Given a triangle ΔABC,

a = BC, b = AC, c = AB are sides of the triangle,

A = CAB, B = ABC, C = BCA are angles of the triangle.

How to use the triangle calculator. Please provide 3 values: a side and 2 other values (for example, an angle and a side, 2 angles or 2 sides). Then please provide "text from the image". Click "Calculate".

Triangle calculator finds solutions for the triangle problems – the lengths of sides and angles of the triangle.

Solving a triangle means to find all angles and sides of the triangle. This calculator finds all elements of a triangle.

## How to solve a triangle

Using this online calculator a triangle can be solved if given three, two sides and angles, by the law of sine and cosine, that is, it shows how to find the angles of a triangle.

The problem answer is calculated automatically by a computer program online, quickly, and conveniently.

This free triangle calculator provides formulas, detailed and accurate solutions and answers to problems for the cosine and sine laws.

The solution shows in detail how to find a third side if given two sides and the angle between them, or how to calculate two sides of a triangle if one side is given.

## Examples of problem solutions

1) **Solve the triangle if given two sides and the opposite angle**, i.e. the angle between them. The sides are given a = 12 cm, b = 8 cm, angle = 60°. In order to solve a problem, it is required to provide the given values of the problem in an online form on this page. In the “a” field is to provide 12, in the “b” is to provide 8, in the “A” angle field is to provide 60. Click the “Calculate” button.

The answer:

- angle B = 35.2644°;

- angle C = 84.7356°;

- side c = 13.80 cm;

- perimeter of the triangle P = 33.80 cm;

- semiperimeter s = 16.90 cm;

- area of the triangle A = 47.80 cm^{2};

- radius of the circle circumscribed around the triangle, R = 6.93 cm;

- radius of the circle inscribed in the triangle is r = 2.83 cm.

So the angle of the triangle was calculated by two sides and the angle.

2) how to **find the angle of a triangle, knowing its sides** or **solve a triangle on three sides**. Three sides are given a = 2 cm, b = 3 cm, c = 4 cm. In the “a” field of the online form is to provide 2, in the “b” field is to provide 3, in the “c” field is to provide 4. Click the “Calculate" button.

Using the cosine law, we get

- angle A = 28.9550°;

- angle C = 104.4775°;

- angle B = 46.5675°;

- perimeter of the triangle P = 9 cm;

- semiperimeter s = 4.5 cm;

- area of the triangle A = 2.9 cm^{2};

- radius of the circle circumscribed around the triangle, R = 2.07 cm;

- radius of the circle inscribed in the triangle is r = 0.65 cm.

So all the angles of the triangle were calculated.

3) **solve a triangle by two angles and a side**. In triangle ABC, side a = 5 cm, two angles B = 30°, C = 45°.

Answer:

- angle A = 105°,

- sides c = 3.66 cm; b = 2.59 cm,

- perimeter of the triangle P = 11.25 cm;

- semiperimeter s = 5.63 cm;

- triangle area A = 4.58 cm^{2};

- radius of the circle circumscribed around the triangle, R = 2.59 cm;

- radius of a circle inscribed in a triangle, r = 0.81 cm.

## Triangles

A triangle is a polygon that consists of three points connected by three segments. The three points of this polygon are vertices of the triangle, and the segments are sides or edges of the triangle. The figure shows the triangle ΔABC, where A, B, C are its vertices, and AB, BC, AC are its sides. The vertices of a triangle give the triangle its designation. The angle at the vertex A is formed by the sides AB and AC. This angle denoted as the angle CAB.

There are different categories of triangles. The category of the triangle depends on its angles and sides.

An equilateral or regular triangle consists of three equal sides and three equal angles. All three angles in the equilateral triangle equal to 60 degrees.

If two sides have the equal length in a triangle, then this is an isosceles triangle.

In the isosceles triangle, two equal sides are called the edges, and the third side is called the base of the triangle.

Laws of isosceles triangles:

1) the angles at the base are equal,

2) if two angles are equal in a triangle, then this is an isosceles triangle,

3) a median, that connects the vertex with the base, is a bisector and height.

Equal sides in triangles are denoted by one, two or three tick marks. Equal angles are denoted by one, two or three arc lines.

There are acute, obtuse, and right triangles.

A triangle is right if one of the three angles of this triangle is 90 degrees. The side opposite the angle of 90 degrees is called the hypotenuse. The hypotenuse is the longest edge in a right triangle. The other two edges are called legs.

If a triangle is not right, so this triangle is oblique and can be either obtuse or acute.

An obtuse triangle is a triangle in which one of the angles is greater than 90 degrees.

An acute triangle is a triangle in which all three angles are less than 90 degrees.

## Triangle Properties

In a triangle, only one angle can be greater than 90 degrees.

In a triangle, the sum of the angles is 180 degrees.

The exterior angle of the triangle is the adjacent angle at this vertex.

The calculation methods to find the exterior angle at the vertex:

a) sum two internal angles not adjacent to it,

b) calculate the difference between 180 degrees and the internal angle of this vertex.

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

## Radius of the inscribed circle

A circle inscribed in a triangle is a circle drawn inside the triangle.

The radius of this circle (r) is a perpendicular drawn from the center of the inscribed circle to one side of the triangle.

The center of the inscribed circle is the intersection point of two bisectors and equidistant from each side of the triangle.

The area and perimeter of the triangle are used to calculate the radius of the inscribed circle.

## Circumradius

The circumradius (R) is the radius of the circumscribed circle passes through the three vertices of the triangle.

To calculate the radius of the circumscribed circle, the area and lengths of all sides of the triangle are used.