A triangle calculator is an online geometry tool that solves unknown side lengths, interior angles, area, and perimeter for any triangle. Enter three known values, such as two sides and one angle or all three sides, to calculate all missing dimensions instantly. The tool uses fundamental geometric laws like the Pythagorean theorem, the Law of Sines, and the Law of Cosines to deliver exact results for right, isosceles, equilateral, and scalene triangles.
Enter triangle parameters to see results
Provide at least 3 values (combination of sides and angles)
Solving a triangle means finding the measures of all three sides and all three interior angles. Geometric rules require at least three known measurements to compute the remaining parts. At least one of these known values must be a side length. Knowing three angles alone only determines the shape, not the size.
Different initial measurements require specific mathematical methods:
When all three side lengths are known, the angles can be found using the Law of Cosines. The lengths must satisfy the triangle inequality theorem, where the sum of any two sides must be greater than the third side.
When two sides and the included angle between them are known, the third side is found using the Law of Cosines. Once all three sides are known, the remaining angles are easily computed.
When two angles and one side are given, finding the third angle is simple because all interior angles sum to 180 degrees. The remaining missing sides are calculated using the Law of Sines.
When two sides and a non-included angle are provided, the calculation uses the Law of Sines. This scenario is called the ambiguous case because it can yield two valid triangles, one unique triangle, or no possible triangle at all.
Calculating triangle properties relies on a few key mathematical formulas.
The three interior angles of any Euclidean triangle always sum to 180 degrees (or pi radians).
Angle A + Angle B + Angle C = 180°
If two angles are known, subtract their sum from 180 degrees to find the final angle.
For a right triangle where one angle is 90 degrees, the relationship between the sides is governed by the Pythagorean theorem formula.
a² + b² = c²
Here, c represents the hypotenuse (the longest side opposite the right angle), while a and b are the two shorter legs.
The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.
a / sin(A) = b / sin(B) = c / sin(C)
This law is useful when dealing with ASA, SAA, or SSA configurations.
The Law of Cosines generalizes the Pythagorean theorem to any triangle, making it essential for SSS and SAS cases.
c² = a² + b² - (2 × a × b × cos(C))
To solve for an angle when three sides are known, rearrange the formula:
cos(C) = (a² + b² - c²) / (2 × a × b)
Understanding how to calculate the boundary and interior space of a geometric shape depends on the available measurements.
The perimeter is the total distance around the outside of the shape. To find it, sum the lengths of all three sides:
Perimeter = a + b + c
When the length of the base and the perpendicular height are known, use the standard area equation:
Area = 0.5 × base × height
When two sides and the angle between them (SAS) are known, calculate area without knowing the height:
Area = 0.5 × a × b × sin(C)
When all three side lengths are available but the height is unknown, use Heron's formula. First, calculate the semiperimeter (s):
s = (a + b + c) / 2
Then calculate the area using:
Area = √(s × (s - a) × (s - b) × (s - c))
Learn how this compares to other two-dimensional boundaries using our area calculator.
Triangles are grouped into distinct categories based on side equality or angle measurements.
| Classification | Primary Characteristic | Key Property |
|---|---|---|
| Equilateral | 3 equal sides | All interior angles equal 60° |
| Isosceles | 2 equal sides | Angles opposite equal sides are equal |
| Scalene | 0 equal sides | All sides and angles have different values |
| Right | 1 angle equals 90° | Governed by the Pythagorean theorem |
| Obtuse | 1 angle is greater than 90° | Can only have one obtuse angle |
| Acute | All angles are less than 90° | All three angles must be acute |
Special right triangles, such as the 30-60-90 or 45-45-90 configurations, have fixed side ratios that allow quick calculations without complex trigonometry. Work through these relationships directly with our right triangle solver.
Beyond basic sides and angles, triangles have several advanced geometric properties.
A median is a straight line segment drawn from a vertex to the midpoint of the opposite side. Every triangle has three medians. The point where all three medians intersect is called the centroid, which serves as the geometric center of gravity.
The inradius is the radius of the largest circle that can fit inside the triangle, touching all three sides. It is computed using the area and semiperimeter:
Inradius = Area / s
The circumradius is the radius of the circle that passes through all three vertices of the triangle. It is calculated using the side length and opposite angle:
Circumradius = a / (2 × sin(A))
You need at least three known values, and at least one of those values must be a side length. Knowing three angles alone is not enough to determine side lengths.
No. The interior angles of a triangle always add up to 180 degrees. If two angles were 90 degrees, their sum would equal 180 degrees, leaving 0 degrees for the third angle, which prevents the shape from closing.
Apply the triangle inequality theorem. The sum of the lengths of any two sides must be strictly greater than the length of the remaining side. For example, sides of lengths 3, 4, and 5 form a valid triangle, but sides 2, 3, and 6 cannot.
The ambiguous case occurs when solving a triangle using two sides and a non-included angle (SSA). Depending on the side lengths, this combination can yield two distinct valid triangles, one right triangle, or no possible triangle.
To convert degrees to radians, multiply the angle by pi and divide by 180. To convert radians to degrees, multiply by 180 and divide by pi.