# Formulas

## Elementary Geometry # Right Triangle

• Legs of a right triangle: $$a$$, $$b$$
Hypotenuse of a right triangle: $$c$$
Acute angles: $$\alpha$$, $$\beta$$
Right angle: $$C$$
Area: $$S$$
Altitude to the hypotenuse: $$h$$
Medians: $${m_a}$$, $${m_b}$$, $${m_c}$$
Circumradius: $$R$$
Inradius: $$r$$
1. A right triangle is a triangle in which one angle is a right angle (equal to $$90^\circ$$).
2. The side opposite of the right angle is called the hypotenuse. The sides adjacent to the right angle are called the legs. In the figure below, $$AC$$ and $$BC$$ are the legs, $$AB$$ is the hypotenuse. The lengths of the legs are $$a,$$ $$b.$$ The length of the hypotenuse is $$c.$$
3. The sum of the acute angles of a right triangle is $$90^\circ$$:
$$\alpha + \beta = 90^\circ$$
4. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
$$\sin \alpha = {\large\frac{a}{c}\normalsize},\;$$ $$\sin \beta = {\large\frac{b}{c}\normalsize}$$
5. The cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:
$$\cos \alpha = {\large\frac{b}{c}\normalsize},\;$$ $$\cos \beta = {\large\frac{a}{c}\normalsize}$$
6. The tangent of an acute angle is the ratio of the opposite leg to the adjacent leg:
$$\tan \alpha = {\large\frac{a}{b}\normalsize},\;$$ $$\tan \beta = {\large\frac{b}{a}\normalsize}$$
7. The cotangent of an acute angle is the ratio of the adjacent leg to the opposite leg:
$$\cot \alpha = {\large\frac{b}{a}\normalsize},\;$$ $$\cot \beta = {\large\frac{a}{b}\normalsize}$$
8. The secant of an acute angle is the ratio of the hypotenuse to the adjacent leg:
$$\sec \alpha = {\large\frac{c}{b}\normalsize},\;$$ $$\sec \beta = {\large\frac{c}{a}\normalsize}$$
9. The cosecant of an acute angle is the ratio of the hypotenuse to the opposite leg:
$$\csc \alpha = {\large\frac{c}{a}\normalsize},\;$$ $$\csc \beta = {\large\frac{c}{b}\normalsize}$$
10. Pythagorean theorem
The sum of the squares of the legs is equal to the square of the hypotenuse:
$${a^2} + {b^2} = {c^2}$$
11. $${a^2} = fc,\;$$ $${b^2} = gc,$$
where $$f$$ and $$g$$ are projections of the legs $$a$$ and $$b,$$ respectively, onto the hypotenuse $$c$$.
12. $${h^2} = fg$$,
where $$h$$ is the altitude drawn to the hypotenuse $$c$$, and $$f$$ and $$g$$ are projections of the legs $$a$$ and $$b$$ onto the hypotenuse.
13. Medians drawn to the legs of a right triangle
$$m_a^2 = {b^2} – {\large\frac{{{a^2}}}{4}\normalsize},\;$$ $$m_b^2 = {a^2} – {\large\frac{{{b^2}}}{4}\normalsize},$$
where $${m_a}$$ and $${m_b}$$ are the medians drawn to the legs $$a$$ and $$b,$$ respectively.
14. Median drawn to the hypotenuse of a right triangle
$${m_c} = {\large\frac{c}{2}\normalsize}\,$$, where $${m_c}$$ is the median to the hypotenuse $$c.$$
15. Radius of the circumcircle of a right triangle
$$R = {\large\frac{c}{2}\normalsize} = {m_c}$$
16. Radius of the incircle of a right triangle
$$r = {\large\frac{{a + b – c}}{2}\normalsize} =$$ $${\large\frac{{ab}}{{a + b + c}}\normalsize}$$
17. $$ab = ch$$
18. Area of a right triangle
$$S = {\large\frac{{ab}}{2}\normalsize} = {\large\frac{{ch}}{2}\normalsize}$$