# Rhombus - Calculator

given values:
= =
calculated values:

With this calculator you can calculate the area, the perimeter, the interior angles, the lengths of the diagonals and the inradius of a rhombus. You can select to how many decimal places the calculation should be rounded. The calculator always uses the rounded values for further calculations.

## Formulas

angles
α + β = 180°
γ = α, δ = β
diagonal lengths
d1 = 2 ∙ a ∙ cos(
 α 2
)
d1 = 2 ∙ a ∙ sin(
 β 2
)
d2 = 2 ∙ a ∙ sin(
 α 2
)
d2 = 2 ∙ a ∙ cos(
 β 2
)
perimeterP = 4 ∙ a
side lengths
a =
 d1² + d2² 2
altitude
h = a ∙ sin(α)
area
A = a ∙ h
A =
 d1 ∙ d2 2
r =
 h 2

## What is a rhombus?

A rhombus is a quadrilateral in which all 4 sides have the same length. Opposite sides of a rhombus are parallel to each other and opposite angles are equal.

## Angles of a rhombus

In a rhombus, opposite angles are equal.

For a quadrilateral, the sum of the 4 interior angles is 360°. It is therefore valid:

α + β + γ + δ = 360°

For a rhombus, γ = α and δ = β applies. Replacing γ and δ in the above equation, we get:

2 ∙ (α + β) = 360°

If you now divide both sides by 2, you get:

α+β=180°

With the help of this formula and the knowledge that opposite interior angles are equal, if one interior angle is known, the remaining 3 interior angles can be calculated.

Example:

The interior angle α of a rhombus has a size of 40°. The angles β, γ and δ are to be calculated.

The angle β can be calculated as follows: β=180°α=180°40°=140°

Since γ = α and δ = β, the following applies to γ and δ:

γ = 40° and δ = 140°

## Calculate perimeter of a rhombus

The perimeter of a quadrilateral is calculated by summing the lengths of all 4 sides. Since all 4 sides of a rhombus have the same length, multiply the side length by 4.

P = 4 ∙ a

Example:

Assume that the sides of a rhombus each have a length of 3 cm. Then the perimeter of the rhombus is calculated as follows:

 P = 4 ∙ a = 4 ∙ 3 cm = 12 cm

## Length of the diagonals of a rhombus

The lengths d1 und d2 of the diagonals of a rhombus can be calculated with the help of the following 4 formulas:

Calculation of the diagonal length d1:

d1=2 ∙ a ∙ sin(
 β 2
)
resp.
d1=2 ∙ a ∙ cos(
 α 2
)

Berechnung der Diagonalenlänge f:

d2=2 ∙ a ∙ sin(
 α 2
)
resp.
d2=2 ∙ a ∙ cos(
 β 2
)

Example:

The sides of a rhombus are each 5 cm long and for the angles α and β α = 70° and β = 110° holds. The lengths d1 and d2 of the diagonals are to be calculated.

Calculation of d1:

d1=2 ∙ a ∙ sin(
 β 2
)
=2 ∙ 5 cm ∙ sin(
 110° 2
)
=10 cm ∙ sin(55°)
8.1915 cm

Calculation of d2:

d2=2 ∙ a ∙ sin(
 α 2
)
=2 ∙ 5 cm ∙ sin(
 70° 2
)
=10 cm ∙ sin(35°)
5.7358 cm

Derivation:

The two diagonals of a rhombus are at right angles to each other. If you draw into a rhombus the two diagonals with the lengths d1 and d2, then these partition the rhombus into 4 right-angled triangles. Each of the 4 right triangles has a hypotenuse of length a and catheti of lengths
 d1 2
and
 d2 2
.
The two diagonals run exactly in the middle of the angles of the rhombus. Thus the acute angles of the 4 triangles are
 α 2
and
 β 2
.

For right triangles, the sine and cosine for one of the acute angles are defined as follows:

sin(Winkel) =
 opposite side hypotenuse
und cos(Winkel) =

The adjacent side of the angle
 α 2
has the length
 d1 2
and the opposite side has the length
 d2 2
. The adjacent side of the angle
 β 2
has the length
 d2 2
and the opposite side has a length of length
 d1 2
.

If you substitute the angles and lengths into the formulas for the sine and cosine for right triangles, you get the following 4 formulas:

• sin( β 2
) =
( d1 2
)
a
• cos( α 2
) =
( d1 2
)
a
• sin( α 2
) =
( d2 2
)
a
• cos( β 2
) =
( d2 2
)
a

If you solve the 4 equations for d1 and d2, respectively, you get:

• d1=2 ∙ a ∙ sin( β 2
)
• d1=2 ∙ a ∙ cos( α 2
)
• d2=2 ∙ a ∙ sin( α 2
)
• d2=2 ∙ a ∙ cos( β 2
)

## Calculate side length with diagonals

If the two diagonal lengths d1 and d2 of a rhombus are known, but the side length a is not, then a can be calculated from d1 and d2. The following formula can be used for this purpose.

a =
 d1² + d2² 2

Example:

The two lengths of the diagonals of a rhombus are d1 = 4 cm and d2 = 3 cm. The side length a is to be calculated.

a=
 d1² + d2² 2
=
 (4 cm)² + (3 cm)² 2
=
 16 cm² + 9 cm² 2
=
 25 cm² 2
=
 5 cm 2
=2.5 cm

Derivation:

If you draw the diagonals d1 and d2 into a rhombus, then the rhombus is divided into 4 right triangles of equal size. Of each of these right triangles, the hypotenuse has length a and the two legs have lengths
 d1 2
and
 d2 2
.

Thus, according to the Pythagorean theorem:

a² = (
 d1 2
)² + (
 d2 2
)²

If you take the root from both sides and simplify the right side, you get:

a=
(
 d1 2
)² + (
 d2 2
)²
=
 d1² 4
+
 d2² 4
=
 1 4
∙ (d1² + d2²)
=
 1 4
(d1² + d2²)
=
 1 4
(d1² + d2²)
=
 (d1² + d2²) 2

## Calculate altitude of a rhombus

The altitude h of a rhombus can be calculated with the following formula:

h = a ∙ sin(α)

The altitude can be used to calculate the area or radius of the incircle.

Derivation:

The angle α can be less than 90°, 90° or greater than 90°.

Case α = 90°:

The simplest case is that the angle α has a magnitude of 90°. Then the rhombus is a square. If you substitute 90° into the formula for calculating the altitude of a rhombus, you get:

h = a ∙ sin(90°) = a ∙ 1 = a

a is also the altitude of a square and thus the formula is correct for the case α = 90°.

Case α < 90°:

If you draw the altitude h so that the altitude ends in the vertex D, then this creates a right-angled triangle. The hypotenuse of this triangle is a and the opposite side of α is the altitude.

For the acute angles in a right triangle holds:

sin(angle) =
 opposite side hypotenuse

If you substitute α, a and h into this formula, you get:

sin(α) =
 h a

Solving this for h gives:

h = a ∙ sin(α)

Case α > 90°:

In the case that α is less than 90°, you can draw the altitude h in such a way that it ends in the vertex C. Then a right triangle is formed, where the hypotenuse has the length a and the opposite side of β has the length h.

This is substituted into the formula for the sine of right triangles:

sin(β) =
 h a

After that, the equation is solved for h again:

h = a ∙ sin(β)

For an angle that is less than or equal to 180°:

sin(angle) = sin(180° - angle)

Moreover, in a rhombus β = 180° − α applies. This allows the above equation for altitude to be transformed even further:

h = a ∙ sin(β) = a ∙ sin(180° − α) = a ∙ sin(α)

## Calculate area of a rhombus

The area of a rhombus can be calculated either with the help of the altitude h or with the help of the two lengths d1 and d1 of the diagonals.

### Calculation with the help of the altitude:

If you want to use the altitude h to calculate the area of a rhombus, you must first calculate the altitude using the formula for the altitude:

h = a ∙ sin(α)

And then multiply the altitude by the side length a:

A = a ∙ h

Example:

The sides of a rhombus each have a length of 5 cm. The angle α is 60°. The area is to be calculated with the help of the altitude of the rhombus:

First, the altitude h must be calculated. For this, the side length a is multiplied by the sine of the angle α:

 h = a ∙ sin(α) = 5 cm ∙ sin(60°) ≈ 4.33013 cm

The result is multiplied by the side length a:

 A = a ∙ h ≈ 5 cm ∙ 4.33013 cm ≈ 21.65065 cm

The two steps can also be combined into one step. For this, the formula for the altitude is inserted into the formula for the area:

A = a² ∙ sin(α)

Derivation:

If you draw the altitude h so that one end ends in a vertex and the other on one of the sides, a right triangle is formed inside the rhombus. This can be moved to the other side of the rhombus, forming a rectangle with side lengths a and h.

This rectangle has the area a ∙ h. Since this rectangle and the rhombus have the same area, because only a part of the rhombus has been moved, the area A of the rhombus is also A = a ∙ h.

### Calculation with the lengths of the diagonals:

If the lengths of the two diagonals are known, then the area can be calculated by multiplying the two diagonal lengths and halving the result:

A =
 d1 ∙ d2 2

Example:

The lengths of the diagonals of a rhombus are d1 = 2 cm and d2 = 4 cm. The area is to be calculated.

A=
 d1 ∙ d2 2
=
 2 cm ∙ 4 cm 2
=
 8 cm² 2
=4 cm²

Derivation:

If you draw the two diagonals into a rhombus, then 4 right-angled triangles are formed inside the rhombus.

For example, if you move the two triangles below from the diagonal with length e so that they form a rectangle together with the other two triangles, then a rectangle with side lengths d1 and
 d2 2
is formed.

This has the area d1
 d2 2
=
 d1 ∙ d2 2
. Since only parts of the rhombus were moved to form the rectangle, A =
 d1 ∙ d2 2
also applies to the area of the rhombus.

The incircle of a rhombus is a circle that lies inside the rhombus and touches each side of the rhombus at exactly one point. The center of the incircle is exactly the same as the center of the rhombus. This is the point where the two diagonals intersect.

The altitude h is as long as the diameter of the incircle.

Thus, the following applies to the inradius r:

r =
 h 2

Example:

The sides of a rhombus each have a length of 5 cm. The angle α is 60°. The radius of the incircle is to be calculated:

The first thing to do is to calculate the altitude h:

 h = a ∙ sin(α) = 5 cm ∙ sin(60°) ≈ 4.33013 cm

Then divide the altitude by 2 to get the incircle radius:

r=
 h 2

 4.33013 cm 2
2.165065 cm²