Trapezoid Calculator
Area, perimeter, height, legs, angles and median of any trapezoid
The area of a trapezoid is the average of its two parallel sides times its height: A = (a + b) ÷ 2 × h. A trapezoid with parallel sides of 6 ft and 16 ft that are 12 ft apart covers (6 + 16) ÷ 2 × 12 = 132 square feet. This calculator also finds the perimeter, the legs and angles of isosceles and right trapezoids, the median, and a missing base or height from the area.
Last updated October 2026
Method: Exact Euclidean geometry. Area = (a + b) ÷ 2 × h and median = (a + b) ÷ 2. Legs and diagonals come from the Pythagorean theorem on each leg's horizontal overhang; with four known sides, the height comes from the triangle formed by the two legs and the difference of the bases. Values are rounded only for display.
Included: Area, perimeter, median (midsegment), height, both legs, both diagonals, all four interior angles, the shape type (isosceles, right or scalene), a missing base or height from the area, and a labeled sketch of the shape.
Not included: Unit conversion, volumes of trapezoidal prisms, the calculus trapezoidal rule, and four-sided shapes without any pair of parallel sides.
⏢ What do you know about the trapezoid?
The perpendicular distance between the two parallel sides.
The label is cosmetic. Enter every length in the same unit; the area comes out in square units.
Area of the trapezoid
Shape: Isosceles trapezoid
📐 Sketch with labeled sides
🧮 Full breakdown
📋 Step by step
- Median. m = (a + b) ÷ 2 = (6 + 16) ÷ 2 = 11 ft.
- Area. A = m × h = 11 × 12 = 132 ft².
- Legs. Each leg overhangs by |b − a| ÷ 2 = 5 ft, so leg = √(h² + 5²) = 13 ft.
- Perimeter. P = a + b + c + d = 6 + 16 + 13 + 13 = 48 ft.
- Angles. Bottom angles 67.3801° and 67.3801°; each top angle is 180° minus the bottom angle on the same leg (112.6199° and 112.6199°).
Trapezoid formulas, worked examples and special cases
A trapezoid is a four-sided figure with a pair of parallel sides, called the bases. The other two sides are the legs, and the height is the perpendicular distance between the bases. Every number in the examples below comes straight from the calculator and can be reproduced with its Try buttons.
The trapezoid formulas
With a top base a, a bottom base b, legs c and d and height h:
A = (a + b) ÷ 2 × h m = (a + b) ÷ 2, so A = m × h P = a + b + c + d Isosceles leg: c = √(h² + ((b − a) ÷ 2)²) Right trapezoid, slanted leg: d = √(h² + (b − a)²) h = 2A ÷ (a + b) b = 2A ÷ h − a The area formula has a short proof. Set an upside-down copy of the trapezoid next to the original and the pair forms a parallelogram with base a + b and height h, so one trapezoid is half of (a + b) × h. A diagonal gives the same result by splitting the shape into two triangles, ½ah + ½bh. Both arguments follow the composing and decomposing approach of the Common Core grade 6 standards. The median, or midsegment, joins the midpoints of the legs, runs parallel to the bases and equals their average, which is why the area is also the median times the height.
Worked example 1: an isosceles garden bed
A raised bed is 6 ft wide at the back and 16 ft wide at the front, the two parallel edges are 12 ft apart, and both angled sides are equal:
A = (6 + 16) ÷ 2 × 12 = 11 × 12 = 132 ft² Because the bed is isosceles, each angled side overhangs the back edge by half the difference of the bases, (16 − 6) ÷ 2 = 5 ft. Each leg is the hypotenuse of a 5 by 12 right triangle, √(12² + 5²) = 13 ft, so the edging runs 6 + 16 + 13 + 13 = 48 ft. The front corners measure arctan(12 ÷ 5) = 67.38°, the back corners 112.62°, and both diagonals 16.28 ft. Filling the bed 8 inches deep takes 132 × 8 ÷ 12 = 88 cubic feet of soil, about 3.26 cubic yards.
Worked example 2: a right-trapezoid deck
A deck is 15 ft long where it meets the house and 9 ft long at its outer edge, 8 ft from the house. One end is square to the house and the other is cut at an angle:
A = (9 + 15) ÷ 2 × 8 = 12 × 8 = 96 ft² The square end is a leg equal to the height, 8 ft. The angled end covers the 15 − 9 = 6 ft difference over the 8 ft depth, so it is √(8² + 6²) = 10 ft. The perimeter is 42 ft, and railing on the three open sides totals 9 + 8 + 10 = 27 ft. The angled corner at the house measures 53.13° and the outer corner beside it 126.87°. The diagonals are 12.04 ft and exactly 17 ft.
Worked example 3: all four sides known
A garden plot has parallel fences of 10 yd and 24 yd and side fences of 13 yd and 15 yd, but nobody measured the distance between the parallel fences. The bases differ by D = 24 − 10 = 14 yd, the top-left corner sits x = (13² − 15² + 14²) ÷ (2 × 14) = 5 yd in from the bottom-left corner, and the height is √(13² − 5²) = 12 yd:
A = (10 + 24) ÷ 2 × 12 = 17 × 12 = 204 yd² The perimeter is 62 yd and the bottom corners measure 67.38° and 53.13°. As a cross-check, the two legs and the 14 yd difference form a 13-14-15 triangle; Heron's formula gives it an area of 84 square yards, so its height is 2 × 84 ÷ 14 = 12 yd.
Solving backward: a missing base or the height
Rearranging the area formula answers the two most common homework variations. For a missing base, b = 2A ÷ h − a: a 120 sq ft bed that is 8 ft deep with a 12 ft back edge needs a front edge of 2 × 120 ÷ 8 − 12 = 18 ft. For the height, h = 2A ÷ (a + b): an area of 132 with bases of 6 and 16 gives 264 ÷ 22 = 12. The area must also exceed half the known base times the height (48 sq ft in the first case), or the missing base would be zero or negative; the calculator flags that case.
How the height shapes an isosceles trapezoid: computed table
Here the bases stay at 10 and 16, so each leg overhangs by 3, and only the height changes. All values come from the calculator's formulas.
| Height h | Each leg | Base angle | Area | Perimeter | Diagonal |
|---|---|---|---|---|---|
| 1 | 3.162 | 18.43° | 13 | 32.325 | 13.038 |
| 2 | 3.606 | 33.69° | 26 | 33.211 | 13.153 |
| 3 | 4.243 | 45.00° | 39 | 34.485 | 13.342 |
| 4 | 5.000 | 53.13° | 52 | 36.000 | 13.601 |
| 5 | 5.831 | 59.04° | 65 | 37.662 | 13.928 |
| 6 | 6.708 | 63.43° | 78 | 39.416 | 14.318 |
| 8 | 8.544 | 69.44° | 104 | 43.088 | 15.264 |
| 10 | 10.440 | 73.30° | 130 | 46.881 | 16.401 |
The area grows in a straight line, 13 square units per unit of height, because the median is (10 + 16) ÷ 2 = 13. The legs behave differently: at h = 3 the base angle is exactly 45°, at h = 4 each leg is exactly 5 (a 3-4-5 triangle), and at h = 10 the leg is only 0.44 longer than the height.
Isosceles, right and scalene trapezoids
An isosceles trapezoid has equal legs, equal base angles and equal diagonals, and it is symmetric about the line through the midpoints of its bases. A right trapezoid has two 90° angles, so one leg is the height itself. Any other trapezoid is scalene. Textbooks also disagree on the definition: the exclusive one requires exactly one pair of parallel sides, the inclusive one at least one pair, which makes every parallelogram a trapezoid. The formulas here work under either.
How to use the trapezoid calculator
Pick the mode that matches your measurements. Bases + height covers the most common case and also computes the legs when you mark the shape as isosceles or right; choose Area only when the legs are irregular. Bases + leg finds the height of an isosceles trapezoid from its slanted sides. All four sides works for any trapezoid with unequal bases. From the area solves for a missing base or the height. The unit label converts nothing, and the sketch marks the height and the median with dashed lines.
Who uses a trapezoid calculator
Students use it for geometry homework. Gardeners lay out tapered beds, and deck builders cut angled ends. Each long face of a hip roof is a trapezoid whose height must be measured along the roof surface, not as the vertical rise: a face with a 40 ft eave, a 20 ft ridge and a 15 ft slant height covers 450 sq ft. Drainage work uses trapezoidal cross-sections: a ditch 4 ft wide at the bottom, 10 ft wide at the top and 3 ft deep has a 21 sq ft cross-section, so 100 ft of it holds 2,100 cubic feet, about 77.78 cubic yards. The trapezoidal rule in calculus uses the same averaging, adding strips of area (y₁ + y₂) ÷ 2 × Δx.
Key terms
- Bases (a and b): the two parallel sides.
- Legs (c and d): the two sides that are not parallel.
- Height (h): the perpendicular distance between the bases, also called the altitude.
- Median or midsegment (m): the segment joining the midpoints of the legs, m = (a + b) ÷ 2.
- Overhang: the horizontal distance a leg covers; (b − a) ÷ 2 for each leg of an isosceles trapezoid.
- Base angles: the two angles along one base. The two angles along the same leg add up to 180°.
What changes the result, and tips
- Where the height is measured. A height taken along a leg inflates the area.
- The shape you assume. Bases and height fix the area but not the legs: with bases of 6 and 16 and a height of 12, an isosceles shape needs 48 ft of edging, a right one 49.62 ft.
- Mixed units. Convert inches to feet before multiplying; 8 in is 0.667 ft.
- Tip: for an irregular plot, measure all four sides and use the four-sides mode instead of guessing the height.
- Tip: check the answer a second way, such as median times height.
Limitations
The calculator needs a true trapezoid with two parallel bases. In the four-sides mode the bases must differ, because the four sides of a parallelogram do not fix its height, and the legs must be able to form a triangle with the difference of the bases: legs of 3 and 4 cannot bridge a gap of 14. Shapes with no parallel sides should be split into triangles. Results are rounded to four decimals for display, and no units are converted.
Related calculators
- For a shape with two pairs of parallel sides, use the Parallelogram Calculator.
- For rectangles, circles, triangles and other shapes side by side, use the Area Calculator.
- A diagonal splits any trapezoid into two triangles, which the Triangle Calculator can solve.
- For the distance around other shapes, use the Perimeter Calculator.
- The leg formulas are the Pythagorean Theorem Calculator applied to each overhang.
- For hip and gable roof faces, continue with the Roofing Calculator.
Sources
Trapezoid formulas are exact geometry, so every figure here was computed from formulas documented in these primary references:
- NIST Digital Library of Mathematical Functions, §4.42 Solution of Triangles: the law of cosines (equation 4.42.5) and Heron's formula (equation 4.42.7), the basis of the four-sides mode.
- Common Core State Standards for Mathematics, Grade 6 Geometry, standard 6.G.A.1: finding the area of special quadrilaterals by composing them into rectangles or decomposing them into triangles, as in the proof above.
- Worked examples and the table: computed with the same formulas the calculator uses and rounded only at the last step.
⚠️ Common mistakes & edge cases
Using a leg as the height
The legs of the garden bed are 13 ft, but its parallel edges are only 12 ft apart. Plugging in 13 gives 143 ft², about 8% too much. Only a right trapezoid has a leg that equals the height.
Forgetting to halve the sum of the bases
(6 + 16) × 12 = 264 is the area of two trapezoids put together. Dividing by 2, or multiplying by the median instead, gives the correct 132 ft².
Using the full difference of the bases for an isosceles leg
Each leg overhangs by half the difference, (16 − 6) ÷ 2 = 5 ft. Using the full 10 ft gives a leg of 15.62 ft instead of 13 ft, which is the slanted side of a right trapezoid, not an isosceles one.
Assuming the bases and height fix the perimeter
The same 132 ft² bed can be isosceles with 48 ft of edging or right-angled with 49.62 ft. Without the legs or the shape type, the perimeter is unknown, which is why the Area only option leaves it blank.
Equal bases in the four-sides mode
With equal bases the shape is a parallelogram that can lean freely, so four side lengths cannot pin down its height or area. Enter the height in the bases-and-height mode instead.
Legs that cannot close the shape
With bases of 10 and 24, legs of 3 and 4 add up to 7 and cannot span the 14-unit difference, so the calculator explains the problem instead of returning a height.
❓ Frequently asked questions
How do you find the area of a trapezoid?
Add the two parallel sides, divide by 2 and multiply by the height: A = (a + b) ÷ 2 × h. The height is the perpendicular distance between the parallel sides. A garden bed with parallel edges of 6 ft and 16 ft that are 12 ft apart covers (6 + 16) ÷ 2 × 12 = 132 square feet.
What is the trapezoid formula?
Area A = (a + b) ÷ 2 × h, where a and b are the parallel bases and h is the height. Perimeter P = a + b + c + d, the sum of all four sides. Median m = (a + b) ÷ 2, so the area is also m × h. Rearranged, the height is h = 2A ÷ (a + b) and a missing base is b = 2A ÷ h − a.
How do you find the perimeter of a trapezoid?
Add all four sides. With only the bases and the height you also need the shape: an isosceles trapezoid with bases 6 and 16 and height 12 has legs of 13 each and a perimeter of 48, while a right trapezoid with bases 9 and 15 and height 8 has legs of 8 and 10 and a perimeter of 42.
How do you find the height of a trapezoid?
From the area, divide twice the area by the sum of the bases: h = 2A ÷ (a + b). An area of 132 with bases 6 and 16 gives 264 ÷ 22 = 12. For an isosceles trapezoid you can also use a leg: h = √(c² − ((b − a) ÷ 2)²), so a 13 ft leg with bases of 6 ft and 16 ft gives 12 ft.
How do you find a missing base of a trapezoid?
Rearrange the area formula to b = 2A ÷ h − a. A 120 square foot trapezoid that is 8 ft high with one base of 12 ft needs another base of 2 × 120 ÷ 8 − 12 = 18 ft. If the result is zero or negative, the area is too small for that base and height, and no such trapezoid exists.
How do you find the leg length of an isosceles trapezoid?
Each leg overhangs the shorter base by half the difference of the bases, so the leg is the hypotenuse of a right triangle: c = √(h² + ((b − a) ÷ 2)²). With bases of 6 and 16 and a height of 12, the overhang is 5 and each leg is √(144 + 25) = 13.
What is the median or midsegment of a trapezoid?
It is the segment that joins the midpoints of the two legs. It runs parallel to the bases and its length is their average, m = (a + b) ÷ 2. For bases of 6 ft and 16 ft the median is 11 ft, and the area equals the median times the height: 11 × 12 = 132 square feet.
Are the diagonals of a trapezoid equal?
Only in an isosceles trapezoid. The garden bed with bases 6 and 16 and legs of 13 has two diagonals of 16.28. The trapezoid with sides 10, 24, 13 and 15 is not isosceles, and its diagonals measure 19.21 and 22.47. A trapezoid with equal diagonals is always isosceles.
Is a parallelogram a trapezoid?
It depends on the definition your course uses. Under the exclusive definition a trapezoid has exactly one pair of parallel sides, so a parallelogram is not a trapezoid. Under the inclusive definition it has at least one pair, so every parallelogram counts. The area formula works either way.
How do you find the area of a right trapezoid?
Use the same formula, A = (a + b) ÷ 2 × h. In a right trapezoid one leg is the height. A deck with parallel edges of 9 ft and 15 ft and a square end of 8 ft covers (9 + 15) ÷ 2 × 8 = 96 square feet, and its angled end is √(8² + 6²) = 10 ft.
💡 Good to know
Shrink the top base to zero and you get a triangle
With a = 0 the formula becomes (0 + b) ÷ 2 × h = ½ × b × h, the triangle area formula.
Equal bases give a parallelogram
With a = b the formula becomes (b + b) ÷ 2 × h = b × h, the parallelogram area formula, which is why the inclusive definition treats parallelograms as trapezoids.
The median is the average width
For material estimates, picture a rectangle as wide as the median: the garden bed behaves like an 11 ft by 12 ft rectangle, 132 ft², though no edge is 11 ft long.