# Parallelogram Calculator

## Parallelogram Shape

a = side a lengths

b = side b lengths (base)

p = shorter diagonal length

q = longer diagonal length

h = height

A, B, C, D = corner angles

K = area

P = perimeter

π = pi = 3.1415926535898

√ = square root

## Calculator Use

Calculate certain variables of a parallelogram depending on the inputs provided. Calculations include side lengths, corner angles, diagonals, height, perimeter and area of parallelograms.

A parallelogram is a quadrilateral with opposite sides parallel. A parallelogram whose angles are all right angles is called a rectangle. And, a parallelogram whose angles are all right angels and whose sides are all equal is called a square. A rhombus (or diamond) is a parallelogram with all 4 sides equal length.

**Units:** Note that units of length are shown for convenience. They do not affect the calculations. The units are in place to give an indication of the order of the calculated results such as ft, ft^{2} or ft^{3}. Any other base unit can be substituted.

## Parallelogram Formulas & Constraints

### Corner Angles: A, B, C, D

- A = C
- B = D
- A + B = 180° = π radians
- for a parallelogram that is not a rectangle or square,
- 0 < A< 90° (0 < A < π/2),
- 90° < B < 180° (π/2 < B < π)

### Area: K

with A and B in radians,

K = bh = ab sin(A) = ab sin(B)

### Height: h

h = a sin(A) = a sin(B)

### Diagonals: p, q

- p = √( a
^{2}+ b^{2}- 2ab cos(A) ) = √( a^{2}+ b^{2}+ 2ab cos(B) ) - q = √( a
^{2}+ b^{2}+ 2ab cos(A) ) = √( a^{2}+ b^{2}- 2ab cos(B) ) - p
^{2}+ q^{2}= 2(a^{2}+ b^{2})

### Perimeter: P

P = 2a + 2b

## Parallelogram Calculations:

The following formulas, based on those above, are used within this calculator for the selected calculation choices.

**Calculate B, C, D | Given A**

Given angle A calculate angles B, C and D- B = 180° - A
- C = A
- D = B

**Calculate A, C, D | Given B**

Given angle B calculate angles A, C and D- A = 180° - B
- C = A
- D = B

**Calculate h, B, C, D | Given A, a**

Given angle A and side a calculate height and angles B, C and D- h = a sin(A)
- B = 180° - A
- C = A
- D = B

**Calculate a, B, C, D | Given A, h**

Given angle A and height calculate side a and angles B, C and D- a = h / sin(A)
- B = 180° - A
- C = A
- D = B

**Calculate b | Given P, a**

Given the perimeter and side a calculate side b- b = (P - 2a) / 2

**Calculate a | Given P, b**

Given the perimeter and side b calculate side a- a = (P - 2b) / 2

**Calculate P | Given a, b**

Given side lengths calculate the perimeter- P = 2a + 2b

**Calculate h | Given K, b**

Given area and side b calculate height- h = K / b

**Calculate b | Given K, h**

Given area and height calculate side b- b = K / h

**Calculate K | Given b, h**

Given side b and height calculate area- K = bh

**Calculate B, p, q, h, P, K | Given a, b, A**

Given side lengths and angle A calculate the diagonals, perimeter, height, area and angles B, C and D- p = √( a
^{2}+ b^{2}- 2ab cos(A) ) - q = √( a
^{2}+ b^{2}+ 2ab cos(A) ) - P = 2a + 2b
- h = a sin(A)
- K = bh
- B = 180° - A
- C = A
- D = B

- p = √( a
**Calculate A, B, q, h, P, K | Given a, b, p**

Given side lengths and diagonal p calculate diagonal q, perimeter, height, area and angles A, B, C and D- A = arccos( (p
^{2}- a^{2}- b^{2}) / (-2ab) ) - q = √( a
^{2}+ b^{2}+ 2ab cos(A) ) - h = a sin(A)
- P = 2a + 2b
- K = ab sin(A)
- B = 180° - A
- C = A
- D = B

- A = arccos( (p
**Calculate A, B, p, h, P, K | Given a, b, q**

Given side lengths and diagonal q calculate diagonal p, perimeter, height, area and angles A, B, C and D- A = arccos( (q
^{2}- a^{2}- b^{2}) / (2ab) ) - p = √( a
^{2}+ b^{2}- 2ab cos(A) ) - h = a sin(A)
- P = 2a + 2b
- K = ab sin(A)
- B = 180° - A
- C = A
- D = B

- A = arccos( (q
**Calculate A, B, p, q, P, K | Given a, b, h**

Given side lengths and height calculate the diagonals, perimeter, area and angles A, B, C and D- A = arcsin(h/a)
- p = √( a
^{2}+ b^{2}- 2ab cos(A) ) - q = √( a
^{2}+ b^{2}+ 2ab cos(A) ) - P = 2a + 2b
- K = ab sin(A)
- B = 180° - A
- C = A
- D = B

**Calculate A, B, p, q, h, P | Given a, b, K**

Given side lengths and area calculate the diagonals, perimeter, height and angles A, B, C and D- A = arcsin(K/ab)
- p = √( a
^{2}+ b^{2}- 2ab cos(A) ) - q = √( a
^{2}+ b^{2}+ 2ab cos(A) ) - h = a sin(A)
- P = 2a + 2b
- B = 180° - A
- C = A
- D = B

**Calculate B, b, p, q, h, P | Given a, A, K**

Given side length a, angle A and area calculate the diagonals, perimeter, height, side length b and angles B, C and D- b = K / (a sin(A))
- p = √( a
^{2}+ b^{2}- 2ab cos(A) ) - q = √( a
^{2}+ b^{2}+ 2ab cos(A) ) - h = a sin(A)
- P = 2a + 2b
- B = 180° - A
- C = A
- D = B

**Calculate B, a, p, q, h, P | Given b, A, K**

Given side length b, angle A and area calculate the diagonals, perimeter, height, side length a and angles B, C and D- a = K / (b sin(A))
- p = √( a
^{2}+ b^{2}- 2ab cos(A) ) - q = √( a
^{2}+ b^{2}+ 2ab cos(A) ) - h = a sin(A)
- P = 2a + 2b
- B = 180° - A
- C = A
- D = B

**Calculate A, B, b, h, P, K | Given a, p, q**

Given side length a and the diagonals calculate the perimeter, height, area, side length b and angles A, B, C and D- b = √( (p
^{2}+ q^{2}- 2a^{2}) / 2 ) - A = arccos( (q
^{2}- a^{2}- b^{2}) / (2ab) ) - h = a sin(A)
- P = 2a + 2b
- K = ab sin(A)
- B = 180° - A
- C = A
- D = B

- b = √( (p
**Calculate A, B, a, h, P, K | Given b, p, q**

Given side length b and the diagonals calculate the perimeter, height, area, side length a and angles A, B, C and D- a = √( (p
^{2}+ q^{2}- 2b^{2}) / 2 ) - A = arccos( (q
^{2}- a^{2}- b^{2}) / (2ab) ) - h = a sin(A)
- P = 2a + 2b
- K = ab sin(A)
- B = 180° - A
- C = A
- D = B

- a = √( (p

## References

Zwillinger, Daniel (Editor-in-Chief).
*CRC Standard Mathematical Tables and Formulae, 31st Edition* New York, NY: CRC Press, p. 322, 2003.

Weisstein, Eric W. "Parallelogram." From
*MathWorld*--A Wolfram Web Resource.
Parallelogram.