**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

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.

$$P=a+b+c$$

where,

a, b, c — the sides of the 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.

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.