Area of Parallelogram Calculator
Enter the base and height of a parallelogram, two sides and the angle, or the diagonals, to find its area.
Updated
Area of Parallelogram Calculator
Our Area of Parallelogram Calculator multiplies the base by the height to give the area in square units. Only know two sides and the angle between them, or the two diagonals? Switch the input and it gives the area along with the height, perimeter, diagonals or sides, the angles and the type of parallelogram.
Quick answer
The area of a parallelogram is base × height: 12 cm × 8 cm = 96 cm².
- Height, not the slanted side: h is measured at a right angle to the base
- Two sides and an angle: A = a × b × sin θ, so 12, 8 and 60° give about 83.14
- Diagonals: A = ½ × d₁ × d₂ × sin φ
- Square units: centimetres give cm², feet give ft²
What Is a Parallelogram?
A parallelogram is a four-sided shape (a quadrilateral) whose opposite sides are parallel and equal in length. Any side can be the base b, and the height h is the straight-across distance from the base to the opposite side. Its area is the space inside, measured in square units such as cm², m² or ft².
Area of a Parallelogram Formula
- Measure the base. b = 12 cm.
- Measure the height at a right angle to the base. h = 8 cm.
- Multiply. 12 × 8 = 96.
- Write the answer in square units. 96 cm².
Why it works: cut the triangle off one slanted end and slide it to the other end. The pieces make a rectangle with the same base and height, so the area is the same b × h.
Parallelogram Area Examples
| Base and height | Area |
|---|---|
| 12 cm and 8 cm | 96 cm² |
| 15 m and 6 m | 90 m² |
| 20 ft and 11 ft | 220 ft² |
| 7.5 in and 4 in | 30 in² |
| 2.4 m and 1.5 m | 3.6 m² |
Area From Two Sides and an Angle
If you know two neighbouring sides a and b and the angle θ between them, the height on side a is b × sin θ, so:
12 × 8 × sin 60°
≈ 96 × 0.866025
≈ 83.14
| Sides 12 and 8 at | Area |
|---|---|
| 30° or 150° | 48 |
| 45° or 135° | ≈ 67.88 |
| 60° or 120° | ≈ 83.14 |
| 90° (rectangle) | 96 |
An angle and its partner 180° − θ give the same area, because the two angles of a parallelogram next to each other always add up to 180° and have the same sine. The area is largest at 90°, when the shape is a rectangle.
Area From the Diagonals
The diagonals of a parallelogram cut each other in half. With diagonals d₁ and d₂ crossing at angle φ:
½ × 10 × 8 × sin 90° = 40
½ × 10 × 8 × sin 60°
≈ 34.64
The sides follow from the diagonals too: a, b = ½√(d₁² + d₂² ± 2d₁d₂ cos φ). Diagonals 10 and 8 at 60° give sides of about 7.81 and 4.58. When the diagonals cross at 90° the shape is a rhombus and the area is simply d₁ × d₂ ÷ 2.
Rectangles, Rhombuses and Squares
- Rectangle: every angle is 90°, so the side is the height and A = length × width.
- Rhombus: all four sides are equal and the diagonals meet at 90°, so A = d₁ × d₂ ÷ 2 also works.
- Square: both at once, so A = side².
All three are parallelograms, so base × height gives the right area for each of them.
Height or Base From the Area
Rearrange the formula for the part you are missing:
96 ÷ 12 = 8 cm
b = A ÷ h
96 ÷ 8 = 12 cm
Common Mistakes
- Using the slanted side as the height. 12 × 10 is not the area if 10 is the slanted side; use the right-angle height.
- Multiplying the two sides. a × b is only right for a rectangle; otherwise multiply by sin θ too.
- Halving the answer. b × h ÷ 2 is a triangle. A parallelogram is the full b × h.
- Calculator in radians. sin 60 in radians is negative; set the angle unit to degrees.
- Mixing units. Convert everything to one unit before you multiply.
Key Terms
- Base
- The side the height is measured from, b.
- Height
- Distance from the base to the opposite side, at a right angle, h.
- Diagonals
- Lines joining opposite corners, d₁ and d₂.
- Angle θ
- The angle between two neighbouring sides.
How to Use the Area of Parallelogram Calculator
- Choose what you know. Pick Base & height, Sides & angle if you know two sides and the angle between them, or Diagonals if you know both diagonals and the angle where they cross.
- Enter the values. Type the lengths, such as 12 and 8, and pick a unit if you want one (mm, cm, m, km, in, ft, yd or mi). Angles can be in degrees or radians. Decimals and fractions such as 3/4 work.
- Read and copy the result. The area appears as you type, with a drawing to scale, the working, the area in other units and, for the other two modes, the height, perimeter, sides or diagonals, angles and type. Press Copy to copy it, or Clear or Esc to start again.
Example: a slanted patio slab has sides of 12 m and 8 m that meet at 60°. Choose Sides & angle, enter 12, 8 and 60 and pick m. The area is about 83.14 m², the height is about 6.93 m and the perimeter is 40 m.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
How do you find the area of a parallelogram?
Multiply the base by the height. A base of 12 cm and a height of 8 cm give 12 × 8 = 96 cm². The height must be measured at a right angle to the base.
What is the formula for the area of a parallelogram?
A = b × h, where b is the base and h is the perpendicular height. From two sides and the angle between them it is A = a × b × sin θ.
How do you find the area of a parallelogram without the height?
Use two neighbouring sides and the angle between them: A = a × b × sin θ. Sides of 12 and 8 at 60° give 96 × sin 60° ≈ 83.14. The height is b × sin θ ≈ 6.93.
How do you find the area of a parallelogram from its diagonals?
Multiply the diagonals, halve it and multiply by the sine of the angle between them: A = ½ × d₁ × d₂ × sin φ. Diagonals of 10 and 8 crossing at 90° give 40.
Is the slanted side the height of a parallelogram?
No. The slanted side is longer than the height unless the angle is 90°. The height is the shortest distance between the base and the opposite side.
How do you find the height of a parallelogram from its area?
Divide the area by the base: h = A ÷ b. An area of 96 cm² on a 12 cm base gives a height of 8 cm.
Why is the area of a parallelogram base times height?
Cut the triangle off one slanted end and move it to the other end. The pieces form a rectangle with the same base and height, so the area is b × h.
Is a rectangle a parallelogram?
Yes. A rectangle is a parallelogram with right angles, a rhombus is one with four equal sides, and a square is both. Base × height works for all of them.
Is the Area of Parallelogram Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Area of Parallelogram Calculator to find the area of any parallelogram from its base and height, two sides and an angle, or its diagonals.