Parallelogram Area Calculator
Area from base and height, two sides and an angle, or the diagonals.
—
- Perimeter—
- Height—
- Base—
- Angles—
How to use this parallelogram calculator
- Pick the method that matches what you have measured. Base and height is the usual one, but two sides with the angle between them is often easier to measure in practice, since the perpendicular height can be awkward to find.
- Enter the measurements and choose a unit. Every box must use the same one.
- Read the area, with the perimeter and derived height where the method makes them available.
What is a parallelogram?
A parallelogram is a four-sided shape with two pairs of parallel sides. Opposite sides are equal in length, opposite angles are equal, and the diagonals cut each other exactly in half.
Several familiar shapes are parallelograms with extra conditions attached:
| Shape | What makes it special | Area formula |
|---|---|---|
| Parallelogram | Two pairs of parallel sides | b × h |
| Rectangle | All angles are 90° | l × w |
| Rhombus | All four sides equal | b × h, or (e × f) / 2 |
| Square | Equal sides and 90° angles | a² |
A square is a rhombus, a rhombus is a parallelogram, and a rectangle is a parallelogram. The relationship only runs one way: a parallelogram is not generally a rectangle.
The parallelogram area formula
Area equals the base multiplied by the perpendicular height. The word perpendicular is doing the real work in that sentence, and ignoring it is the most common mistake with this shape.
Why height is not the slanted side
The height is the straight-up distance between the two parallel sides, measured at a right angle. It is not the length of the sloping side. On a strongly slanted parallelogram the sloping side can be much longer than the height, and using it inflates the area considerably.
There is a neat way to see why base times perpendicular height is correct. Cut a right-angled triangle off one end of the parallelogram and slide it round to the other end. The pieces fit together into a rectangle with the same base and the same height, and the area has not changed because nothing was added or removed. A parallelogram is simply a rectangle that has been pushed over.
This also explains something that surprises people: you can shear a parallelogram as far as you like, making the sloping sides enormously long, and as long as the base and the perpendicular height stay the same, the area does not change at all.
When you know two sides and the angle
If the perpendicular height is hard to measure, measure the two side lengths and the angle between them instead. The sine of the angle converts the sloping side into its perpendicular height, since h = b × sin(θ).
This also shows why a rectangle is the special case: at 90 degrees the sine is 1, and the formula collapses to a × b. At shallower angles the sine is less than 1, so the same four sides enclose less area. A hinged four-bar frame holds its maximum area when square to itself and loses area as it is pushed over.
When you know the diagonals
Where e and f are the diagonals and θ is the angle between them. For a rhombus the diagonals always cross at right angles, so the sine is 1 and this simplifies to the familiar (e × f) / 2.
A worked example
A parallelogram has a base of 12 cm and a perpendicular height of 7 cm.
Now suppose instead you measured the sides as 12 cm and 8 cm with an angle of 60 degrees between them:
Both are legitimate routes to the area. Which you use depends entirely on which measurements you can take accurately.
Perimeter of a parallelogram
Because opposite sides are equal, the perimeter is twice the sum of two adjacent sides. Note that the perimeter needs the actual side lengths, so it cannot be found from base and height alone. Two parallelograms can have identical areas and completely different perimeters, which is exactly what happens as you shear one further over.
Where this comes up
- Land and flooring. Plots and rooms are rarely perfect rectangles, and a slanted boundary makes a parallelogram.
- Structural bracing. A rectangular frame with no diagonal brace deforms into a parallelogram under load, which is precisely why triangular bracing is used instead.
- Vectors and physics. The parallelogram rule adds two vectors by drawing them as adjacent sides, and the area represents the cross product of the two.
- Computer graphics. A shear transformation turns rectangles into parallelograms, and the determinant of the transformation matrix is exactly the area scaling factor.
- Italic type. Slanting a letterform is a shear, which is why italic text occupies the same horizontal space as upright text of the same width.
Frequently asked questions
What is the formula for the area of a parallelogram?
Area equals base times perpendicular height, written A = b x h. If you know two sides and the angle between them instead, use A = a x b x sin(theta).
Is the height the same as the slanted side?
No, and this is the most common error. The height is the perpendicular distance between the two parallel sides, measured at a right angle. The slanted side is longer, and using it gives an area that is too large.
How do I find the area without the height?
Measure the two side lengths and the angle between them, then use A = a x b x sin(theta). The sine converts the sloping side into its perpendicular height, so the two methods agree.
What is the area of a parallelogram with base 12 and height 7?
It is 84 square units. Multiply the base by the perpendicular height: 12 x 7 = 84.
Is a rectangle a parallelogram?
Yes. A rectangle is a parallelogram whose angles are all 90 degrees. Squares and rhombuses are also parallelograms. The reverse is not true, since a general parallelogram has no right angles.
How do I find the perimeter of a parallelogram?
Add two adjacent sides and double the result: P = 2(a + b). Opposite sides are equal, so two measurements are enough. The perimeter cannot be worked out from base and height alone.
Can two parallelograms have the same area but different perimeters?
Yes. Shearing a parallelogram further over makes the sloping sides longer while the base and perpendicular height stay the same, so the perimeter grows while the area does not change at all.
How do I find the area from the diagonals?
Use A = (e x f x sin(theta)) / 2, where e and f are the diagonals and theta is the angle between them. For a rhombus the diagonals meet at right angles, so this simplifies to half the product of the diagonals.