Triangle Calculator

Please enter three values, including at least one side, into the following six fields and click the “Calculate” button. When radians are used as the angle unit, they can have values like pi/2, pi/4, and so on.

A triangle is a polygon with three vertices. A vertex is a location where two or more curves, lines, or edges intersect; in the example of a triangle, the three vertices are connected by three line segments known as edges. A triangle is commonly identified by its vertices. A triangle having vertices a, b, and c is commonly abbreviated as Δabc. Furthermore, triangles are typically characterized using the lengths of their sides and internal angles. For example, an equilateral triangle is one with equal lengths on all three sides, but an isosceles triangle has equal lengths on two sides, Scalene occurs when none of the sides of a triangle are of equal length, as shown below.

Tick marks on the edge of a triangle are a popular notation for the length of the side, with the same amount of ticks representing equal length. Similar notation exists for a triangle’s internal angles, which are represented by varying numbers of concentric arcs centered at its vertices, As shown in the triangles above, a triangle’s length and internal angles are closely related, so an equilateral triangle has three equal internal angles and three equal length sides. The triangle presented in the calculator is not shown to scale; while it appears equilateral (and contains angle markings that would normally be read as equal), it is not necessarily equilateral and is only a representation of a triangle. When actual values are supplied, the calculator’s output will show how the input triangle should look.

Triangles are classed according to their interior angles as either right or oblique. A right triangle is one in which one of the angles is 90°, and it is represented by two line segments forming a square at the vertex that forms the right angle. The hypotenuse is a right triangle’s longest edge, the one opposite the right angle. Any triangle that is not a right triangle is considered an oblique triangle, which can be either obtuse or acute. An obtuse triangle has one angle greater than 90°, whereas an acute triangle has all angles less than 90°, as seen below.

Triangle facts, theorems, and laws

A triangle cannot contain more than one vertex with an internal angle greater than or equal to 90°; otherwise, it is not a triangle.

A triangle’s inner angles always add up to 180°, whereas its exterior angles are equal to the sum of its two non-adjacent interior angles. Another method for calculating a triangle’s exterior angle is to subtract the angle at the vertex of interest from 180°.

The sum of any two sides of a triangle is always greater than the length of the third side.

The Pythagorean theorem is a theorem that applies to right triangles. For each right triangle, the square of the hypotenuse length equals the sum of the squares of the other two sides. It follows that any triangle whose sides meet this requirement is a right triangle. Special examples of right triangles, such as 30° 60° 90, 45° 45° 90°, and 3 4 5 right triangles, make calculations easier. The Pythagorean theorem can be stated as follows:

a2 + b2 = c2

EX: Given a = 3, c = 5, find b:

32 + b2 = 52

9 + b2 = 25

b2 = 16

b = 4

Law of sines : The ratio of the length of one side of a triangle to the sine of its opposite angle is constant.  Given sufficient knowledge, the law of sines can be used to calculate unknown angles and sides of a triangle. The law of sines can be stated as follows for sides a, b, and c, as well as angles A, B, and C, as represented in the above calculator. Thus, if b, B, and C are known, c can be determined by comparing b/sin(B) and c/sin(C).  It should be noted that there are cases where a triangle fits specific conditions, allowing for two alternative triangle configurations given the identical amount of data.

Area of a Triangle

The area of a triangle can be calculated using a variety of equations, depending on the information available. The most well-known equation for computing the area of a triangle involves its base (b) and height (h). The term “base” refers to any side of a triangle whose height is measured by the length of the line segment drawn from the vertex opposite the base to a point on the base that forms a perpendicular.

area =

1

2

b × h

area =

1

2

× 5 × 6 = 15

Given the lengths of the two sides and the angle between them, the following formula can be used to calculate the triangle’s area. The variables utilized are in reference to the triangle depicted in the calculator above. Given a = 9, b = 7, and C = 30 degrees:

Median, inradius, and circumradius

Median

The median of a triangle is the length of a line segment that connects a triangle’s vertex to the midway of the opposite side. A triangle can have three medians that all cross at the triangle’s centroid. Please see the figure below for clarity.

The medians of the triangle are represented by the lines ma, mb, and mc. The length of each median is calculated as follows:

Where a, b, and c are the lengths of the triangle’s sides, as illustrated in the image above.

For example, if a=2, b=3, and c=4, the median ma can be determined as follows:

Inradius

The inradius is the radius of the greatest circle that can fit within the specified polygon, which in this case is a triangle. The inradius runs perpendicular to each side of the polygon. The inradius of a triangle can be calculated by creating two angle bisectors that intersect at the triangle’s center. The inradius is the perpendicular distance from the incenter to one of the triangle’s sides. Any side of the triangle can be used as long as the perpendicular distance between the side and the incenter is known, as the incenter is equidistant from each side of the triangle.

This calculator calculates the inradius using the triangle’s area (Area) and semiperimeter (s), as well as the following formulas:

Circumradius

The circumradius is the radius of a circle that travels across all of the vertices of a polygon, specifically a triangle. The center of this circle, where all the perpendicular bisectors of each side of the triangle meet, is the triangle’s circumcenter and the point from which the circumradius is measured. The triangle’s circumcenter does not have to lie within its boundaries. It is worth noting that all triangles have a circumcircle (a circle that runs through each vertex), and thus a circumradius.

This calculator uses the following formula to compute the circumradius:

circumradius = 

a

2sin(A)

Where an is a side of the triangle, and A is the angle opposite side a.

Although side A and angle A are employed, the formula can be used to any of the sides and their respective opposing angles.