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CalcMax

Parallelogram Area Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

24.0000 cm²

Area

Perimeter
22.0000 cm

A parallelogram calculator takes a base, a height and one sloping side and returns the area and the perimeter. A parallelogram is any four-sided figure with two pairs of parallel sides: the rectangle, the rhombus and the square are all special cases of it, and the shape you get by pushing a rectangle sideways is the general one. Its area is the easiest formula in geometry, base times height, and it is the same formula a rectangle uses — which is the fact worth stopping on, because a pushed-over rectangle obviously holds less than the upright one did and the formula says otherwise. The resolution is that the height is not the sloping side. It is the perpendicular distance between the base and the side opposite it, measured straight up rather than along the slope, and it is the number the area actually depends on. Push the rectangle over and the sloping side gets longer while the perpendicular height stays the same, and a rectangle and its pushed-over twin have exactly the same area. The perimeter is where the slope does matter, and it is the reason this page asks for three numbers rather than two. A parallelogram has two sides of one length and two of another, so its perimeter is twice the sum of those two lengths — and the two lengths are the base and the sloping side, not the base and the height. Height and sloping side are different numbers for every parallelogram except a rectangle, where they coincide. That gives the page its one refusal: a sloping side shorter than the height describes no parallelogram, because the height is the sloping side multiplied by the sine of the angle it makes with the base, and the sine of an angle never exceeds one. A base of 6 with a height of 4 and a sloping side of 1 is three numbers that cannot be drawn, and the page says so rather than returning a number. Everything is in centimetres and square centimetres regardless of the unit dropdowns, and both outputs are exact whenever the inputs are.

Parallelograms from a base, a height and a sloping side

Base (cm)Height (cm)Adjacent side (cm)Area (cm²)Perimeter (cm)
6452422
3451216
125136050
101101040
2.523511
11114
10011100202
503016

Eight parallelograms, and the first row is the one the page loads with. The second row is the one to read against the triangle page: a base of 3 and a height of 4 give an area of 12 here, and the triangle on the same base and height has an area of 6, which is the two-to-one relation that the diagonal cut makes obvious. The fourth row is a long low one, ten by one with a sloping side of ten — a shape squashed almost flat, which is the clearest place on the table to see that the area is governed by the perpendicular height while the perimeter is governed by the two side lengths and barely notices the height at all. The seventh row pushes that further: a base of 100 with a height of 1 and a sloping side of 1 is a sliver a hundred centimetres long and one high, whose area is 100 square centimetres and whose perimeter is 202. The sixth row is the boundary the page allows: a sloping side equal to the height, which makes the angle a right angle and the shape a square. The eighth and last row is a height of zero, where the top side has collapsed onto the bottom one: the area is 0, honestly, and the perimeter is still 16, because the four sides are all still there to walk around. Two numbers that look inconsistent on a row where the shape has flattened into a line is the sort of thing worth meeting on a table before meeting it on the results panel. Every figure here is recomputed from its three inputs when the page is built, in centimetres, and all of them are exact since nothing on this page introduces a π or a square root.

Formula

Area = b × h Perimeter = 2(b + a)

Base
The length of one of the two parallel sides the parallelogram stands on, in centimetres. Either of the two parallel pairs can be called the base; what matters is that the height is measured perpendicular to whichever one you pick
Height
The perpendicular distance from the base to the opposite side, in centimetres — measured straight up, not along the sloping side. This is the number the area depends on, and it is the one people confuse with the sloping side, which is a different length for every parallelogram that is not a rectangle
Adjacent side
The length of one of the two sloping sides, in centimetres. The area ignores it entirely and the perimeter cannot do without it: the perimeter is twice the base plus twice this. It must be at least as long as the height, since it is the height divided by the sine of the angle it makes with the base
Area
The space inside, in square centimetres: base times height, the same formula a rectangle uses. It is exactly twice the area of the triangle that shares the same base and height, which is what the second row of the reference table shows
Perimeter
The distance all the way around, in centimetres: twice the base plus twice the adjacent side, since the two pairs of sides are equal. The height plays no part in it at all, which is why a page that asked only for a base and a height could not have given you this number
Centimetres and square centimetres
The units of the two outputs, always, whichever unit the dropdowns are set to. The perimeter is a length and the area is a length squared, so the two are not comparable and doubling one does not double the other
Four decimal places
How wide both outputs are written. Four rather than two because the page runs on whole numbers in its reference table but is not restricted to them, and a base entered in metres through the dropdown arrives as a decimal

The area formula is the one that matters most, and it is worth knowing that it is the same base-times-height formula every parallelogram shares, rectangle included. The place it earns its keep is in showing that a slanted shape is not smaller than an upright one of the same base and height: a field pushed out of square by a leaning fence loses nothing in area, which is not what the eye reports. If you need the area of a plot whose sides you have measured along the ground, though, be careful — the height is the perpendicular distance, which is rarely the number you get from a tape measure, and the sloping side is not a substitute. For the perimeter, the same plot needs the two side lengths, and that is where this page asks for the third number. The refusal is worth meeting deliberately rather than by accident: if you enter a sloping side shorter than the height, the two numbers contradict each other, and the commonest cause is that a base and a sloping side have been measured but the perpendicular height has been guessed or taken as the other side. The triangle beside this page uses the same base and height and returns half the area, and the rectangle page returns the same area with the adjacent side equal to the height — those two are the quickest checks that this page is behaving.

Worked examples

  1. A base of 6, a height of 4 and a sloping side of 5

    1. Area: 6 × 4 = 24 square centimetres
    2. Perimeter: 2 × (6 + 5) = 22 centimetres
    3. The height of 4 is not used in the perimeter, and the sloping side of 5 is not used in the area

    The input the page loads with, and the pair of calculations that shows why the page needs three numbers: the area comes from the base and the height and the perimeter from the base and the sloping side, and neither output uses all three. The shape is a 3-4-5 rectangle pushed over, which is why the numbers are so tidy.

  2. A base of 3, a height of 4 and a sloping side of 5

    1. Area: 3 × 4 = 12 square centimetres
    2. Perimeter: 2 × (3 + 5) = 16 centimetres
    3. Twelve is exactly twice the area of the triangle with the same base and height, which is 3 × 4 ÷ 2 = 6

    The row that makes the doubling visible. Take the parallelogram and cut it along a diagonal and you get two triangles with the same base and the same height, so the parallelogram's area is exactly twice the triangle's — 12 against 6. That is where the two in the triangle's formula comes from, read backwards.

  3. A base of 12, a height of 5 and a sloping side of 13

    1. Area: 12 × 5 = 60 square centimetres
    2. Perimeter: 2 × (12 + 13) = 50 centimetres
    3. The 5-12-13 triangle is the second smallest whole-number right triangle, and it is why these numbers are round

    A long low parallelogram, and the row where the two outputs are furthest apart: the area is 60 while the perimeter is 50, which is a reminder that the two are different quantities measured in different units and there is no rule that one exceeds the other. The 5, 12 and 13 in the three fields are the sides of a right triangle, which is the tidiest way to build a parallelogram whose slope is exactly the perpendicular.

  4. A base of 1, a height of 1 and a sloping side of 1

    1. Area: 1 × 1 = 1 square centimetre
    2. Perimeter: 2 × (1 + 1) = 4 centimetres
    3. The sloping side equals the height, so the angle is 90 degrees and the shape is a square

    The boundary the page allows rather than refuses. When the sloping side equals the height, the sine of the angle is one, the angle is a right angle, and the parallelogram is a square — a real member of the family rather than a degenerate one. The rule the page applies is that the sloping side must be at least the height, not more than it, and this is the case that makes the difference.

  5. A base of 5, a height of 0 and a sloping side of 3

    1. Area: 5 × 0 = 0 square centimetres
    2. Perimeter: 2 × (5 + 3) = 16 centimetres
    3. Both figures follow from their own formulas, and they describe different things about the same flattened shape

    The case where the two outputs look inconsistent and are not. A height of zero means the top side has come down onto the bottom one and the shape has flattened into a line segment, so it encloses no area at all — zero is the right answer. The perimeter, meanwhile, is the total length of the four sides, which is what the formula computes and what a perimeter means: go along the base, back along the sloping side, along the base again, back along the sloping side. It is not the length of the flattened result, which would be 2 × 5 = 10. The page gives the four-sided figure's edge length, and that is worth knowing before one of the two numbers looks like a mistake.

Limitations

The height is the perpendicular distance between the base and the opposite side, and the page cannot tell if what you measured was the sloping side instead. Entering the sloping side in the height field gives an area that is too large, and it is the commonest mistake this page sees, because the sloping side is the one a tape measure reaches. The perimeter uses the base and the adjacent side and takes no account of the height, so if you supply a height and no adjacent side the page cannot run at all. A sloping side shorter than the height is refused, because it describes no parallelogram: the height equals the sloping side times the sine of the angle it makes with the base, and no angle has a sine above one. The outputs are in centimetres and square centimetres whatever unit the dropdowns are set to. Four decimal places is a display width rather than a claim about precision. The page assumes a true parallelogram with two pairs of parallel sides, so it will not check a quadrilateral whose opposite sides are unequal, and it gives no angles, no diagonals and no way to enter the angle between the sides directly. A height of zero is accepted and returns an area of zero with a non-zero perimeter, which is arithmetically correct and physically a line segment rather than a shape — the perimeter then describes the path around the four sides rather than the length of the flattened result.

Frequently asked questions

Why does the page ask for three numbers when the area only needs two?
Because the perimeter needs a number the area does not. The area is base times height; the perimeter is twice the base plus twice the sloping side, and height and sloping side are different lengths for every parallelogram that is not a rectangle. A base and a height fix the area but leave the shape free to be squashed, and squashing it changes the perimeter while the area stays put.
Is the height the same as the sloping side?
No, and this is the mistake the page sees most. The height is the perpendicular distance from the base straight up to the opposite side; the sloping side is the length of the edge itself. They are equal only for a rectangle. On a parallelogram leaning at 30 degrees the sloping side is twice the height, so entering the sloping side in the height field would double the area.
Why will it not accept a sloping side shorter than the height?
Because no such parallelogram exists. The height is the sloping side multiplied by the sine of the angle the side makes with the base, and a sine never goes above one, so the sloping side is always at least as long as the height. A sloping side equal to the height means the angle is 90 degrees and the shape is a rectangle, which is why the page allows equality rather than refusing it.
Why is the area the same as a rectangle's?
Because a parallelogram and a rectangle with the same base and the same perpendicular height enclose the same area, however far the parallelogram is leaning. Slide the triangle off one end of the parallelogram and it fits exactly onto the other end, turning it into a rectangle of the same base and height. Leaning a rectangle over makes the sloping sides longer without changing what it encloses.
The area is zero but the perimeter is not — which is wrong?
Neither. A height of zero means the top side has come down onto the bottom one, so the shape is a line segment and encloses no area; zero is correct. The perimeter is the total length of the four sides — along the base, back along the sloping side, along the base again, back along the sloping side — and that path is still 16 centimetres long when the base is 5 and the side is 3. It is not the length of the flattened result, which would be 10.
How do I get the area of a triangle instead?
Halve this page's area, provided the triangle shares the same base and perpendicular height. Cutting a parallelogram along a diagonal gives two triangles of equal area, which is exactly where the half in the triangle formula comes from. If your triangle's sides are what you measured rather than its base and height, the triangle pages elsewhere on this site take the three sides directly.

References

Related calculators

Perimeter CalculatorPerimeter of common shapes from their side lengths.Hexagon CalculatorArea, perimeter and diagonals of a regular hexagon.Rectangle CalculatorRectangle area, perimeter and diagonal from length and width.Trapezoid CalculatorTrapezoid area, perimeter and height from its sides.