Parallelogram Area Calculator
Calculate the area of a parallelogram from its base and perpendicular height, or from two adjacent sides and the angle between them. You can also solve for a missing base or height, calculate perimeter, and view the formula step by step.
Calculate a Parallelogram
Choose the information you know, enter your measurements, and calculate.
Parallelogram Diagram
The height is measured at 90° to the base. Changing the slant does not change the area if the base and perpendicular height remain the same.
Calculation Steps
- Identify the base and perpendicular height.
- Multiply the base by the height.
- Area = 12 × 5 = 60 cm².
Quick Formula
Base multiplied by perpendicular height.
Use two adjacent sides and their included angle when height is unknown.
Perimeter is twice the sum of the two adjacent side lengths.
Parallelogram Area Formula
The area of a parallelogram is the product of a chosen base and the perpendicular distance between that base and the opposite parallel side. This is the same base-times-height relationship used for a rectangle.
A = b × h
Where A is area, b is the base length, and h is the perpendicular height.
Area when the height is not known
If two adjacent side lengths and the angle between them are known, the height can be found using trigonometry. For base b, adjacent side s, and included angle θ, the perpendicular height is s × sin(θ).
Use the angle in degrees. The angle must be between the two measured sides.
Solving for a missing measurement
| Unknown | Formula | Requirements |
|---|---|---|
| Area | A = b × h | Base and perpendicular height |
| Height | h = A ÷ b | Area and nonzero base |
| Base | b = A ÷ h | Area and nonzero height |
| Area from sides and angle | A = b × s × sin(θ) | Two adjacent sides and included angle |
| Perimeter | P = 2(b + s) | Two adjacent side lengths |
Worked Examples
Example 1: Base and height
Base = 12 cm
Height = 5 cm
A = 12 × 5
Area = 60 cm²Example 2: Sides and angle
Base = 8 m
Side = 5 m, angle = 30°
A = 8 × 5 × sin(30°)
Area = 20 m²Example 3: Find the height
Area = 72 cm²
Base = 9 cm
h = 72 ÷ 9
Height = 8 cmHow to Calculate the Area of a Parallelogram
- Choose the base. Any side can be selected as the base.
- Measure the perpendicular height. Measure the shortest distance at a right angle to the base and its opposite parallel side.
- Use the appropriate formula. Multiply base by height, or use two adjacent sides and their included angle.
- Include square units. If the measurements are in centimetres, report the area in square centimetres.
- Check the result. The area should be nonnegative and consistent with the dimensions supplied.
If you only know the two side lengths and the angle between them, the calculator uses the sine of the angle to derive the perpendicular height. A smaller angle generally produces a shorter height and therefore a smaller area for the same side lengths.
Properties of a Parallelogram
- Both pairs of opposite sides are parallel.
- Opposite sides have equal lengths.
- Opposite angles are equal.
- Adjacent interior angles add up to 180°.
- The diagonals bisect each other.
- A rectangle, rhombus, and square are special types of parallelograms.
A parallelogram with a 90° interior angle is a rectangle. A rhombus is a parallelogram with four equal sides. The same base-times-height area formula applies to both.
Common Mistakes to Avoid
- Using the sloping side as height: the height must be perpendicular to the base.
- Using the wrong angle: the sides-and-angle formula requires the included angle between the two adjacent sides.
- Mixing units: convert all length measurements to the same unit before calculating.
- Forgetting square units: area is expressed in units such as cm², m², or ft².
- Using an invalid angle: for a non-degenerate parallelogram, the included angle must be greater than 0° and less than 180°.
Frequently Asked Questions
1. What is the area formula for a parallelogram?
The standard formula is A = b × h, where b is the base and h is the perpendicular height. Multiply these two measurements to get the area.
2. How do I find the area without the height?
If you know two adjacent side lengths and the angle between them, use A = b × s × sin(θ). Enter the included angle in degrees.
3. Why is the height different from the side length?
The side length follows the sloping edge, while height is the perpendicular distance between the parallel sides. They are equal only in special cases such as a rectangle when measuring the corresponding side.
4. How do I find a missing base?
Divide the known area by the corresponding perpendicular height: b = A ÷ h. Make sure the height is not zero.
5. How do I find a missing height?
Divide the known area by the base: h = A ÷ b. The base must be greater than zero.
6. How do I calculate the perimeter?
Use P = 2(a + b), where a and b are the two adjacent side lengths. Perimeter is a length, not an area.
7. Can a parallelogram have an angle greater than 90°?
Yes. A parallelogram can have an obtuse interior angle. Adjacent interior angles add to 180°, and the sides-and-angle area formula works for included angles between 0° and 180°.
8. What units should I use for area?
Area uses square units. For example, a base in metres and height in metres produce an area in square metres. If you use another length unit, the result is in that unit squared.
Learn More About Parallelogram Area
The base-times-height rule follows from the fact that a triangular piece can be moved from one side of a parallelogram to the other to form a rectangle with the same base and height. This rearrangement preserves area.
For further reading, see these geometry resources: