Here are the main triangle formulas: area, perimeter, height, inradius, circumradius, and formulas for equilateral, isosceles and right triangles.
Triangle formulas at a glance
- Area from base and height: \(S=\frac{a \times h}{2}\)
- Perimeter: \(P=a+b+c\)
- Semiperimeter: \(s=\frac{a+b+c}{2}\)
- Heron's formula: \(S=\sqrt{s(s-a)(s-b)(s-c)}\)
- Height from area: \(h=\frac{2S}{a}\)
- Inradius: \(r=\frac{S}{s}=\frac{2S}{P}\)
- Circumradius: \(R=\frac{abc}{4S}\)
A triangle is a plane figure with three vertices, three sides and three interior angles. The sum of the interior angles of a triangle is always 180°.
Sum of interior angles:
$$\alpha + \beta + \gamma = 180^{\circ}$$
Types of triangles
Triangles are classified by angles and by sides. By angles, triangles are:
- Right triangle - one angle is 90°
- Acute triangle - all angles are smaller than 90°
- Obtuse triangle - one angle is greater than 90°
- Oblique triangle - an acute or obtuse triangle
By sides, triangles are:
- Equilateral triangle - all sides are equal and all angles are 60°
- Isosceles triangle - two sides are equal
- Scalene triangle - all sides have different lengths
Area of a triangle
The area of a triangle can be calculated in several ways. The most common formula uses the base and height.
1. From base and height:
$$S= \frac{a \times h}{2}$$
where,
a - base;
h - height.
Example: if the base is 8 cm and the height is 5 cm, then \(S=\frac{8 \times 5}{2}=20\) cm2.
2. From 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.
Use Heron's formula when all three side lengths are known but the height is not known.
3. From the inradius and perimeter:
$$S= \frac{r \times P}{2}$$
where,
r - inradius;
P - perimeter of the triangle.
4. From 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. From 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 angles adjacent to side a.
The last formula can be applied to any side and the two adjacent angles of that side.
Triangle height
If the area and base are known, the height can be found from the area formula:
$$h=\frac{2S}{a}$$
where,
S - area of the triangle;
a - base.
Example: if the area is 30 cm2 and the base is 10 cm, then \(h=\frac{2 \times 30}{10}=6\) cm.
Perimeter of a triangle
The perimeter of a triangle is the sum of all three side lengths.
$$P=a+b+c$$
where,
a, b, c - the sides of the triangle.
Example: if the sides are 4 cm, 5 cm and 6 cm, then \(P=4+5+6=15\) cm.
Incircle and circumcircle
The inradius of a triangle can be found from the area and semiperimeter:
$$r=\frac{S}{s}=\frac{2S}{P}$$
The circumradius of a triangle can be found from the sides and area:
$$R=\frac{abc}{4S}$$
where,
r - inradius;
R - circumradius;
s - semiperimeter.
Equilateral triangle
The height of an equilateral triangle:
$$h= \frac{a \sqrt{3}}{2}$$
where,
a - side of the triangle.
The area of an equilateral triangle:
$$S= \frac{\sqrt{3}}{4}a^{2}$$
The perimeter of an equilateral triangle:
$$P=3a$$
Isosceles triangle
The height of an isosceles triangle:
$$h= \sqrt{b^{2}-\left(\frac{a}{2}\right)^{2}}$$
where,
a - base;
b - equal side.
The area of an isosceles triangle from the base and equal side:
$$S=\frac{a}{4}\sqrt{4b^{2}-a^{2}}$$
The perimeter of an isosceles triangle:
$$P=a+2b$$
Right triangle
The sides of a right triangle are the legs a and b, and the hypotenuse c. A right triangle satisfies the Pythagorean Theorem.
$$a^{2}+b^{2}=c^{2}$$
The area of a right triangle:
$$S=\frac{ab}{2}$$
The perimeter of a right triangle:
$$P=a+b+c$$
Frequently asked questions
How do you find the area of a triangle?
If the base and height are known, use \(S=\frac{a \times h}{2}\). If all three sides are known, use Heron's formula \(S=\sqrt{s(s-a)(s-b)(s-c)}\).
How do you find the height of a triangle?
If the area and base are known, use \(h=\frac{2S}{a}\). For an isosceles triangle, use \(h=\sqrt{b^{2}-\left(\frac{a}{2}\right)^{2}}\).
How do you calculate the perimeter of a triangle?
Add the three sides: \(P=a+b+c\). For an equilateral triangle, \(P=3a\); for an isosceles triangle, \(P=a+2b\).
Does a triangle have volume?
A triangle is a plane figure, so it has no volume. Volume is calculated for three-dimensional solids, such as a triangular prism.
