how do you find the area of a parallelogram

To find the area of a parallelogram, the most common formula is: Area = base × height

• The base (b) is the length of one side of the parallelogram.
• The height (h) is the perpendicular distance from the base to the opposite side (not the side length if the parallelogram is slanted).

This formula works because a parallelogram can be rearranged into a rectangle with the same base and height, so their areas are equal

How to apply the formula:

1. Identify the base of the parallelogram.
2. Measure the height, which must be perpendicular to the base.
3. Multiply the base length by the height.

Example:

If the base is 10 cm and the height is 8 cm, then the area is:

Area=10×8=80 cm2\text{Area}=10\times 8=80\text{ cm}^2Area=10×8=80 cm2

Alternative formulas if height is unknown:

• If you know the lengths of two adjacent sides aaa and bbb and the angle θ\theta θ between them, use:

Area=a×b×sin⁡(θ)\text{Area}=a\times b\times \sin(\theta)Area=a×b×sin(θ)

• If you know the lengths of the diagonals d1d_1d1​ and d2d_2d2​ and the angle ϕ\phi ϕ between them, use:

Area=12×d1×d2×sin⁡(ϕ)\text{Area}=\frac{1}{2}\times d_1\times d_2\times \sin(\phi)Area=21​×d1​×d2​×sin(ϕ)

These formulas come from trigonometry and are useful when the height is not directly known

. Summary:

Known values| Formula
---|---
Base and height| Area=b×h\text{Area}=b\times hArea=b×h
Two sides and included angle| Area=a×b×sin⁡(θ)\text{Area}=a\times b\times \sin(\theta)Area=a×b×sin(θ)
Diagonals and angle between| Area=12×d1×d2×sin⁡(ϕ)\text{Area}=\frac{1}{2}\times d_1\times d_2\times \sin(\phi)Area=21​×d1​×d2​×sin(ϕ)

This allows you to calculate the area of any parallelogram using whichever measurements are available.