The right-angle triangle has, as the name suggests, always a right angle.

The right angle is called angle C.

The angles in triangles are always named with uppercase letters, and the sides with lowercase letters.

Side a and side b are called legs (or catheti) and form the right angle.

Side c is called the hypotenuse and will always be the longest side in the triangle. 

Note that side a is opposite angle A. 

The same applies to side b and side c, and this will always be the case.

Pythagorean theorem

The Pythagorean theorem describes the relationship between the sides of a right triangle.

Pythagoras states that "the square of side a + the square of side b = the square of side c".

$$ a^2 + b^2 = c^2 $$

If we have a triangle where \(a = 3\) and \(b = 4\), we get the following:

$$ \begin{align} 3^2 + 4^2 &= c^2 \Leftrightarrow \\ 9 + 16 &= 25 \Leftrightarrow \\ c^2 &= 25 \Leftrightarrow \\ c &= \sqrt{25} = 5 \end{align} $$

From the theorem, two formulas can be derived to calculate side a and b:

$$ a = \sqrt{c^2 - b^2} $$

$$ b = \sqrt{c^2 - a^2} $$

Figure

The figure shows the squares on the sides of the triangle.

You can read about calculating angles in the Trigonometry section.

Trigonometry and right triangles

In the Trigonometry section you can read about Cosine, Sine, and Tangent.

Cosine, sine, and tangent can be used to calculate angles and sides in right triangles.

REMEMBER that you must not use these formulas for triangles that are not right-angled.

For other triangles you must use the cosine and sine laws.

When we talk about the sides of the triangle, we refer to the "adjacent" and "opposite" legs.

If you look at the drawing you can see that side b is adjacent to angle A, so it is the adjacent leg.

Side a is instead on the opposite side of angle A, so it is the opposite leg.

Formulas

In right triangles the following relationships hold:

$$ \cos(v) = \mathit{adjacent\ leg \over hypotenuse}$$

$$ \sin(v) = \mathit{opposite\ leg \over hypotenuse}$$

$$ \tan(v) = \mathit{opposite\ leg \over adjacent\ leg}$$

By using inverse trigonometric functions (arcsin, arccos, arctan), you can calculate the angle directly:

$$ v = \cos^{-1}\biggl(\mathit{adjacent\ leg \over hypotenuse}\biggr)$$

$$ v = \sin^{-1}\biggl(\mathit{opposite\ leg \over hypotenuse}\biggr)$$

$$ v = \tan^{-1}\biggl(\mathit{opposite\ leg \over adjacent\ leg}\biggr)$$