How to Find the Perimeter of a Trapezoid
P = a + b + c + d
Add all four sides. A trapezoid with parallel sides of 10 and 24 and legs of 13 and 15 has a perimeter of 62. The work only starts when one of those four numbers is missing, which is most of the time.
If somebody hands you four side lengths there is nothing to teach here, so this lesson is about the version you actually meet: a shape described by its parallel sides and a height, or by a sketch with two lengths marked and two blank. The legs are the missing pieces, and Pythagoras produces them one at a time.
The area will never tell you the perimeter
Worth settling this before anything else, because it is the question people arrive with. You cannot get from an area to a perimeter, not on a trapezoid and not on anything else. Fix the parallel sides at 10 and 24 and the height at 12, and the area is locked at 204 no matter what the legs do. Slide the top side sideways and the area does not move a millimetre while the legs stretch.
Three shapes make the point. The symmetric one has legs of 13.89 and a perimeter of 61.78. Stand one leg upright and the other has to reach further, giving 12 and 18.44, a perimeter of 64.44. Slide the top past the corner and the legs become 13.42 and 23.32, a perimeter of 70.74. Same 204 square units in all three, and 9 units of difference in what it costs to fence.
Count what you have
Four sides means one addition. Fewer than four means the legs have to be built out of the height first.
Split the overhang
The difference between the parallel sides is shared between the two ends. On a symmetric shape each end takes half.
Pythagoras, then add
Each leg is the square root of the height squared plus its own overhang squared. Then add all four sides.
A worked check you can repeat on paper
The two overhangs adding to 14 is the check to run before anything else. They have to account for the whole difference between the parallel sides, so if your two right triangles do not span 14 between them, one of the measurements is wrong and no amount of careful arithmetic afterwards will save it.
The midline is in that table to be dismissed. It is 17, it is genuinely useful for the area, and it has nothing whatsoever to do with the perimeter. Averaging the parallel sides throws away exactly the information the perimeter needs.
The three cases you actually get asked
An isosceles trapezoid is the friendliest. The two legs match, so the overhang splits evenly and each end takes half the difference. Parallel sides of 8 and 18 with a height of 12 give overhangs of 5, so each leg is √(144 + 25), which is 13, and the perimeter is 8 plus 18 plus 26, or 52. One square root does the whole job.
A right trapezoid gives you one leg free, because the perpendicular one is the height itself. The other leg takes the entire difference as its overhang. Parallel sides of 8 and 20 with a height of 5 give legs of 5 and √(25 + 144), which is 13, so the perimeter is 46.
A scalene trapezoid is the worked example, and it needs both overhangs named before either leg can be built. If you were given the four sides but not the height, that is the reverse trip: find the overhang from the sides, then the height. Either way the perimeter is still just the sum.
| Shape | Parallel sides and height | Legs | Perimeter |
|---|---|---|---|
| Worked example, scalene | 10, 24, h 12 | 13 and 15 | 62.00 |
| Symmetric version of it | 10, 24, h 12 | 13.89 each | 61.78 |
| Right trapezoid version | 10, 24, h 12 | 12 and 18.44 | 64.44 |
| Strongly leaning version | 10, 24, h 12 | 13.42 and 23.32 | 70.74 |
| Isosceles, a different shape | 8, 18, h 12 | 13 each | 52.00 |
| Right trapezoid, a different shape | 8, 20, h 5 | 5 and 13 | 46.00 |
Fencing a plot, where this becomes a bill
Trapezoidal plots are common wherever a straight road meets land that was divided up along a different line, so the fencing question comes up more than the textbook suggests. Read the worked example in metres and the plot covers 204 square metres and needs 62 m of fence. At 18 per metre that is a bill of 1116.
The first four rows of the table are the same 204 square metres of ground fenced four different ways, from 61.78 m to 70.74 m. That spread is close to 15 percent, which matters on a real invoice, and it is entirely down to shape rather than size. If the boundaries are already fixed you have no say in it, but it is worth knowing that the symmetric layout is always the cheapest to enclose and that the worked plot is within 0.35 percent of that best case.
Mistakes that catch people out
- Trying to reach the perimeter from the area. It cannot be done. Three trapezoids in this lesson all cover 204 square units with perimeters of 61.78, 64.44 and 70.74. An area is one number and a perimeter needs four.
- Adding the height in. The height is not a side. On the worked trapezoid it is 12, which is close enough to the legs of 13 and 15 to slip past unnoticed and land you on 74 instead of 62.
- Assuming the legs match. They only match on an isosceles trapezoid. Halving the difference of the parallel sides is a shortcut that quietly gives the wrong answer on every other shape.
- Doubling one leg. The same mistake wearing different clothes. Two legs of 13 give 60, not 62, and the sketch will not look wrong enough for you to notice.
- Using the midline. The midline of 17 is exactly the right tool for the area and exactly no use here. Averaging the parallel sides discards the difference, and the difference is what sets the legs.
Try these
- A trapezoid has sides of 5, 9, 7 and 11. Find its perimeter.
- An isosceles trapezoid has parallel sides of 6 and 16 and a height of 12. Find its perimeter.
- A right trapezoid has parallel sides of 8 and 20 and a height of 5. Find its perimeter.
- A trapezoid has a perimeter of 62 and three sides measuring 10, 24 and 13. Find the fourth.
- Double every side of the worked trapezoid. What happens to the perimeter, and what happens to the area?
Show the answers
- 32. No height, no Pythagoras, no shape needed. Four sides is always just an addition.
- The difference of 10 splits into overhangs of 5, so each leg is √(144 + 25), which is 13. The perimeter is 6 plus 16 plus 26, which is 48.
- One leg is the height of 5. The other spans the whole difference of 12, so it is √(25 + 144), which is 13. The perimeter is 46.
- 62 minus 47 is 15. Perimeter questions run backwards without any rearranging.
- The perimeter doubles to 124, because every side doubled. The area quadruples to 816, because it is built from two lengths multiplied together.
Common questions
What is the formula for the perimeter of a trapezoid?
P = a + b + c + d, the sum of all four sides. Parallel sides of 10 and 24 with legs of 13 and 15 give a perimeter of 62. There is no shortcut version, because every side has to be counted individually.
How do you find the perimeter without knowing the legs?
Build them from the height. Each leg is the square root of the height squared plus its own overhang, and the two overhangs add up to the difference between the parallel sides. With a height of 12 and overhangs of 5 and 9 the legs come out at 13 and 15.
Can you find the perimeter from the area?
No. Trapezoids with parallel sides of 10 and 24 and a height of 12 all have an area of 204, and their perimeters range from 61.78 upwards with no limit. The area cannot see which way the shape leans, and the perimeter can see nothing else.
What is the perimeter of an isosceles trapezoid?
Halve the difference between the parallel sides to get the overhang, find one leg with Pythagoras, then double it. Parallel sides of 8 and 18 with a height of 12 give legs of 13 each and a perimeter of 52.
Does the height count as a side?
Only on a right trapezoid, where one leg happens to be perpendicular and so is the height as well. On every other trapezoid the height is an internal measurement and adding it to the perimeter is simply counting something that is not an edge.
Which trapezoid needs the least fencing for a given area?
The symmetric one. For parallel sides of 10 and 24 and a height of 12, the isosceles shape needs 61.78 against 64.44 for the right trapezoid and 70.74 for a strongly leaning one, all enclosing the same 204 square units.
Related lessons and calculators
- Area of a trapezoid, the other measurement of the same shape, and the one the midline was built for.
- Height of a trapezoid, which is where the overhangs come from when only the four sides are known.
- Missing side of a right triangle, the Pythagoras step that turns a height and an overhang into a leg.
- Missing side of a triangle from the perimeter, the same subtraction on a shape with one side fewer.
- Area of 2D shapes calculator, for trapezoids given as four sides without the arithmetic.
- Area formulas for every common shape, the index this lesson sits inside.
- Wolfram MathWorld on the perimeter, for the general definition across shapes and curves.