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Parallelogram

Parallelogram

A quadrilateral with two pairs of parallel sides.

 

Parallelogram with vertices A, B, C, D; height h drawn inside, base b labeled along bottom. Area = hb = (AB)(AD)sinA =...

 

 

See also

Height of a parallelogram, base, area of a parallelogram, sine, polygon

Key Formula

A=b×hA = b \times h
Where:
  • AA = Area of the parallelogram
  • bb = Length of the base (any one side)
  • hh = Height (perpendicular distance from the base to the opposite side)

Worked Example

Problem: A parallelogram has a base of 10 cm and a height of 6 cm. Find its area.
Step 1: Identify the base and height. The base is the side you choose to measure along, and the height is the perpendicular distance to the opposite side.
b=10 cm,h=6 cmb = 10 \text{ cm}, \quad h = 6 \text{ cm}
Step 2: Apply the area formula for a parallelogram.
A=b×hA = b \times h
Step 3: Substitute the values and compute.
A=10×6=60 cm2A = 10 \times 6 = 60 \text{ cm}^2
Answer: The area of the parallelogram is 60 cm².

Another Example

This example uses the sine-based area formula, which is needed when the perpendicular height is not given but an angle between two sides is known.

Problem: A parallelogram has sides of length 8 cm and 5 cm, and the angle between them is 30°. Find its area.
Step 1: When the height is not given directly, you can use the alternative area formula involving the sine of the included angle.
A=a×b×sin⁡(θ)A = a \times b \times \sin(\theta)
Step 2: Identify the two adjacent sides and the included angle.
a=8 cm,b=5 cm,θ=30°a = 8 \text{ cm}, \quad b = 5 \text{ cm}, \quad \theta = 30°
Step 3: Recall that sin(30°) = 0.5.
sin⁡(30°)=0.5\sin(30°) = 0.5
Step 4: Substitute and compute.
A=8×5×0.5=20 cm2A = 8 \times 5 \times 0.5 = 20 \text{ cm}^2
Answer: The area of the parallelogram is 20 cm².

Frequently Asked Questions

Is a rectangle a parallelogram?
Yes. A rectangle has two pairs of parallel sides, so it meets the definition of a parallelogram. A rectangle is a special case where all four angles are 90°. Every rectangle is a parallelogram, but not every parallelogram is a rectangle.
What is the difference between a parallelogram and a trapezoid?
A parallelogram has two pairs of parallel sides, while a trapezoid (trapezium in British English) has exactly one pair of parallel sides. Because a parallelogram satisfies a stricter condition, every parallelogram could be considered a special trapezoid under the inclusive definition, but not vice versa.
What are the properties of a parallelogram?
Opposite sides are parallel and equal in length. Opposite angles are equal, and consecutive angles add up to 180°. The diagonals bisect each other, meaning they cut each other exactly in half. The area equals base times height.

Parallelogram vs. Rectangle

ParallelogramRectangle
DefinitionQuadrilateral with two pairs of parallel sidesParallelogram with all four angles equal to 90°
AnglesOpposite angles equal; consecutive angles supplementary (not necessarily 90°)All angles are exactly 90°
DiagonalsBisect each other but are generally unequal in lengthBisect each other and are equal in length
Area formulaA = b × h (h is perpendicular height)A = l × w (length times width, since sides are already perpendicular)
Special case?General form; includes rectangles, rhombuses, and squaresA special type of parallelogram

Why It Matters

Parallelograms appear throughout geometry courses when you study area, coordinate proofs, and vector addition. In physics, the parallelogram law of vector addition uses the shape to find the resultant of two forces or velocities. Understanding parallelogram properties also helps you classify other quadrilaterals—rectangles, rhombuses, and squares are all special cases of parallelograms.

Common Mistakes

Mistake: Using the slanted side length as the height when calculating area.
Correction: The height must be the perpendicular distance from the base to the opposite side, not the length of the slanted side. If the slanted side and an angle are given, use h = side × sin(angle) to find the true height.
Mistake: Assuming the diagonals of a parallelogram are equal in length.
Correction: The diagonals of a general parallelogram bisect each other but are not equal. Equal diagonals occur only in the special case of a rectangle.

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