What is the area of my trapezoid?
Type the two parallel sides and the height to get the area, or type all four sides to get the height, area and angles. The trapezoid calculator works with exact fractions and shows how each number is found.
- Area
- 28
A trapezoid with bases 10 and 4 and height 4 has an area of 28.
- Midsegment
- 7
- Height
- 4
Area: 28. A trapezoid with bases 10 and 4 and height 4 has an area of 28.
How to calculate
Computes a trapezoid’s area, midsegment and perimeter from its bases and height, or its height, area and angles from all four sides.
Example with the default inputs (I know Bases and height, Base a (bottom) 10, Base b (top) 4, Height h 4): A trapezoid with bases 10 and 4 and height 4 has an area of 28.
Method: Area = (a + b) ÷ 2 × h; midsegment = (a + b) ÷ 2; perimeter = a + b + c + d; from four sides: x = (c² − d² + (a − b)²) ÷ (2(a − b)), h = √(c² − x²).
- a and b are the parallel sides (bases); c joins their left ends and d their right ends.
- All lengths are in the same unit; the area is in that unit squared.
Worked examples
Each example is checked against the calculator on every build.
- I know Bases and height, Base a (bottom) 14, Base b (top) 11, Height h 6 gives Area 75, Midsegment 12.5.Source: OpenStax, Prealgebra 2e, §9.4 Use Properties of Rectangles, Triangles, and Trapezoids, Example 9.38 (A = ½h(b + B): height 6, bases 14 and 11, area 75). https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- I know Bases and height, Base a (bottom) 8.2, Base b (top) 5.6, Height h 3.4 gives Area 23.46, Midsegment 6.9.Source: OpenStax, Prealgebra 2e, §9.4, Example 9.40 (height 3.4 yd, bases 8.2 and 5.6 yd: 23.46 square yards). https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- I know All four sides, Base a (bottom) 10, Base b (top) 4, Leg c (left) 5, Leg d (right) 5 gives Height 4, Area 28, Perimeter 24, Angle between a and c 53.130102, Angle between b and c 126.869898.Source: OpenStax, Prealgebra 2e, §9.4 (area of a trapezoid) and §9.3 (Pythagorean theorem). https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- I know All four sides, Base a (bottom) 7, Base b (top) 3, Leg c (left) 3, Leg d (right) 5 gives Height 3, Area 15, Angle between a and c 90, Angle between a and d 36.869898.Source: OpenStax, Prealgebra 2e, §9.4 (area of a trapezoid) and §9.3 (Pythagorean theorem). https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
- I know Bases and height, Base a (bottom) 10, Base b (top) 4, Height h 4, Leg c (left) 5, Leg d (right) 5 gives Area 28, Perimeter 24.Source: OpenStax, Prealgebra 2e, §9.4 Use Properties of Rectangles, Triangles, and Trapezoids. https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids
How it works
Base a is the bottom parallel side and base b the top one. Leg c joins their left ends, leg d their right ends.
Bases and height.
- Area = (a + b) ÷ 2 × h
- Midsegment = (a + b) ÷ 2
- Perimeter = a + b + c + d, shown when you type both legs
When you type both legs, they must fit: each leg is at least the height, and the horizontal spans p = √(c² − h²) and q = √(d² − h²) satisfy p + q = |a − b| or |p − q| = |a − b| (checked to 1 part in 10⁹). Otherwise there is no answer.
All four sides. Put base a on the x-axis from (0, 0) to (a, 0) and the top base from (x, h) to (x + b, h). Then c² = x² + h² and d² = (x + b − a)² + h². Subtracting gives:
- x = (c² − d² + (a − b)²) ÷ (2(a − b))
- h = √(c² − x²)
- Area = (a + b) ÷ 2 × h; perimeter = a + b + c + d
- Angle between a and c = atan2(h, x); angle between a and d = atan2(h, (a − b) − x), both in degrees
- Angle between b and c = 180° − the angle between a and c; angle between b and d = 180° − the angle between a and d
When there is no answer.
- All four sides with a = b: a parallelogram, whose height the sides do not fix.
- All four sides with c² − x² ≤ 0: the legs and the difference of the bases cannot form a triangle.
- Bases and height with legs that do not fit (above).
- All four sides with a leg left empty.
Exact arithmetic. Each number is read as the exact decimal you typed. Area, midsegment, perimeter and x are exact fractions; the height is an exact square root when c² − x² is the square of a fraction, and a 64-bit float otherwise. Angles are floats.
Output format. Lengths and the area show at most 6 decimals, angles at most 4, each rounded half up from its value, with trailing zeros left off.
Assumptions
- All lengths are in the same unit, more than 0 and at most 10⁹; the area is in that unit squared.
Worked examples by hand
Bases 14 and 11, height 6 (OpenStax Example 9.38). (14 + 11) ÷ 2 = 12.5 (midsegment); 12.5 × 6 = 75.
Bases 8.2 and 5.6, height 3.4 (OpenStax Example 9.40). (8.2 + 5.6) ÷ 2 = 6.9; 6.9 × 3.4 = 23.46, exactly.
Bases 10 and 4, legs 5 and 5. x = (25 − 25 + 6²) ÷ (2 × 6) = 3; h = √(25 − 9) = 4; area = 7 × 4 = 28; perimeter = 24; angle between a and c = atan2(4, 3) = 53.1301°, angle between b and c = 126.8699°.
Bases 7 and 3, legs 3 and 5. x = (9 − 25 + 4²) ÷ 8 = 0, so leg c is upright; h = 3; area = 5 × 3 = 15; angles 90° and atan2(3, 4) = 36.8699°.
Bases 10 and 4, height 4, legs 5 and 5. Area 28; p = q = √(25 − 16) = 3 and p + q = 6 = 10 − 4, so the legs fit; perimeter 24.
Other questions people ask
What is the formula for the area of a trapezoid?
Area = (a + b) ÷ 2 × h, where a and b are the parallel sides (the bases) and h is the height between them. A trapezoid with bases 14 and 11 and height 6 has an area of (14 + 11) ÷ 2 × 6 = 75.
How do I find the height of a trapezoid from its sides?
Slide the two legs together to form a triangle with sides c, d and a − b. Its height is the trapezoid’s height. In numbers, x = (c² − d² + (a − b)²) ÷ (2(a − b)) and h = √(c² − x²). For bases 10 and 4 and legs 5 and 5: x = 3 and h = 4.
What is the midsegment of a trapezoid?
The midsegment (or median) joins the midpoints of the two legs. It is parallel to the bases and its length is their average, (a + b) ÷ 2. The area is the midsegment times the height.
Why can I not find the height when the bases are equal?
Equal parallel sides make a parallelogram, which can lean by any amount with the same four sides. The sides do not fix its height, so the page asks for the height instead.
How do I find the perimeter of a trapezoid?
Add all four sides: a + b + c + d. In "Bases and height" mode, type both legs to see it; the page checks that the legs fit the bases and the height.
Is a trapezium the same as a trapezoid?
In British English a trapezium is a four-sided shape with one pair of parallel sides, which is what American English calls a trapezoid. The formulas are the same.