Area of a Trapezoid Calculator
Calculate the area and perimeter of a trapezoid from its two parallel bases and height. See also Area of Rectangle Calculator and Area of Parallelogram Calculator.
How to Calculate the Area of a Trapezoid
A trapezoid (or trapezium) has two parallel sides called bases and two non-parallel sides called legs. To find the area, add the two bases together, multiply by the height (the perpendicular distance between the bases), and divide by 2. If you also know the leg lengths, you can calculate the perimeter by adding all four sides.
Trapezoid Area Formula
A = ½ × (a + b) × h
Perimeter = a + b + side₁ + side₂
Where a and b are the parallel bases and h is the perpendicular height between them.
Example
Find the area of a trapezoid with bases 8 and 12, height 5:
A = ½ × (a + b) × h
A = ½ × (8 + 12) × 5
A = ½ × 20 × 5
A = 50 square units
Trapezoid Area Reference Table
| Base a | Base b | Height | Area |
|---|---|---|---|
| 3 | 5 | 2 | 8.00 |
| 4 | 6 | 3 | 15.00 |
| 5 | 8 | 4 | 26.00 |
| 6 | 10 | 5 | 40.00 |
| 8 | 12 | 5 | 50.00 |
| 10 | 14 | 6 | 72.00 |
| 10 | 16 | 7 | 91.00 |
| 12 | 18 | 8 | 120.00 |
| 15 | 20 | 10 | 175.00 |
| 20 | 30 | 12 | 300.00 |
Real-World Applications of Trapezoid Area
Property Lots
Many residential lots are trapezoidal — wider at the street and narrower at the back (or vice versa). Calculating the lot area helps determine property value, landscaping costs, and building setbacks.
Swimming Pool Cross-Sections
The side profile of a swimming pool that transitions from shallow to deep forms a trapezoid. Knowing this cross-sectional area helps calculate water volume needed to fill the pool.
Bridge and Road Embankments
Road cross-sections through hills or along embankments often have trapezoidal shapes. Civil engineers use the trapezoid area formula to estimate earthwork volumes for cut-and-fill operations.
Architectural Features
Trapezoidal windows, tabletops, and decorative panels are common in modern design. Calculating area is essential for ordering glass, materials, or estimating weight.
Solved Examples
Example 1: Trapezoidal Property Lot
A property lot has a front width of 20 meters (street side), a back width of 15 meters, and a depth of 30 meters. Find the lot area.
a = 20 m (front), b = 15 m (back), h = 30 m (depth)
A = ½ × (a + b) × h
A = ½ × (20 + 15) × 30
A = ½ × 35 × 30
A = 525 square meters
Example 2: Swimming Pool Cross-Section
A pool's side view shows a trapezoid: shallow end depth 1.2 m, deep end depth 2.4 m, and pool length 10 m. Find the cross-sectional area.
a = 1.2 m (shallow), b = 2.4 m (deep), h = 10 m (length)
A = ½ × (1.2 + 2.4) × 10
A = ½ × 3.6 × 10
A = 18 square meters (cross-sectional area)
Example 3: Road Embankment
A road embankment cross-section has a top width of 6 m, bottom width of 14 m, and height of 3 m. Find the area of soil in the cross-section.
a = 6 m (top), b = 14 m (bottom), h = 3 m
A = ½ × (6 + 14) × 3
A = ½ × 20 × 3
A = 30 square meters per meter of road length
Practice Questions
1. A trapezoidal window has parallel sides of 80 cm and 60 cm with a height of 50 cm. What is its area?
Answer: A = ½ × (80 + 60) × 50 = 3,500 cm²
2. A dam cross-section: top 4 m, bottom 12 m, height 8 m. Find the area.
Answer: A = ½ × (4 + 12) × 8 = 64 m²
3. A trapezoid has area 150 cm², parallel sides 10 cm and 20 cm. Find the height.
Answer: h = 2A/(a+b) = 2×150/(10+20) = 300/30 = 10 cm
4. A garden bed shaped as a trapezoid has bases 3 m and 5 m, height 2.5 m. How much topsoil (area) is needed?
Answer: A = ½ × (3 + 5) × 2.5 = 10 m²
5. An irrigation channel has a trapezoidal cross-section: top 2 m, bottom 1 m, depth 0.8 m. Find the area.
Answer: A = ½ × (2 + 1) × 0.8 = 1.2 m²
Key Takeaways
- Trapezoid area: A = ½ × (a + b) × h, where a and b are the two parallel sides (bases).
- The height (h) must be the perpendicular distance between the two parallel sides, not the slant side.
- If you know the area and both bases, the height is: h = 2A / (a + b).
- A trapezoid becomes a rectangle when both bases are equal (a = b), and becomes a triangle when one base = 0.
- Real-world trapezoids appear in property lots, pool cross-sections, road embankments, and architectural features.
- For the perimeter, you need all four sides: P = a + b + side₁ + side₂.
Frequently Asked Questions
What is a trapezoid?
A trapezoid (called a trapezium in British English) is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the non-parallel sides are called legs.
What is an isosceles trapezoid?
An isosceles trapezoid has legs of equal length. Its base angles are equal, and its diagonals are equal in length. The area formula is the same: A = ½(a+b)h.
How do I find the height if I only know the sides?
If you know all four sides of a trapezoid, you can find the height using the formula: h = √(c² − ((b−a)² + c² − d²)/(2(b−a)))²), where c and d are the legs. For an isosceles trapezoid: h = √(c² − ((b−a)/2)²).
Is a rectangle a special trapezoid?
Depending on the definition used, yes. Under the inclusive definition (at least one pair of parallel sides), a rectangle is a special trapezoid where both pairs of sides are parallel and all angles are 90°.