Geometry
How To Calculate The Area Of A Parallelogram
Parallelogram area looks like rectangle area until you notice the sides are slanted, and using the slanted side instead of the height is the mistake almost everyone makes once.
Quick Answer
area = base x perpendicular height
- A
- Area enclosed, in square units
- b
- Base, the length of one parallel side
- h
- Perpendicular height between the parallel sides
Multiply the base by the perpendicular height, which is the straight up distance between the two parallel sides. A base of ten with a height of six gives an area of 60, the same as a rectangle of those dimensions.
What Is The Area Of A Parallelogram?
The area of a parallelogram is its base multiplied by its perpendicular height. A parallelogram with a base of ten and a height of six encloses an area of sixty, exactly the same as a rectangle ten by six.
That equivalence is the whole idea. Slide the triangular overhang from one end to the other and a parallelogram becomes a rectangle with the same base and the same height. Cutting and moving a piece does not change the area, so the two formulas coincide.
The height is the perpendicular distance between the two parallel sides, not the length of the slanted side. In a strongly slanted parallelogram the side can be far longer than the height, and using it inflates the area dramatically.
Perimeter is different and does use the slanted side. It is twice the base plus the side, because a parallelogram has two of each. Area and perimeter therefore need different inputs, which is why a single diagram with all three lengths labelled is such a common source of confusion.
A rectangle is the special case where the slant is zero, so the height equals the side. Every formula for a parallelogram reduces to the rectangle formula when the sides are at right angles, which is a good way to remember which measurement goes where.
The height can also be found from the side and the angle: it is the side multiplied by the sine of the angle between the side and the base. That sine is always at most one, which is another way of seeing that the height never exceeds the side length.
Diagonals of a parallelogram bisect each other, though they are not equal unless the shape is a rectangle. They split the shape into four triangles of equal area, a fact that is useful when a problem gives diagonal lengths instead of a height.
Area is measured in square units and is always positive. If you get a negative number, you have subtracted something that should have been multiplied, or taken an absolute value of a displacement where a length was needed.
In real life the shape turns up in roof planes, ramps, shear walls, and any panel cut on a slant. Estimating sheet material for a slanted section with the rectangle formula gives a number that is too large, sometimes by a wide margin.
A rhombus is a parallelogram with all four sides equal, so the same area formula applies with the base and the height. Its area also equals half the product of its diagonals, which is a useful shortcut when the diagonals are the measurements you have.
Finally, be careful with units. The base and the height must be in the same unit before multiplying, and the result is that unit squared. A base in metres and a height in centimetres gives an answer that is out by a factor of a hundred.
Formula
A = base x height
Height is the perpendicular distance between the parallel sides, never the slanted side length.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Area | square metres | Surface enclosed by the parallelogram. |
| b | Base | metres | Length of one of the parallel sides. |
| h | Height | metres | Perpendicular distance between the parallel sides. |
P = 2 x (base + side)
Uses the slanted side, unlike the area formula. Two of each side, added and doubled.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Perimeter | metres | Distance around the shape. |
| b | Base | metres | Length of a parallel side. |
| s | Side | metres | Length of a slanted side. |
h = side x sin(angle)
Lets you recover the height when only the slanted side and the interior angle are known.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| h | Height | metres | Perpendicular distance between the parallel sides. |
| s | Side | metres | Slanted side length. |
| angle | Angle | degrees | Angle between the side and the base. |
How To Calculate The Area Of A Parallelogram
- 1
Pick one of the parallel sides as the base
Either pair works, as long as you then use the height measured between that pair. Mixing a base from one pair with a height from the other gives a meaningless answer.
- 2
Measure the perpendicular height
This is the straight up distance between the two parallel sides, not the length of the slanted edge. Drop a perpendicular from the top side to the base line and measure that.
- 3
Convert both to the same unit
Multiplication cannot combine metres with centimetres. Convert first, or the area will be out by a factor of a hundred.
- 4
Multiply base by height
Ten times six is sixty square units. There is no one half and no division, unlike the triangle formula the shape is often mistaken for.
- 5
Add the slanted side only if perimeter is wanted
Perimeter is twice the base plus the side, which for a base of ten and a side of seven is thirty four. Do not put the side into the area calculation.
Examples
Example 1: A parallelogram with base ten and height six
- Base
- 10
- Height
- 6
- Side
- 7
| Step | Calculation | Result |
|---|---|---|
| Perimeter | 2 x (10 + 7) | 34 |
| Sine of the slant angle | 6 / 7 | 0.8571 |
| Area | 10 x 6 | 60 |
Result: A base of ten and a height of six enclose an area of 60 square units, with a perimeter of 34 units once the slanted side is counted.
Example 2: A slanted roof panel
- Base
- 25
- Height
- 8
- Side
- 11
| Step | Calculation | Result |
|---|---|---|
| Perimeter | 2 x (25 + 11) | 72 |
| Sine of the slant angle | 8 / 11 | 0.7273 |
| Area | 25 x 8 | 200 |
Result: A slanted panel with a base of 25 and a perpendicular height of 8 covers 200 square units, even though its slanted side measures eleven.
Example 3: A small rhombus with base 4.5 and height 2
- Base
- 4.5
- Height
- 2
- Side
- 4.5
| Step | Calculation | Result |
|---|---|---|
| Perimeter | 2 x (4.5 + 4.5) | 18 |
| Height to base ratio | 2 / 4.5 | 0.4444 |
| Area | 4.5 x 2 | 9 |
Result: A rhombus of side 4.5 with a perpendicular height of 2 encloses an area of 9 square units, which is the base times the height just as for any parallelogram.
Calculator
Area
60
- Perimeter
- 34
- Height to base ratio
- 0.6
- Sine of the slant angle
- 0.8571
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Area Of A Parallelogram calculator page.
Common Mistakes
Using the slanted side as the height
The height is the perpendicular distance between the parallel sides, which is shorter than the slant. Substituting the side inflates the area, often by thirty percent or more.
Taking a base from one pair and a height from the other
Each base has its own matching height. Pairing them across the shape gives an area that is wrong without looking obviously wrong.
Using the triangle formula with a half
A parallelogram is not a triangle. There is no factor of one half, and including it halves the answer.
Treating the diagonal as the height
Diagonals run corner to corner and are longer than the height. They are not part of the area formula at all.
Mixing units between base and height
A base in metres and a height in centimetres produces an answer out by a factor of a hundred. Convert before multiplying.
Reporting the answer in plain units
Area is square units. Writing metres for a surface is the same error as confusing the perimeter with the area.
Assuming a slanted shape has less area
A parallelogram and a rectangle with the same base and height have identical areas. The slant redistributes the surface rather than reducing it.
FAQ
What is the formula for the area of a parallelogram?
Base multiplied by perpendicular height. The height is measured straight up between the two parallel sides, not along the slanted edge.
Why is the slanted side not used in the area?
Because area depends on the perpendicular distance between the parallel sides. The slanted side is longer than that distance, so using it overstates the area.
Is a parallelogram's area the same as a rectangle's?
Yes, when the base and height match. Cutting the triangular overhang off one end and moving it to the other turns the parallelogram into that rectangle.
How do you find the height if you only know the side?
Multiply the side by the sine of the angle it makes with the base. Without the angle, the height cannot be recovered from the side alone.
How is the perimeter different?
Perimeter uses both the base and the slanted side, since the shape has two of each. It is twice the sum of the base and the side.
What is the area of a rhombus?
The same base times height formula applies. A rhombus is a parallelogram with equal sides, and its area is also half the product of its diagonals.
References
- [1]Wikipedia, Parallelogram — https://en.wikipedia.org/wiki/Parallelogram
- [2]Wikipedia, Area — https://en.wikipedia.org/wiki/Area
- [3]Math is Fun, Parallelogram area — https://www.mathsisfun.com/geometry/parallelogram.html
- [4]Wikipedia, Rhombus — https://en.wikipedia.org/wiki/Rhombus