triangle’s side lengths and internal angles is essential.

Our free Triangle Calculator is a comprehensive trigonometry solver. By inputting any three values (including at least one side length), the tool automatically calculates all remaining sides, angles (in degrees or radians, including fractions of π), area, perimeter, medians, inradius, and circumradius.

How Triangles Are Classified

Triangles are categorized based on their geometric symmetry, side lengths, and interior angle measurements:

Classification by Side Lengths

• Equilateral Triangle: All three sides are of equal length, and all three interior angles are exactly 60° (or π/3 radians).
• Isosceles Triangle: Two sides have equal lengths, and the angles opposite those sides are equal.
• Scalene Triangle: All three sides have different lengths, and all three interior angles are unique.

Classification by Angles

• Right Triangle: Contains exactly one 90° angle (π/2 radians), with the side opposite the right angle called the hypotenuse. Right triangles satisfy the Pythagorean theorem.
• Oblique Triangle: Any triangle that does not contain a 90° angle. Oblique triangles are split into:
  • Acute Triangle: All three interior angles are less than 90°.
  • Obtuse Triangle: One interior angle is greater than 90°.

Core Geometric Theorems and Trigonometric Laws

To solve a triangle, our calculator applies fundamental geometric laws:

1. Interior Angle Sum and Triangle Inequality

The interior angles of any triangle in Euclidean space always add up to exactly 180° (or π radians). Furthermore, under the Triangle Inequality Theorem, the sum of the lengths of any two sides must always be strictly greater than the length of the remaining side (e.g., a + b > c).

2. The Pythagorean Theorem (Right Triangles)

For right-angled triangles, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b):

a² + b² = c²

For example, given a = 3 and c = 5: 3² + b² = 5² ⇒ 9 + b² = 25 ⇒ b² = 16 ⇒ b = 4.

3. The Law of Sines

The ratio of a side’s length to the sine of its opposite angle is constant:

a / sin(A) = b / sin(B) = c / sin(C)

This law is critical for solving triangles when you know two angles and one side (AAS/ASA) or two sides and a non-included angle (SSA). Note that the SSA configuration can occasionally yield an ambiguous case with two possible triangle layouts.

4. The Law of Cosines

Used to find interior angles when all three side lengths are known (the SSS case):

• A = arccos[ (b² + c² - a²) / 2bc ]
• B = arccos[ (a² + c² - b²) / 2ac ]
• C = arccos[ (a² + b² - c²) / 2ab ]

Formulas for Calculating Triangle Area

Depending on what values are known, the calculator utilizes three distinct formulas to estimate area:

1. Standard Base and Height: If base (b) and perpendicular height (h) are known:

   Area = ½ × b × h

2. Trigonometric Method (SAS): If two sides and their included angle are known:

   Area = ½ × a × b × sin(C) = ½ × b × c × sin(A) = ½ × a × c × sin(B)

3. Heron’s Formula (SSS): If only the three side lengths (a, b, c) are known, Heron’s formula calculates the area without requiring height:

   Area = √[ s(s - a)(s - b)(s - c) ]

   Where the semiperimeter (s) is: s = (a + b + c) / 2

Medians, Inradius, and Circumradius

Our calculator solves advanced geometric segments inside the triangle:

1. Median Lengths

A median is a line segment drawn from a vertex to the midpoint of the opposite side. The three medians of a triangle intersect at a single point called the centroid (the arithmetic center of mass). The lengths of the medians (m) are:

• m_a = √[ (2b² + 2c² - a²) / 4 ]
• m_b = √[ (2a² + 2c² - b²) / 4 ]
• m_c = √[ (2a² + 2b² - c²) / 4 ]

2. Inradius (r)

The inradius is the radius of the largest circle (incircle) that can fit entirely inside the triangle. Its center (the incenter) is located at the intersection of the triangle’s three angle bisectors. The inradius is calculated as:

Inradius = Area / s

3. Circumradius (R)

The circumradius is the radius of the circle (circumcircle) that passes through all three vertices of the triangle. The circumcenter lies at the intersection of the perpendicular bisectors of the sides. It is solved as:

Circumradius = a / [ 2 × sin(A) ] (where any side-angle opposite pair can be substituted).

Evaluate other geometric structures with our Area Calculator or solve right-triangle hypotenuse problems on the Pythagorean Theorem Calculator.

Frequently Asked Questions (FAQ)

What is the ambiguous case in solving triangles?

The ambiguous case occurs during the Side-Side-Angle (SSA) configuration, where you are given two sides and a non-included acute angle. Depending on the length of the side opposite the angle relative to the adjacent side and height, there may be zero possible triangles, exactly one right triangle, or two distinct oblique triangles that fit the data.

Can a triangle have two obtuse angles?

No. The interior angles of a Euclidean triangle must always sum to exactly 180°. Since an obtuse angle is strictly greater than 90°, having two obtuse angles would result in a sum greater than 180°, making it impossible to close the polygon.

What is the difference between the centroid and the incenter?

The centroid is the intersection point of the triangle’s three medians, representing the physical center of gravity. The incenter is the intersection point of the three angle bisectors, representing the center of the incircle (the largest circle that fits inside the triangle, touching all three sides perpendicularly).

How does Heron’s formula work for right triangles?

Heron’s formula works for all Euclidean triangles, including right triangles. While using the standard 1/2 base times height formula is simpler for right triangles, applying Heron’s formula with the three side lengths will yield the exact same area result.