Trapezoid Area Calculator
Find the area, perimeter, height, and midsegment of any trapezoid — standard, isosceles, right, or irregular — with complete step-by-step working. Also called a trapezium calculator (UK). Supports all four trapezoid types using side lengths or coordinates.
Leave one field empty and enter height instead — the calculator will find the missing value.
Enter the four vertices in order (clockwise or counter-clockwise). Uses the Shoelace formula for exact area.
Trapezoid Area Formula — A = ((a+b)/2) × h
The trapezoid area formula is A = ((a + b) / 2) × h, where a and b are the two parallel sides (called the bases) and h is the perpendicular height — the straight-line distance between the two parallel sides measured at right angles. The formula works for any trapezoid: standard, isosceles, right, or irregular trapezoid.
Intuition: The term (a+b)/2 is the midsegment — the average width of the trapezoid. Multiplying the average width by the height gives the area, exactly as for a rectangle. This is why the formula A = ((a+b)/2) × h can be written as A = m × h (area = midsegment × height).
Special Cases of the Trapezoid Area Formula
| Shape | Condition | Formula reduces to |
|---|---|---|
| Rectangle | a = b | A = a × h (length × width) |
| Parallelogram | a = b | A = base × height |
| Triangle | a = 0 | A = (b/2) × h = (1/2) × base × height |
| Trapezoid | a ≠ b, a > 0 | A = ((a+b)/2) × h |
How to Find the Area of a Trapezoid — Step-by-Step
To find the area of a trapezoid, follow these five steps every time. The most common mistake is using the leg length instead of the perpendicular height — always verify that h is truly perpendicular.
- Identify the two parallel sides — call them a (shorter, top) and b (longer, bottom). The parallel sides are the bases.
- Measure the perpendicular height h — this is the straight vertical distance between the two parallel sides, NOT the length of the slanted legs. The perpendicular height makes a right angle with both bases.
- Add the two bases: a + b
- Divide by 2: (a + b) / 2 — this is the midsegment length
- Multiply by h: ((a + b) / 2) × h = area in square units
Critical distinction: h is the perpendicular height (vertical distance between bases) — NOT the slant length of the legs. The #1 error in trapezoid area problems is using the leg length as h. If you have the leg length but not h, use the Pythagorean theorem: h = √(leg² − ((b−a)/2)²).
Example 1 — a = 5 cm, b = 11 cm, h = 4 cm
- A = ((a+b)/2) × h
- = ((5+11)/2) × 4
- = (16/2) × 4
- = 8 × 4
- A = 32 cm²
- Midsegment: m = (5+11)/2 = 8 cm. Verify: A = 8 × 4 = 32 cm² ✓
Example 2 — a = 6 m, b = 14 m, h = 8 m
- A = ((6+14)/2) × 8 = (20/2) × 8 = 10 × 8
- A = 80 m²
Example 3 — Reverse: A = 50 cm², a = 6 cm, b = 14 cm → find h
- h = 2A / (a + b) = 2 × 50 / (6 + 14) = 100 / 20
- h = 5 cm
Trapezoid vs Trapezium — US and UK Terminology
This trapezium calculator handles what is called a trapezium in the UK, Australia, and most of the world — a quadrilateral with exactly one pair of parallel sides. The terminology difference causes significant confusion:
| Country | "Trapezoid" | "Trapezium" |
|---|---|---|
| United States | One pair of parallel sides ✓ | NO parallel sides |
| UK / Australia / Canada | NO parallel sides | One pair of parallel sides ✓ |
This trapezoid area calculator follows US convention (trapezoid = one pair of parallel sides) — but the formulas and calculations are identical regardless of which term you use. If you searched for trapezium calculator, you are in the right place — this calculator computes the area of a trapezium (UK) / trapezoid (US) using A = ((a+b)/2) × h.
Isosceles Trapezoid — Equal Legs and Symmetric Properties
An isosceles trapezoid has two legs (non-parallel sides) of equal length. This symmetry gives it special properties: equal base angles, equal diagonals, and a vertical line of symmetry through the midpoints of both parallel sides.
How to derive the height: Drop a perpendicular from each top corner to the bottom base. Each perpendicular creates a right triangle with: hypotenuse = c (the leg), horizontal leg = (b−a)/2 (the horizontal excess on each side), and vertical leg = h. By the Pythagorean theorem: h² + ((b−a)/2)² = c², so h = √(c² − ((b−a)/2)²).
Example — Isosceles: a=4, b=10, c=5
- half_diff = (b−a)/2 = (10−4)/2 = 3
- h = √(5² − 3²) = √(25−9) = √16 = 4
- Area = ((4+10)/2) × 4 = 7 × 4 = 28 cm²
- Perimeter = 4 + 10 + 5 + 5 = 24 cm
- Leg angle θ = arccos(3/5) = 53.13°
Irregular Trapezoid Area Calculator — Using Coordinates
When the height of a trapezoid is not directly measurable — or when you have an irregular trapezoid defined by the coordinates of its four corners — use the Shoelace formula (also called the Surveyor's formula) to find the exact area.
The Shoelace formula works for any polygon — not just trapezoids. It requires the vertices to be listed in order (either all clockwise or all counter-clockwise). The irregular trapezoid tool above also detects whether the entered shape actually has parallel sides.
Example — Shoelace for vertices (0,0), (4,0), (4,3), (0,3)
- Sum = (0×0 − 4×0) + (4×3 − 4×0) + (4×3 − 0×3) + (0×0 − 0×3)
- = 0 + 12 + 12 + 0 = 24
- A = |24| / 2 = 12 sq units ✓ (matches 4×3 rectangle)
Trapezoid Perimeter — All Four Sides
The trapezoid perimeter formula is simply P = a + b + c + d — the sum of all four sides. The challenge is that for a standard trapezoid, if only a, b, and h are known, the leg lengths c and d must be computed.
| Trapezoid type | Leg c | Leg d |
|---|---|---|
| Isosceles | c = √(h² + ((b−a)/2)²) | d = c (equal) |
| Right trapezoid | c = h (perpendicular leg) | d = √(h² + (b−a)²) |
| General | Given or from geometry | Given or from geometry |
Common Mistakes in Trapezoid Calculations
Mistake 1 — Using the leg length instead of perpendicular height
- ❌ Wrong: using the slanted leg c as h in A = ((a+b)/2) × h
- ✅ Correct: h is the perpendicular distance between the two parallel sides — it makes a right angle with both bases. Always confirm h is truly perpendicular.
Mistake 2 — Forgetting to divide by 2
- ❌ Wrong: A = (a+b) × h = 16 × 4 = 64
- ✅ Correct: A = ((a+b)/2) × h = 8 × 4 = 32
Mistake 3 — Confusing which sides are parallel
- a and b must be the two parallel sides (the bases). The legs c and d are the non-parallel sides. If you use legs instead of bases in the formula, the area will be wrong.
Mistake 4 — Wrong vertex order for Shoelace formula
- The Shoelace formula requires vertices in consecutive order around the perimeter (CW or CCW). Entering them in the wrong order (e.g., diagonally opposite vertices first) gives a wrong area for an irregular trapezoid.
Mistake 5 — Confusing midsegment with diagonal
- The midsegment connects the midpoints of the TWO LEGS — it's parallel to both bases and has length m = (a+b)/2. A diagonal connects opposite VERTICES — entirely different.
Trapezoid Worked Examples — 8 Problems
1. Standard trapezoid: a=5, b=11, h=4
- A = ((5+11)/2) × 4 = 8 × 4 = 32 cm²
- m = (5+11)/2 = 8 cm
2. Isosceles: a=6, b=14, c=5
- h = √(5² − ((14−6)/2)²) = √(25−16) = √9 = 3
- A = ((6+14)/2) × 3 = 10 × 3 = 30 cm²
- P = 6+14+5+5 = 30 cm
3. Right trapezoid: a=5, b=11, h=4
- A = ((5+11)/2) × 4 = 32 cm²
- Slant leg d = √(4² + (11−5)²) = √(16+36) = √52 ≈ 7.21 cm
- P = 5+11+4+7.21 = 27.21 cm
4. Reverse — find height: A=50, a=6, b=14
- h = 2A/(a+b) = 2×50/(6+14) = 100/20 = 5 cm
5. Irregular — coordinates: (0,0),(6,0),(5,4),(1,4)
- Shoelace: (0×0−6×0)+(6×4−5×0)+(5×4−1×4)+(1×0−0×4)
- = 0 + 24 + 16 + 0 = 40
- A = 40/2 = 20 sq units
6. Perimeter — unknown leg: a=4, b=10, h=3, c=5 (isosceles)
- d = c = 5 (isosceles), P = 4+10+5+5 = 24 cm
7. Midsegment given: A=40, h=8 → find midsegment
- m = A/h = 40/8 = 5 units
- And (a+b)/2 = 5, so a+b = 10
8. Real-world — road cross-section: a=8m, b=16m, h=3m
- A = ((8+16)/2) × 3 = 12 × 3 = 36 m² (cross-sectional area)
- Volume per 100m length = 36 × 100 = 3,600 m³