# Triangle formulas

Triangle is determined in Euclidean space by three points that do not lie on the same line, and these points are called the vertices of the triangle. A triangle is a shape formed by segments connecting the vertices of the triangle, also known as the sides of the triangle. The triangle lies in a plane, i.e., it is a planar figure.

At least two angles of the triangle are acute angles (i.e., < 90°). One angle can be acute, right, or obtuse angle.

There is always a relationship between the sum of the interior angles of a triangle:

$$\alpha + \beta + \gamma = 180^{\circ}$$

Triangles can be classified by angles and sides. Triangles are classified by angles as follows:

• Right triangle - a triangle with one angle being a right angle, i.e., 90°
• Acute triangle - a triangle with all angles smaller than 90 degrees
• Obtuse triangle - a triangle with one angle greater than 90 degrees
• Oblique triangle - this term is sometimes used for obtuse and acute triangles

Triangles are classified by sides as follows:

• Equilateral triangle - a triangle with all sides of equal length. All angles of an equilateral triangle are also of equal size, 60°.
• Isosceles triangle - a triangle with two sides of equal length
• Scalene triangle - a triangle with all sides of different lengths

## Area of a Triangle

1. Calculated through base and height:

$$S= \frac{a \times h}{2}$$

where,

a— base;
h— height.

2. through three sides (Heron's formula)

\begin{align} S &=\sqrt{s \times (s-a) \times (s-b) \times (s-c)} \\ s &=\frac {a+b+c}{2} \\ \end{align}

where,

a, b, c — the sides of the triangle.

3. through the radius of the inscribed circle and the perimeter of the triangle:

$$S= \frac{r \times P}{2}$$

where,

r— radius of the inscribed circle;
P— perimeter of the triangle.

4. through two sides and the included angle:

$$S= \frac{ab\;\textrm{sin}\,\gamma }{2} = \frac{bc\;\textrm{sin}\,\alpha }{2} = \frac{ac\;\textrm{sin}\,\beta }{2}$$

where,

a, b, c — the sides of the triangle;
α, β, γ — the interior angles of the triangle.

5. through one side and its adjacent angles:

$$S= \frac{a^{2}}{2\;(\textrm{cot}\,\beta+\textrm{cot}\,\gamma)}=\frac{a^{2}(\textrm{sin}\,\beta)(\textrm{sin}\,\gamma)}{2\,\textrm{sin}\,(\beta+\gamma)}$$

where,

a— a side of the triangle;
β, γ— the adjacent angles of side a of the triangle.

The last formula can be applied to all sides of the triangle and their respective adjacent angles.

## Perimeter of a Triangle

$$P=a+b+c$$

where,

a, b, c — the sides of the triangle.

## Equilateral Triangle

The height of an equilateral triangle can be found using the formula:

$$h= \frac{a \sqrt{3}}{2}$$

where,

a— a side of the triangle.

The area of an equilateral triangle can be found using the formula:

$$S= \frac{\sqrt{3}}{4}a^{2}$$

where,

a— a side of the triangle.

## Isosceles Triangle

The height of an isosceles triangle can be found using the formula:

$$h= \sqrt{b^{2}-\left(\frac{a}{2}\right)^{2}}$$

where,

a— base;
b— legs, equal sides.