What this calculator does
An isosceles trapezium (also spelled trapezoid in the US) is a four-sided shape with one pair of parallel sides and two legs of equal length, arranged symmetrically. Because both legs are the same length, this calculator only needs one leg measurement rather than two, unlike a general trapezoid where the two legs can differ.
Symmetry also means the height and both base angles can be worked out from just three numbers: the two parallel sides and the shared leg length. That is the whole point of a dedicated isosceles page rather than always reaching for the general four-input trapezoid calculator.
The formula
Half the difference between the two parallel sides, and the leg length, form the two shorter sides of a right triangle whose hypotenuse is the leg itself. The height is the third side of that triangle, found with the Pythagorean theorem: height = √(leg² − half-difference²). Area is then the usual (a+b)/2 × height, perimeter is both parallel sides plus twice the leg, and the base angle comes from the inverse cosine of half-difference ÷ leg.
| Term | Meaning |
|---|---|
| a, b | The two parallel sides (a is treated as the longer one; the calculator sorts this automatically if entered the other way round). |
| c | The leg length, the same on both non-parallel sides. |
| Base angle | The angle between the longer parallel side and a leg; the angle at the shorter side is 180° minus this. |
The inputs explained
| Field | What to enter |
|---|---|
| Longer parallel side (a) | One of the two parallel sides. It does not matter which one you call a and which b; the calculator works out which is longer. |
| Shorter parallel side (b) | The other parallel side. |
| Leg length (equal on both sides) | The length of each leg, since both legs are equal in an isosceles trapezium. |
When to use it
Cutting a trapezium panel or tabletop
Knowing the exact height and base angles lets you mark out an isosceles trapezium panel accurately from just the two parallel edges and one leg measurement, without needing to measure both legs separately.
Checking a design is actually isosceles
If a shape is meant to be symmetric, entering the two parallel sides and one leg and confirming the base angles are equal on both sides is a quick sanity check.
Working out roof or ramp geometry
Many sloped structures, such as a trapezoidal roof section or garden bed edge, are isosceles by design, and the base angle here is the slope angle at the wider edge.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How area and angle change as the leg gets longer
The two parallel sides held at 12 and 8, across a range of leg lengths.
| Leg length | Area | Height | Base angle (at the longer side) |
|---|---|---|---|
| 3 | 22.361 | 2.236 | 48.19° |
| 4 | 34.641 | 3.464 | 60.00° |
| 5 | 45.826 | 4.583 | 66.42° |
| 6 | 56.569 | 5.657 | 70.53° |
| 8 | 77.460 | 7.746 | 75.52° |
| 10 | 97.980 | 9.798 | 78.46° |
How area changes as the parallel sides move further apart
A fixed 6-unit leg, across a range of shorter parallel side lengths against a 12-unit longer side.
| Shorter parallel side | Area | Height | Base angle (at the longer side) |
|---|---|---|---|
| 4 | 35.777 | 4.472 | 48.19° |
| 6 | 46.765 | 5.196 | 60.00° |
| 8 | 56.569 | 5.657 | 70.53° |
| 10 | 65.077 | 5.916 | 80.41° |
| 11 | 68.760 | 5.979 | 85.22° |
| 11.9 | 71.698 | 6.000 | 89.52° |
Questions
What makes a trapezium isosceles rather than just a general trapezoid?
In an isosceles trapezium the two non-parallel legs are equal in length and the shape is symmetric about a line through the midpoints of the parallel sides. A general trapezoid allows the two legs to be different lengths, which is why the site’s general trapezoid calculator asks for both legs separately.
Why does the calculator say the sides cannot form a trapezium?
The leg has to be long enough to bridge half the difference between the two parallel sides. If the leg is shorter than that half-difference, there is no way to close the shape, so lengthen the leg or bring the two parallel sides closer together.
Are the two base angles at the same parallel side always equal?
Yes. That equal-base-angle property is a direct consequence of the shape being isosceles, and it is one of the standard ways to prove a trapezium is isosceles in geometry.
Does the order I enter the two parallel sides matter?
No. The calculator automatically treats whichever value is larger as the longer parallel side, so entering them in either order gives the same area, height and perimeter.
For a trapezoid with two independently-sized legs, use the general trapezoid calculator. For other symmetric shape variants, see the isosceles triangle calculator.