To find the area of a parallelogram, you can use several formulas depending on the information available:
Basic Formula Using Base and Height
The most common method is to multiply the length of the base by the perpendicular height (altitude):
Area=base×height\text{Area}=\text{base}\times \text{height}Area=base×height
- Base (b): Any one side of the parallelogram chosen as the base.
- Height (h): The perpendicular distance from the base to the opposite side.
This formula requires the height to be perpendicular to the base. The area is expressed in square units (e.g., cm², m²)
Using Adjacent Sides and the Included Angle
If the height is not known but you know the lengths of two adjacent sides 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)
where aaa and bbb are the lengths of adjacent sides, and θ\theta θ is the angle between those sides
Using Diagonals and the Angle Between Them
If you know the lengths of the diagonals d1d_1d1 and d2d_2d2 and the angle xxx between them, the area can be found by:
Area=12×d1×d2×sin(x)\text{Area}=\frac{1}{2}\times d_1\times d_2\times \sin(x)Area=21×d1×d2×sin(x)
This formula is useful when diagonal lengths and the angle between diagonals are given
Using Vectors (Advanced)
If the sides are given as vectors a\mathbf{a}a and b\mathbf{b}b, the area is the magnitude of their cross product:
Area=∣a×b∣\text{Area}=|\mathbf{a}\times \mathbf{b}|Area=∣a×b∣
Similarly, using diagonal vectors d1\mathbf{d_1}d1 and d2\mathbf{d_2}d2:
Area=12∣d1×d2∣\text{Area}=\frac{1}{2}|\mathbf{d_1}\times \mathbf{d_2}|Area=21∣d1×d2∣
This method applies in coordinate geometry or physics contexts
Summary
- With base and height: Area=b×h\text{Area}=b\times hArea=b×h
- With adjacent sides and angle: Area=a×b×sin(θ)\text{Area}=a\times b\times \sin(\theta)Area=a×b×sin(θ)
- With diagonals and angle between them: Area=12×d1×d2×sin(x)\text{Area}=\frac{1}{2}\times d_1\times d_2\times \sin(x)Area=21×d1×d2×sin(x)
These formulas allow you to find the area of a parallelogram depending on what measurements you have available
