Area of a parallelogram calculator
The area of a parallelogram calculator gives 40 square units for a base of 8 and a perpendicular height of 5, the same as an 8 by 5 rectangle.
Area
- Area
- 40 square units
- Formula
- A = b x h = 8 x 5
- If the units were feet
- 40 sq ft = 4.4444444 sq yd = 0.0009182736 acres
- Doubling every length
- 160 square units
How the Area Calculator works
The area of a parallelogram is A = b × h: the length of the base times the perpendicular height, the straight-line distance between the base and the side opposite it. For a base of 8 and a height of 5 the area is 40 square units. If the measurements are in feet, that is 40 square feet.
The reason is a cut and a slide. Drop a perpendicular from one top corner to the base and cut along it. The piece that comes off is a right triangle; slide it across to the other end and it fits the gap exactly, leaving a rectangle 8 wide and 5 tall. Nothing was added or removed, so the parallelogram had the area of that rectangle all along. Leaning a shape over moves its corners but does not change how much of the plane it covers.
That is also why the slanted side never appears in the formula, and why using it is the classic mistake. Suppose the slanted side of this parallelogram is 6. Multiplying 8 by 6 gives 48, which is the area of a rectangle, not of this shape. The height of 5 is shorter than the side of 6 because the side leans; the more it leans, the bigger the gap. If you know the sides and the angle between them instead of the height, the height is the slanted side times the sine of that angle, so A = ab sin θ. Sides of 8 and 6 with a height of 5 mean sin θ = 5/6, an angle of about 56.44°.
Any side can be the base, as long as the height is measured to that side. Take the side of 6 as the base and the height to it must be 40 / 6, about 6.67, so the product is still 40. That check is a quick way to catch a measurement taken to the wrong side.
The perimeter is a different matter: it needs the slanted side, 2(a + b), which this formula does not use. With sides of 8 and 6 it would be 28. Base and height alone do not fix the perimeter, and many parallelograms of different perimeters share the same 40.
The family around it: a rectangle is a parallelogram with right angles, so its height is its side. A rhombus is a parallelogram with four equal sides, and base times height still works, as does half the product of its diagonals. A triangle is half a parallelogram with the same base and height, which is where its ½ comes from, and a trapezoid, with only one pair of parallel sides, averages them.
Area Calculator FAQ
What is the formula for the area of a parallelogram?
A = b × h, base times perpendicular height. A parallelogram with base 8 and height 5 has an area of 40 square units.
Why is the slanted side not used?
Because cutting off the triangle at one end and moving it to the other turns the parallelogram into a rectangle of the same base and height. The slanted side only matters for the perimeter.
How do I find the area from two sides and an angle?
A = ab sin θ, where θ is the angle between sides a and b. The height is the slanted side times sin θ, so this is still base times height.
Is the area of a parallelogram the same as a rectangle?
It equals the rectangle with the same base and perpendicular height. It is less than the rectangle made from its two side lengths, unless the angles are right angles.