Parallelogram Calculator

Table of Contents

• Parallelogram definition
• Parts of a Parallelogram
  • Base length (a)
  • Side length (b)
  • Height (ha)
  • Area (A)
  • Perimeter (P)
  • Smaller angle (α)
  • Larger angle (β)
• Diagonals
• Example
• Frequently asked questions

Parallelogram definition

A parallelogram is a quadrilateral, or a four-sided polygon, where both pairs of opposite sides are parallel and equal in length. It can be represented as a flat shape with four sides, two pairs of parallel lines, and four angles.

Every parallelogram has these properties:

• Opposite sides are parallel and equal in length.
• Opposite angles are equal, and two neighbouring angles add up to 180°.
• The diagonals cut each other in half.
• Each diagonal divides it into two equal triangles.

Rectangles, squares and rhombuses are all special parallelograms.

Parts of a Parallelogram

The seven parts of a parallelogram mentioned in this tool are base length, side length, height, area, perimeter, smaller angle and larger angle.

Base length (a)

The term base refers to the length of one side and "height" to the length of a perpendicular segment between that side and the opposite parallel side. Any side of a parallelogram can be a base.

It can be found with the following formulas:

• If you know the area and height of the parallelogram:

  a = A / ha

• If you know the perimeter and the side length:

  a = (P - 2 * b) / 2

• If you know the area, side length and one of the angles:

  a = A / b / sin(β), or a = A / b / sin(α)

Side length (b)

The side length of a parallelogram is the length of its adjacent side to the base. It is usually denoted by b.

You can determine it with these formulas:

• If you know the perimeter and base length:

  b = (P - 2 * a) / 2

• If you know the height and one of the angles:

  b = ha / sin(α), or b = ha / sin(β)

• If you know the area, base length and one of the angles:

  b = A / a / sin(β), or b = A / a / sin(α)

Height (ha)

The height of a parallelogram is the distance between its two parallel sides. It is perpendicular to the base and is usually denoted by h.

• If you know the base length and area:

  ha = A / a

• If you know the side length and one of the angles:

  ha = b * sin(β), or ha = b * sin(α)

Area (A)

The area of a parallelogram can be calculated by multiplying its base length and height. The formula to calculate the area of a parallelogram is given by:

A = a * ha

Perimeter (P)

The perimeter of a parallelogram is the total length of its four sides. The formula to calculate the perimeter of a parallelogram is given by:

P = 2 * (a + b)

Smaller angle (α)

In a parallelogram, there exist two pairs of angles. Among these, the lesser angles refer to the smaller ones. Since the total sum of angles within a parallelogram equals 360°, and there are two pairs of congruent angles, it follows that the sum of the lesser angle and its counterpart is 180°.

Thus, the calculation for the smaller angle is as follows:

α = 180 - β

However, if you do not know the size of the larger angle, you can calculate the smaller angle using the height and side length of the parallelogram:

α = asin(ha / b)

Larger angle (β)

Similarly to the smaller angle mentioned above, the larger angle can be determined by subtracting the smaller angle from 180°.

β = 180 - α

Diagonals

A parallelogram has a long diagonal (p) and a short one (q). They follow from the two sides and the angle between them:

p = √(a² + b² + 2 * a * b * cos(α))

q = √(a² + b² - 2 * a * b * cos(α))

Together, p² + q² = 2 * (a² + b²). The calculator doesn't show the diagonals, so use these formulas with its side and angle values.

Example

A parallelogram with a base of 8, a side of 5 and a smaller angle of 60°:

1. Height:

   ha = 5 * sin(60°) = 4.33

2. Area:

   A = 8 * 4.33 = 34.64

3. Perimeter:

   P = 2 * (8 + 5) = 26

4. Larger angle:

   β = 180 - 60 = 120°

5. Diagonals: with cos(60°) = 0.5, the products 2 * 8 * 5 * 0.5 are 40.

   p = √(64 + 25 + 40) = √129 = 11.36

   q = √(64 + 25 - 40) = √49 = 7

Frequently asked questions

How do you find the angles of a parallelogram?

From a side and the height on the base: α = asin(ha / b), and the other angle is β = 180 - α. For example, a side of 5 and a height of 4 give α = asin(0.8) = 53.13° and β = 126.9°. The four angles always add up to 360°.

Is the area of a parallelogram base times side?

No, it is base times height, the perpendicular distance between the base and the opposite side: A = a * ha. Base times side gives the area only when the angles are 90°, in a rectangle.