How to Find the Base of a Parallelogram

This is a one step division that goes wrong for one reason: the word base is ambiguous on a parallelogram. Either pair of parallel sides can play the part, each has its own height, and the two pairs give completely different numbers while agreeing on the area. Before dividing anything, settle which side you are actually after.

Which base, and which height

The worked parallelogram has sides of 15 and 10 and covers 120 square units. Take the 15 as the base and the height measured to it is 8, because 15 times 8 is 120. Take the 10 as the base and the height measured to that one is 12, because 10 times 12 is also 120. Both statements are true about the same shape at the same time.

Notice that the second height of 12 is longer than the side it belongs to. Nothing is wrong there. On a leaning shape the distance across a short side can easily beat the side itself, and it is one of the reasons people reject a correct answer for looking odd.

The relationship worth carrying away is that base times height is the same for both pairs. Write it as b₁h₁ = b₂h₂ and you have a way of getting from one base to the other without touching the area at all: if you know the shape is 10 by 12 and you want the base that goes with a height of 8, that base is 10 times 12 over 8, which is 15.

A worked check you can repeat on paper

Route six is the one nobody expects. Every parallelogram obeys d₁² + d₂² = 2(a² + b²), which says the two diagonals between them carry exactly as much as the four sides. Here the diagonals of 22.47 and 12.04 square to 505 and 145, adding to 650, and 2 times (225 plus 100) is also 650. Rearranged, that gives a base of √(650 over 2 minus 100), which is √225, which is 15. So two diagonals and one side are enough, with no area, no height and no angle in sight.

When the height is missing

The awkward version of this question gives you an area and a side but no height, which is not enough on its own. A side of 10 could sit at any angle, and the base that goes with it changes as the shape leans. What rescues it is an angle, because the area of a parallelogram is also ab sin θ, so the base is the area divided by the other side times the sine of the angle between them.

An area of 120 with a side of 10 at 53.13 degrees gives 120 over 8, which is 15. Change nothing but the angle and watch the base move: at 30 degrees the same area and side need a base of 24, and at 90 degrees, where the shape is a rectangle, the base is only 12. One area, one side, three different bases, all correct.

What you are given	Route to the base	Worked value
Area and the matching height	area divided by height	15
Area, the other side and the angle	area over the side times the sine	15
Perimeter and the other side	half the perimeter, minus that side	15
The other base and height pair	b₂h₂ divided by h₁	15
The four corner coordinates	distance between two adjacent corners	15
Both diagonals and one side	the parallelogram law, rearranged	15

Only the first row is a single division. The rest exist because real problems rarely hand over a clean matching pair, and knowing three of these means you are almost never stuck.

Mistakes that catch people out

• Dividing by the sloping side. 120 over 10 gives 12, and 12 is a perfectly real length on this shape, just not the one asked for. That is what makes this slip so hard to spot: the wrong answer is not absurd, it is simply an answer to a different question.
• Pairing a base with the wrong height. The 15 goes with the 8 and the 10 goes with the 12. Cross them over and 15 times 12 gives 180 instead of 120, a shape half as big again as the real one.
• Expecting the height to be shorter than the side. Here the height of 12 is longer than the side of 10 it belongs to. That is normal on a leaning shape and is not a sign of an error.
• Assuming the perpendicular lands inside the shape. Lean a parallelogram far enough and the foot of the height falls outside the base, on the extension of it. The measurement still counts and the formula still works.
• Dividing the perimeter by four. That only works when all four sides match, which means a rhombus. A parallelogram has two different sides, so half the perimeter minus the known side is the correct move.

Try these

Common questions

How do you find the base of a parallelogram?

Divide the area by the perpendicular height. An area of 120 with a height of 8 gives a base of 15. The height must be the one measured to the side you want.

Which side of a parallelogram is the base?

Whichever one the given height is perpendicular to. Both pairs of parallel sides can serve as a base, and each has its own height. On the worked shape it is 15 with 8, or 10 with 12.

Can the height be longer than the base?

Yes. The worked parallelogram has a height of 12 measured to a side of 10. On a leaning shape the distance across a short side is often greater than the side itself.

How do you find the base without knowing the height?

Use the area, the other side and the angle between them, since the area is also ab sin θ. An area of 120 with a side of 10 at 53.13 degrees gives 120 over 8, which is 15. The angle is not optional here, because the same area and side fit many different shapes.

How do you find the base from the perimeter?

Halve the perimeter, then subtract the side you know. A perimeter of 50 with one side of 10 gives 25 minus 10, which is 15. Dividing by four only works on a rhombus.

Can the diagonals give you the base?

Yes, if you also know one side. The parallelogram law says the squares of the diagonals add to twice the squares of the sides, so diagonals of 22.47 and 12.04 with a side of 10 give a base of 15.

Related lessons and calculators

• Area of a parallelogram, the formula this lesson divides through, and why the lean changes nothing.
• Base of a triangle from area and height, the same division with a 2 in the way, since a triangle is half a parallelogram.
• Area of a rhombus, the parallelogram whose four sides match, where dividing the perimeter by four does work.
• Height of a trapezoid, the same rearrangement on the shape with only one pair of parallel sides.
• Area of 2D shapes calculator, for parallelograms and the rest without the arithmetic.
• Area formulas for every common shape, the index this lesson sits inside.
• Wolfram MathWorld on the parallelogram, including the parallelogram law used in route six.