Triangle Calculator
Calculate the properties of any triangle using sides and angles. Supports various triangle types and provides comprehensive geometric analysis.
Triangle Calculator
Please provide 3 values including at least one side to the following 6 fields, and click the "Calculate" button.
About Triangles
A triangle is a polygon that has three vertices. A vertex is a point where two or more curves, lines, or edges meet; in the case of a triangle, the three vertices are joined by three line segments called edges. A triangle is usually referred to by its vertices. Hence, a triangle with vertices a, b, and c is typically denoted as Δabc.
Triangle Classification
Triangles tend to be described based on the length of their sides, as well as their internal angles:
By Sides:
- Equilateral: All three sides have equal lengths
- Isosceles: Two sides have equal lengths
- Scalene: No sides have equal lengths
By Angles:
- Acute: All angles are less than 90°
- Right: One angle is exactly 90°
- Obtuse: One angle is greater than 90°
Triangle Facts and Theorems
- It is not possible for a triangle to have more than one vertex with internal angle greater than or equal to 90°, or it would no longer be a triangle.
- The interior angles of a triangle always add up to 180° while the exterior angles of a triangle are equal to the sum of the two interior angles that are not adjacent to it.
- The sum of the lengths of any two sides of a triangle is always larger than the length of the third side (Triangle Inequality Theorem).
- Pythagorean theorem: For any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides: a² + b² = c²
- Law of sines: The ratio of the length of a side of a triangle to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C)
- Law of cosines: For any triangle with sides a, b, c and opposite angles A, B, C: c² = a² + b² - 2ab·cos(C)
Area Calculations
There are multiple different equations for calculating the area of a triangle, dependent on what information is known:
Base and Height:
Area = ½ × base × height
Two Sides and Included Angle:
Area = ½ × a × b × sin(C)
Heron's Formula (Three Sides):
Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Special Properties
Median
The median of a triangle is defined as the length of a line segment that extends from a vertex of the triangle to the midpoint of the opposing side. A triangle can have three medians, all of which will intersect at the centroid.
Inradius
The inradius is the radius of the largest circle that will fit inside the given triangle. The inradius is perpendicular to each side of the triangle.
Circumradius
The circumradius is defined as the radius of a circle that passes through all the vertices of a triangle. The center of this circle is the circumcenter of the triangle.
Centroid
The centroid is the arithmetic mean position of all the points in the triangle. It is the intersection point of the three medians.