Calculate the area of a parallelogram from base and height, two sides and the included angle, or two diagonals and the angle between them, in any unit.
Parallelogram Area Formula
The area of a parallelogram can be found three ways depending on what you know. The most common method uses the base and the perpendicular height.
A = b * h
If you know two adjacent sides and the angle between them, use:
A = a * b * sin(theta)
If you know the two diagonals and the angle between them, use:
A = (1/2) * d1 * d2 * sin(theta)
- A = area of the parallelogram (square units)
- b = length of the base
- h = perpendicular height measured at a right angle from the base to the opposite side
- a = length of the side adjacent to the base
- theta = the included angle between the two sides, or the angle between the two diagonals
- d1, d2 = the lengths of the two diagonals
The base and height method multiplies the base by the perpendicular height, not the slanted side. The side and angle method works when you have the two side lengths but not the height, since a*sin(theta) gives that height. The diagonal method works when you only have the crossing diagonals and the angle they form. The calculator can also solve in reverse: enter the area and the height to get the base, or enter the area and the base to get the height.
Choosing the Right Method
Use the table below to pick the inputs that match what you already measured.
| You know | Method | Formula |
|---|---|---|
| Base and perpendicular height | Base and height | A = b * h |
| Two sides and the angle between them | Two sides and included angle | A = a * b * sin(theta) |
| Two diagonals and their angle | Two diagonals and angle | A = 0.5 * d1 * d2 * sin(theta) |
Area always comes out in square units. The table below shows how the unit you enter maps to the area unit.
| Length unit | Area unit |
|---|---|
| inches (in) | square inches (in²) |
| feet (ft) | square feet (ft²) |
| centimeters (cm) | square centimeters (cm²) |
| meters (m) | square meters (m²) |
Example Problems
Example 1. A parallelogram has a base of 8 cm and a perpendicular height of 5 cm. Using A = b * h, the area is 8 * 5 = 40 square centimeters.
Example 2. A parallelogram has sides of 6 in and 9 in with an included angle of 30 degrees. Using A = a * b * sin(theta), the area is 6 * 9 * sin(30) = 54 * 0.5 = 27 square inches.
Frequently Asked Questions
Do I use the slanted side or the height in the base times height formula? You use the perpendicular height, which is the straight-line distance between the base and the opposite side measured at a right angle. The slanted side is longer than the height, so using it would overstate the area. If you only have the slanted side and an angle, switch to the two sides and included angle method instead.
Why is the area of a parallelogram the same as a rectangle with the same base and height? You can cut a right triangle off one end of a parallelogram and slide it to the other end to form a rectangle. The base and height stay the same, so the area is identical. That is why A = b * h works for both shapes.
What happens to the area when the angle is 90 degrees? When the included angle is 90 degrees, sin(90) = 1, so A = a * b. The parallelogram is a rectangle in that case, and the two sides act as the base and height directly.
