Area Calculator guide
How to Find the Area of a Parallelogram
The area of a parallelogram is the base multiplied by the perpendicular height: A = b × h. The height is the straight-across distance between the two parallel sides, not the length of the slanted side. The perimeter is 2 × (base + slanted side).
The formula
A = b × h
That is the same formula as a rectangle, which is not a coincidence — and the next section explains why.
For a parallelogram with a base of 10 and a height of 6: A = 10 × 6 = 60 square units.
Why it is the same as a rectangle
Take your parallelogram and cut straight down from the top-left corner, at a right angle to the base. You now have a triangle and a lumpy four-sided piece.
Slide that triangle around to the right-hand end, and it fits exactly into the gap. What you are left with is a rectangle, with the same base and the same height.
Nothing was added and nothing was thrown away, so the area did not change. A parallelogram has the same area as the rectangle you can rearrange it into, which is why the formula is identical.
The mistake almost everyone makes once
In our example the slanted side is 7 and the height is 6. Using the slanted side gives 10 × 7 = 70, and the correct answer is 60. That is an error of nearly 17%.
The reason it is wrong is easiest to see at an extreme. Imagine pushing the parallelogram over further and further, so it gets flatter and flatter. The slanted sides stay exactly 7 long the whole time, but the shape is clearly getting smaller, and eventually it flattens to nothing.
The side length did not change, so it cannot be what determines the area. The height did change, and it is.
Finding the height from the side
If you were given the slanted side and an angle rather than a height, you can work the height out: h = s × sin(θ), where θ is the angle between the slanted side and the base.
At 90° the sine is 1 and h equals s exactly — which is the rectangle case, and a useful check that the formula is behaving.
For our parallelogram, a side of 7 and a height of 6 implies an angle of about 59°, since sin(59°) ≈ 0.857 and 7 × 0.857 ≈ 6.
Perimeter
Opposite sides of a parallelogram are equal, so the perimeter is 2 × (b + s) — base plus slanted side, doubled.
For our example: 2 × (10 + 7) = 34.
Note that the height plays no part here. Area needs the height and perimeter needs the side, which is why the calculator asks for all three measurements.
