What is a Triangle?
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.
Triangles tend to be described based on the length of their sides, as well as their internal angles:
- Equilateral Triangle: All three sides have equal lengths, and all three internal angles are 60°.
- Isosceles Triangle: Two sides have equal lengths.
- Scalene Triangle: None of the sides have equal lengths.
Triangles classified based on their internal angles fall into two categories: right or oblique.
- Right Triangle: One of the angles is exactly 90°. The longest edge, opposite the right angle, is the hypotenuse.
- Obtuse Triangle: One of the angles is greater than 90°.
- Acute Triangle: All of the angles are less than 90°.
How to Use This Calculator
To fully solve any triangle, you need to know exactly 3 values out of the 6 components (3 sides and 3 angles). At least one of those values must be a side length (since knowing only 3 angles gives you the shape, but not the scale of the triangle).
Simply enter your 3 known values into the corresponding input fields. Make sure to select whether your angles are in Degrees or Radians, and press Calculate Triangle.
Triangle Facts, Theorems, and Laws
Several fundamental geometric laws make solving missing triangle components possible:
- The interior angles of a triangle always add up to exactly 180° (or π radians).
- 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).
The Pythagorean Theorem
The Pythagorean theorem is a theorem specific to right triangles. 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.
Example: Given a = 3, c = 5, finding b gives b = √(5² - 3²) = 4
The Law of Sines
The ratio of the length of a side of a triangle to the sine of its opposite angle is constant. This makes it possible to find unknown angles and sides given enough information.
The Law of Cosines
Given the lengths of all three sides of any triangle, each angle can be calculated using the following equations based on the Law of Cosines:
A = arccos((b² + c² - a²) / 2bc)
B = arccos((a² + c² - b²) / 2ac)
C = arccos((a² + b² - c²) / 2ab)
Area of a Triangle
There are multiple methods for calculating the area of a triangle depending on what information is known:
- Base and Height: area = ½ × b × h
- Two Sides and Angle (SAS): area = ½ × a × b × sin(C)
- Heron's Formula (SSS): Requires the semiperimeter s = (a + b + c) / 2.
Median, Inradius, and Circumradius
Median: The median of a triangle is a line segment that extends from a vertex to the midpoint of the opposing side. All three medians intersect at the centroid.
Inradius: The radius of the largest circle that will fit perfectly inside the triangle (inscribed). It is calculated as area / semiperimeter.
Circumradius: The radius of a circle that passes through all three vertices of the triangle. It is calculated as a / (2 × sin(A)).
Frequently Asked Questions
If you only provide three angles, you only define the shape of the triangle (which dictates its proportions), but not its size. There are infinitely many triangles with the same three angles (similar triangles). Providing at least one side locks in the physical scale.
The calculator applies the Triangle Inequality Theorem (the sum of any two sides must be greater than the third) and verifies angle rules. If your inputs violate geometric laws, the calculator will immediately show an error indicating that a valid triangle cannot be formed from those values.
A scalene triangle is a triangle in which all three sides have different lengths, and all three internal angles have different measurements. It is the most common and generalized type of triangle.
The perimeter is simply the total distance around the outside of the triangle. It is calculated by adding the lengths of all three sides together (Perimeter = a + b + c).
When you know two sides and a non-included angle (SSA), there is sometimes more than one valid triangle that can be drawn with those exact measurements. This is known as the ambiguous case. This calculator will find the primary valid configuration if one exists.