How to Find the Area of a Parallelogram
A = b × h
Multiply the base by the perpendicular height, which is the straight up distance between the two parallel sides. A base of 15 with a height of 8 gives 120 square units. The sloping side is not the height, and using it instead is the mistake almost everyone makes at least once.
A parallelogram is a rectangle that has been pushed over. Nothing was added when it leaned and nothing was taken away, so its area is still base times height. The only thing the lean changes is which measurement counts as the height, and that is where the whole difficulty of this shape lives.
The height is not the sloping side
Take the worked example. The base measures 15, the sloping side measures 10, and the perpendicular height between the parallel sides is 8. Those last two are different numbers describing different distances, and only the 8 belongs in the formula. Use the 10 and you get 150 instead of 120, which is 25 percent too much.
The reason the two differ is that the sloping side is a hypotenuse. It travels 6 sideways while it climbs 8, so it measures √(36 + 64), which is 10. The height only counts the climb. The steeper the lean, the further apart these two numbers get, and the worse the error becomes. On a nearly flat parallelogram the sloping side can be many times the height.
Pick a base
Any side will do, as long as the height you use is measured to that same side. Here it is the 15.
Measure straight across
Perpendicular from that base to the parallel side opposite. Here that is 8, not the sloping 10.
Multiply
15 times 8 is 120 square units. No halving, because this is the full rectangle, not a triangle.
Why leaning changes nothing
Cut a triangle off one end of the parallelogram with a single vertical slice, then slide that triangle round to the other end. It fits exactly, because the two sloping sides are parallel and the same length. What you are left with is a rectangle, same base, same height, and obviously the same area, because you only moved material rather than adding or removing it.
This is the same move that explains the triangle and the trapezoid, which is why it is worth doing once with scissors rather than taking it on trust. A triangle is half a parallelogram. A trapezoid is half of a bigger parallelogram made from two copies. Everything in this family reduces to base times height and then divides by something.
A worked check you can repeat on paper
Route five is how a computer does it. Treat the two edges leaving one corner as vectors, and the area is the determinant they form, which for (15, 0) and (6, 8) is 15 times 8 minus 0 times 6. It needs no height, no angle and no trigonometry, only the corner coordinates, and it is the reason mapping software can measure a plot of land without anyone walking a perpendicular.
Two side lengths do not fix the area
This is the part worth internalising, because it explains why so many searches for an area formula come up empty. A triangle is rigid: give me three sides and the shape is settled, which is why Heron works. A parallelogram is not. Give me sides of 15 and 10 and I can still open or close it like a hinge, and the area moves with the angle.
| Angle between the sides | Height | Area |
|---|---|---|
| 90 degrees, a rectangle | 10.00 | 150.0 |
| 60 degrees | 8.66 | 129.9 |
| 53.13 degrees, the worked example | 8.00 | 120.0 |
| 30 degrees | 5.00 | 75.0 |
| 20 degrees | 3.42 | 51.3 |
Every row has the same two side lengths and the same perimeter of 50. The area runs from 150 all the way down toward zero as the shape flattens, and it peaks at the right angle, where the sloping side finally is the height. So if all you have is four side lengths, you cannot get an area, and no formula will rescue you. You need one more piece of information: the height, one angle, or a diagonal.
When the angle is what you have, use A = a b sin θ. For sides of 15 and 10 at 53.13 degrees that is 150 times 0.8, which is 120, matching the height route exactly. The sine is doing one job here: converting the sloping side into the height, since h equals the sloping side times sin θ.
Mistakes that catch people out
- Using the sloping side as the height. The one to watch for. In the worked example it turns 120 into 150. The error grows as the shape flattens, and on a very shallow parallelogram it can more than double the answer.
- Mixing a base with the wrong height. A parallelogram has two bases and two heights, one pair for each direction. The long base pairs with the short height. Pair a long base with the height belonging to the short side and the answer is nonsense.
- Halving out of habit. The triangle formula halves, the trapezoid formula halves, and the hand keeps going. A parallelogram does not halve, because it already is the full rectangle.
- Expecting the perimeter to say something. Two parallelograms with identical perimeters can have wildly different areas, as the table above shows. Perimeter and area are independent here in a way they are not for a triangle.
- Using the angle at the wrong corner. A parallelogram has two different angles, and they add to 180 degrees. It happens not to matter for this formula, because sin θ and sin(180 − θ) are equal, but it matters for almost everything else you might do with the shape.
Try these
- Base 9, perpendicular height 6. Find the area.
- Base 20, perpendicular height 7. Find the area.
- A parallelogram has an area of 96 and a base of 12. Find its height.
- Two sides of 8 and 5 meet at 40 degrees. Find the area.
- A rhombus has sides of 9 and an angle of 60 degrees. Find its area two different ways.
Show the answers
- 9 times 6 is 54 square units.
- 20 times 7 is 140 square units.
- Rearrange to h = A / b. That is 96 over 12, so the height is 8.
- 8 times 5 times sin 40 degrees is 40 times 0.6428, which is 25.71 square units.
- By sides and angle, 9 times 9 times sin 60 degrees is 81 times 0.8660, which is 70.15. By diagonals, the two diagonals measure 9 and 15.59, and half their product is 70.15. A rhombus is a parallelogram, so both routes are open to it.
Common questions
What is the formula for the area of a parallelogram?
Base times perpendicular height, written A = b h. A base of 15 with a height of 8 gives 120 square units. There is no halving, unlike the triangle.
Is the height the same as the side?
Only when the parallelogram happens to be a rectangle. Otherwise the sloping side is longer, because it travels sideways as well as up. In the worked example the side is 10 and the height is 8, and swapping them overstates the area by 25 percent.
How do you find the area from two sides and an angle?
Use A = a b sin θ, where θ is the angle where the two sides meet. Sides of 15 and 10 at 53.13 degrees give 150 times 0.8, which is 120. The sine converts the sloping side into the perpendicular height.
Can you find the area from the four sides alone?
No, and this is a real limitation rather than a missing formula. A parallelogram hinges, so the same four sides can enclose any area from near zero up to a maximum at the right angle. You need the height, an angle or a diagonal as well.
Why is a parallelogram the same area as a rectangle?
Because you can cut a triangle off one end with a vertical slice and slide it to the other end, where it fits exactly. That turns the parallelogram into a rectangle of the same base and height without adding or removing anything.
Is a rhombus a parallelogram?
Yes, one with all four sides equal. Base times height still works on it, and so does a b sin θ. It also gains a shortcut the general parallelogram does not have: half the product of its diagonals, because in a rhombus the diagonals cross at right angles.
Related lessons and calculators
- Area formulas for every common shape, where this one sits alongside the rest, with a calculator covering fifteen shapes.
- Area of a trapezoid, the same formula with the two parallel sides allowed to differ. Set them equal and you are back here.
- Area of a triangle, which is exactly half of a parallelogram, by the same cut and slide argument.
- Area from two sides and the included angle, the triangle version of the a b sin θ route used above.
- Area of 2D shapes calculator, for running these without doing the arithmetic yourself.
- the National Council of Teachers of Mathematics, for guidance on how this material is sequenced in the curriculum.
- Area of a rhombus, the equal sided member of this family, which gains a diagonal shortcut the general case lacks.
- Height of a trapezoid, worked back from an area, four sides, or a single angle.
- Base of a parallelogram, the same formula rearranged, and a straight answer to which side the base actually is.