Area Of Trapezoid Calculator

What this calculator does

A trapezoid is a four-sided shape with one pair of parallel sides that are different lengths, so it is wider at one end than the other. The side of a ramp, the profile of a bucket, a plot of land that tapers. The two parallel sides are called the bases, and the gap between them is the height.

Give the tool the two bases and the height, and it returns the area inside.

Using the calculator

1. Type Base 1 and pick its unit.
2. Type Base 2 and pick its unit.
3. Type the height and pick its unit, then press Calculate.

All three values have to be positive, and the area comes back in square units of Base 1's unit. The three can be in different units, since the tool converts them to match before working out the area.

The formula | area = ((base1 + base2) ÷ 2) × height

The area of a trapezoid is:

area = ((base1 + base2) ÷ 2) × height

Add the two parallel sides, halve them to get their average, then multiply by the height. The averaging is the clever part. Because the shape tapers from one base to the other, its effective width is the average of the two ends. Once you have that average width, the trapezoid behaves like a rectangle that wide, so you just multiply by the height.

Where the halving comes from

Here is a tidy way to see why the formula works, dividing by two and all. Take a second trapezoid identical to your first, flip it upside down, and slot it against the original. The two fit together into a parallelogram, one whose base is the two bases added together, base1 + base2, and whose height is the same as before.

That parallelogram's area is (base1 + base2) × height. Your single trapezoid is exactly half of it, which is where the ÷ 2 comes from.

The one formula that contains the others

This is the quietly remarkable thing about the trapezoid. Its formula holds the other area formulas inside it.

Make the two bases equal, base1 = base2, and ((base1 + base2) ÷ 2) × height collapses to base × height, which is a parallelogram or a rectangle. Now shrink one base all the way to zero, and the formula becomes ½ × base × height, which is a triangle. So a rectangle, a parallelogram and a triangle are all just special trapezoids, with the bases set to particular values. Understand this one and you have the others for free.

Units, the height, and rounding

The area comes out in square units of Base 1's unit, and if the three measurements are in different units the tool lines them up first. As with every shape here, the height is the perpendicular distance between the two parallel bases, straight across, not the length of a slanted side. A tidy result is shown in full, and a long decimal is rounded to three places.

A worked example | bases 8 and 12 cm, height 6 cm

Say the two bases are 8 cm and 12 cm, and the height is 6 cm.

1. Average the bases: (8 + 12) ÷ 2 = 10.
2. Multiply by the height: 10 × 6 = 60.

So the area is 60 cm². The average of 8 and 12 is 10, so the trapezoid holds exactly as much as a 10 by 6 rectangle.

Questions people ask

What is the area of a trapezoid with bases 8 and 12 and height 6?

It is 60 square units. Average the two bases and multiply by the height: ((8 + 12) ÷ 2) × 6.

Why do you average the two bases?

Because the trapezoid tapers from one base to the other, so its effective width is the average of the two. Multiplying that average by the height gives the area, like a rectangle of that width.

Why divide by 2?

Because two identical trapezoids fit together into a parallelogram of base (base1 + base2) and the same height. One trapezoid is half of that parallelogram.

Which sides are the bases?

The two parallel sides, the ones that never meet however far you extend them. The height is measured straight across between them, not along the slanted sides.

Is the height the slanted side?

No. The height is the perpendicular distance between the two parallel bases. The slanted sides are longer and are not used in the area.

References

A note on where this comes from. The area follows from rearranging two trapezoids into a parallelogram, a standard piece of classical plane geometry. The same shape lends its name to the trapezoidal rule in calculus, which estimates the area under a curve by slicing it into thin trapezoids and adding them up, the very averaging idea used here, applied over and over. For further reading, see Trapezoid.

1. Euclid, Elements (c. 300 BCE), the classical treatment of areas by decomposing shapes into triangles and parallelograms.
2. The trapezoidal rule, a method of numerical integration that approximates the area under a curve using trapezoids.