Parallelogram Calculator

Calculation With:

Corner Angle b =

Perimeter h =

pi π =

Units:

The online Parallelogram Calculator allows you to calculate all key parameters of a parallelogram based on given inputs. Before using the tool, let's explore the geometry of this quadrilateral.

What Is a Parallelogram?

A quadrilateral with opposite sides parallel to each other is called a parallelogram.

parallelogram

In the figure:

a and b are the side lengths (b is considered the base)
p and q are the lengths of the longer and shorter diagonals, respectively
h represents the height
P and K are the perimeter and area, respectively
A, B, C, D are the vertices with angles ∠A, ∠B, ∠C, ∠D

Important Formulas

Angles of Corners

Let ∠A, ∠B, ∠C, ∠D be the angles at vertices A, B, C, D respectively:

$$ ∠A + ∠B = 180^\circ $$

$$ ∠A = 180^\circ - ∠B, \quad ∠B = 180^\circ - ∠A $$

Also:

$$ ∠A = ∠C, \quad ∠B = ∠D $$

For non-rectangular parallelograms:

$$ 0 < ∠A < 90^\circ, \quad 90^\circ < ∠B < 180^\circ $$

Area

Given base b and height h, or sides and angles:

$$ K = b \cdot h = a \cdot b \cdot \sin(A) = a \cdot b \cdot \sin(B) $$

Height

$$ h = a \cdot \sin(A) = a \cdot \sin(B) $$

Diagonals

Long diagonal p (A to C):

$$ p = \sqrt{a^2 + b^2 - 2ab \cos(A)} = \sqrt{a^2 + b^2 + 2ab \cos(B)} $$

Short diagonal q (B to D):

$$ q = \sqrt{a^2 + b^2 + 2ab \cos(A)} = \sqrt{a^2 + b^2 - 2ab \cos(B)} $$

Also:

$$ p^2 + q^2 = 2(a^2 + b^2) $$

Perimeter

$$ P = 2a + 2b $$

Common Conversions

Given ∠A:

$$ ∠B = 180^\circ - ∠A, \quad ∠C = ∠A, \quad ∠D = ∠B $$

Given ∠B:

$$ ∠A = 180^\circ - ∠B, \quad ∠C = ∠A, \quad ∠D = ∠B $$

Given ∠A and a:

$$ h = a \cdot \sin(A), \quad ∠B = 180^\circ - ∠A, \quad ∠C = ∠A, \quad ∠D = ∠B $$

Given ∠A and h:

$$ a = \frac{h}{\sin(A)}, \quad ∠B = 180^\circ - ∠A, \quad ∠C = ∠A, \quad ∠D = ∠B $$

Given P and a:

$$ b = \frac{P - 2a}{2} $$

Given P and b:

$$ a = \frac{P - 2b}{2} $$

Given K and b:

$$ h = \frac{K}{b} $$

Given K and h:

$$ b = \frac{K}{h} $$

Given b and h:

$$ K = b \cdot h $$

Given a, b, and ∠A:

$$ p = \sqrt{a^2 + b^2 - 2ab \cos(A)}, \quad q = \sqrt{a^2 + b^2 + 2ab \cos(A)} $$

Given a, b, and p:

$$ ∠A = \arccos\left(\frac{p^2 - a^2 - b^2}{-2ab}\right) $$

Given a, b, and q:

$$ ∠A = \arccos\left(\frac{q^2 - a^2 - b^2}{2ab}\right) $$

Given a, b, and h:

$$ ∠A = \arcsin\left(\frac{h}{a}\right) $$

Given a, b, and K:

$$ ∠A = \arcsin\left(\frac{K}{ab}\right) $$

Given a, ∠A, and K:

$$ b = \frac{K}{a \cdot \sin(A)} $$

Given a, p, and q:

$$ b = \frac{\sqrt{p^2 + q^2 - 2a^2}}{2} $$

Given b, p, and q:

$$ a = \frac{\sqrt{p^2 + q^2 - 2b^2}}{2} $$

Example Calculations

Example 1: Find perimeter

Given: a = 2 cm, b = 4 cm

$$ P = 2a + 2b = 2*2 + 2*4 = 12 \text{ cm} $$

Example 2: Find remaining angles

Given: ∠A = 113°

$$ ∠B = 180° - 113° = 67° $$

$$ ∠C = ∠A = 113°, \quad ∠D = ∠B = 67° $$

Converting to radians: $$ ∠B = 1.139 \text{ rad}, \quad ∠C = 1.972 \text{ rad}, \quad ∠D = 1.139 \text{ rad} $$

Example 3: Find area

Given: base b = 6 cm, height h = 4 cm

$$ K = b \cdot h = 6*4 = 24 \text{ cm²} $$

Example 4: Find side a

Given: P = 6.2 cm, b = 2 cm

$$ a = \frac{P - 2b}{2} = \frac{6.2 - 4}{2} = 1.1 \text{ cm} $$

How the Calculator Works

Input:

Select “Calculation With” from the dropdown
Choose the parameter(s) to calculate remaining properties
Enter the values
Click “Calculate”

Output:

Sides, angles, and diagonals
Height
Perimeter
Area
Results displayed in a table

FAQs

Is a parallelogram a rectangle?

No, because the angles are not necessarily 90°.

What happens if one side changes?

It will no longer be a parallelogram.

Types of Quadrilaterals

Rectangle
Square
Parallelogram
Trapezium
Kite
Rhombus

Conclusion

Parallelograms are widely used in architecture and design. Accurate calculations are essential, and the online Parallelogram Calculator ensures precision and efficiency.

References

Wikipedia: Diagonal

Khan Academy: Quadrilateral Overview

Lumen Learning: Geometry Lessons