# Right triangles, Pythagorean theorem - Calculator given values:
= =
calculated values:

With this calculator you can calculate important values of a right triangle. The Pythagorean theorem, the altitude theorem (geometric mean theorem), leg theorem and the definition of the sine for right triangles are used for this. The number of decimal places to which the results should be rounded can be set. The calculator always uses the rounded values for further calculations. The angles cannot be rounded up to 100 decimal places, because for their calculation the standard functions of the used programming language are used.

The first figure in the box with the results shows the calculated triangle, where the size has been adjusted, but the ratios of the sides have remained the same. The figures in the boxes for the calculation of the individual values can differ completely from the calculated triangle. These figures only serve to illustrate which sides and angles are used in the calculation and which are to be calculated.

## Formulas Pythagorean theoremc² = a² + b²
Leg geometric mean theorem
 c a
=
 a p
(a² = c ∙ p)
 c b
=
 b q
(b² = c ∙ q)
Altitude theorem
 q h
=
 h p
(h² = p ∙ q)
Sine
sin(α) =
 a c
sin(β) =
 b c
Area
A =
 a ∙ b 2
A =
 c ∙ h 2
Perimeter
P = a + b + c

## Pythagorean theorem  In a right triangle, the side opposite the right angle is called the hypotenuse and the two sides between which the 90° angle is located are called legs. Very often, in the context of the Pythagorean theorem, a and b are the lengths of the two legs and c is the length of the hypotenuse. If this is the case, the Pythagorean Theorem states that the following equation holds:

a² + b² = c²

With this equation, if the length of 2 of the 3 sides are known, the length of the third side can be calculated.

### Calculate length of hypotenuse:

To calculate c, the length of the hypotenuse, the equation must be solved for c. This is achieved by calculating the square root of both sides of the equation. Then you get:

c = a² + b²

Example:

The following applies to the side lengths of the two legs: a = 5 and b = 3. c is to be calculated. c = a² + b² = 5² + 3² = 25 + 9 = 34 ≈ 5.83095189

### Calculate the length of a leg:

Assume that the side lengths c and b are given and that the side length a is to be calculated. First subtract b² and then calculate the square root. Then you get for a:

a = c² − b²

If b is to be calculated, the equation must be solved for b:

b = c² − a²

Example:

As an example, b = 3 and c = 5 are given and the side length a is to be calculated. Then you calculate: a = c² − b² = 5² − 3² = 25 − 9 = 16 = 4

## Leg geometric mean theorem For "leg geometric mean theorem", the hypotenuse is partitioned into 2 segments. Therefor the altitude of the triangle is drawn in (Line between the right angle of the triangle and the hypotenuse, so that the angle between the line for the altitude and the hypotenuse of the triangle is 90°). At the point where the line for the altitude hits the hypotenuse is the boundary between the two segments. In the following, the length of the segment adjacent to the side with length a is called p, and the length of the segment adjacent to the side with length b is called q.

The two smaller triangles, with side lengths h, p, a and side lengths h, q, b, are similar to the large triangle. This means that although they differ in size, the ratios of the side lengths and the angles are the same. Of the small left triangle, the side with length b is the hypotenuse and the ratio of the sides with lengths b and q in the small triangle is equal to the ratio of the sides with lengths c and b in the large triangle. Accordingly, it holds:

 c b
=
 b q

For the small right triangle, a is the hypotenuse and the ratio of the sides with lengths a and p in the small triangle is the same as the ratio of the sides with lengths c and a in the large triangle.

 c a
=
 a p

If both sides of the first equation are multiplied by b and q, and both sides of the second equation are multiplied by a and p, then we get:

a² = p ∙ c
b² = q ∙ c

To determine the length of the legs, calculate:

a = p ∙ c or b = q ∙ c

To compute the length of the segments, calculate:

p =
 a² c
or
q =
 b² c

If the length of one of the two segments and if the length of the hypotenuse c is known, then the length of the other segment can be determined with the help of the equation c = p + q.

To determine the length of the hypotenuse c, calculate:

c =
 p a²
or
c =
 q b²

## Altitude theorem

For the geometric mean theorem (altitude theorem), the altitude h and the two segments p and q of the hypotenuse c are drawn in the same way as for the leg theorem. The altitude theorem now states that the following equation is valid:

 q h
=
 h p

If now both sides of the equation are multiplied by h and p we get:

h² = p ∙ q

If the lengths of both segments of the hypotenuse are known, then the altitude can be calculated and if the altitude of the triangle and the length of one of the two segments of the hypotenuse are known, then the length of the other segment can be calculated.

If p and q are known and you want to calculate the altitude, then calculate:

h = p ∙ q

If the altitude and the length of one of the two segments is known, then to determine the length of the other segment, calculate:

p =
 h² q
or
q =
 h² p

## Angles In a right triangle, the two acute angles are usually called α and β and the 90° angle γ. α and β each have an adjacent and opposite side. Usually the length of the opposite side of α is called a and the length of the opposite side of β is typically called b.  The following applies to the sine of an angle:

sin(angle) =
 opposite hypotenuse

Thus, for the two angles α and β it applies:

sin(α) =
 a c
sin(β) =
 b c

To transform the equations so that only the angle is on the left side, the arcsine (the inverse function of the sine) must be applied to both sides of the equation.

α = arcsin(
 a c
)
β = arcsin(
 b c
)

Once you have calculated the angle α or β, the other angle can be calculated quite easily with the help of the following equation:

α + β = 90°