Trapezoid Area Calculator
Calculate the area of any trapezoid — right, isosceles, or scalene — by entering the two parallel bases and the perpendicular height. Supports all major length units, shows full step-by-step working, and displays results in every area unit simultaneously.
Built by He Loves Math with the formula derivation, worked examples, and all the geometry you need to understand trapezoids — not just calculate them.
Quick Answer: The Trapezoid Area Formula
The area of a trapezoid with parallel bases \(b_1\) and \(b_2\) and perpendicular height \(h\) is:
Equivalently written as: \( A = \frac{(b_1 + b_2) \cdot h}{2} \). The key requirement is that h must be the perpendicular height — the shortest distance between the two parallel bases — not the length of a slanted leg.
Trapezoid Area Calculator
Enter Base 1, Base 2, and the perpendicular Height. Each dimension can use a different unit. Select your preferred area output unit and click Calculate Area.
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What Is a Trapezoid?
A trapezoid (called a trapezium in British English) is a four-sided polygon — a quadrilateral — with exactly one pair of parallel sides. The two parallel sides are called the bases, and the two non-parallel sides are called the legs or lateral sides.
This one-pair criterion is the defining property of a trapezoid. A rectangle, square, rhombus, and parallelogram all have two pairs of parallel sides, which is why they are not trapezoids (though the inclusive definition used in some curricula accepts parallelograms as a special case of trapezoid).
The word "trapezoid" comes from the Greek trapézion (τραπέζιον), meaning "a little table" — reflecting the shape's resemblance to a small table viewed from the side. The shape appears constantly in architecture, engineering, art, and nature, which makes the area formula one of the most practically useful in plane geometry.
Naming note: In American English, a trapezoid = one pair of parallel sides. In British English, the same shape is called a trapezium, and a "trapezoid" historically meant a quadrilateral with no parallel sides. Modern international mathematics education generally uses the American convention. Both terms refer to the same formula: \( A = \frac{1}{2}(b_1 + b_2)h \).
The Trapezoid Area Formula — Full Derivation
The formula for the area of a trapezoid is one of the most elegant in plane geometry because it can be derived in multiple independent ways. Understanding why the formula works — not just what it is — makes it far easier to remember and apply correctly.
The Standard Formula
Where \(b_1\) and \(b_2\) are the lengths of the two parallel bases and \(h\) is the perpendicular height (the perpendicular distance between the planes containing the two bases).
Derivation Method 1 — Parallelogram Doubling
This is the most visually intuitive proof. Take any trapezoid with bases \(b_1\), \(b_2\) and height \(h\). Make an exact copy of it, then flip the copy 180° and join it to the original along one of its slanted sides. The result is always a parallelogram.
The parallelogram has base \( b_1 + b_2 \) and height \( h \), so its area is:
Since the parallelogram is made of two identical trapezoids, each trapezoid has area:
Derivation Method 2 — Splitting into Two Triangles
Draw a diagonal across the trapezoid, splitting it into two triangles. Triangle 1 has base \(b_1\) and height \(h\); Triangle 2 has base \(b_2\) and height \(h\). Their areas sum to:
Derivation Method 3 — Average Base × Height
The formula can also be read conceptually: the trapezoid has an average width of \( \frac{b_1 + b_2}{2} \) (the arithmetic mean of the two bases). Multiplying the average width by the height gives the area — exactly as a rectangle's area is width × height, but with the width averaged across the varying cross-section.
The midsegment (or median) of a trapezoid — the line segment connecting the midpoints of the two legs — has exactly this length \( m = \frac{b_1+b_2}{2} \). So the area of a trapezoid equals its midsegment length times its height: \( A = m \times h \).
Types of Trapezoids
All trapezoids share the same area formula, but they have distinct geometric properties depending on the relationship between their legs and angles.
Scalene Trapezoid
No special relationships. The two legs have different lengths and the two pairs of base angles are unequal. The standard formula applies with no simplifications.
Isosceles Trapezoid
The two legs are equal in length. It is symmetric about its midsegment. Base angles are equal in pairs. Diagonals are equal. If leg = a: \( h = \sqrt{a^2 - \left(\frac{b_1-b_2}{2}\right)^2} \).
Right Trapezoid
Has exactly two right angles. One leg is perpendicular to both bases — making it equal to the height h. This perpendicular leg can be measured directly as the height without any additional calculation.
Worked Examples
Example 1 — Basic Trapezoid
A trapezoid has bases \(b_1 = 14\text{ cm}\), \(b_2 = 8\text{ cm}\), and height \(h = 6\text{ cm}\). Find the area.
Example 2 — Mixed Units
A garden bed is shaped like a trapezoid. Base 1 = 4.5 m, Base 2 = 2.8 m, Height = 90 cm. Find the area in m².
Example 3 — Finding a Missing Base
A trapezoid has area 120 cm², base 1 = 18 cm, and height = 8 cm. Find base 2.
Example 4 — Isosceles Trapezoid with Slanted Legs
An isosceles trapezoid has bases 20 cm and 12 cm, with equal legs of 5 cm. Find the height and area.
Perimeter of a Trapezoid
The perimeter of a trapezoid is simply the sum of all four sides. Unlike the area, there is no simplification — all four sides must be measured or calculated:
Where \(\ell_1\) and \(\ell_2\) are the two legs (non-parallel sides). For an isosceles trapezoid where both legs have equal length \(a\):
For a right trapezoid with perpendicular leg \(h\) and slanted leg \(a\):
Key Properties of Trapezoids
Midsegment (Median)
The midsegment connects the midpoints of the two legs. Its length equals the average of the two bases:
The midsegment is parallel to both bases and divides the trapezoid into two smaller trapezoids of equal area. The area formula can be stated as \( A = m \times h \).
Diagonal Properties
The diagonals of a general trapezoid bisect each other in the ratio \( b_1 : b_2 \). They divide the trapezoid into four triangles; the two triangles adjacent to the legs have equal area, while the triangles adjacent to the bases have areas in the ratio \( (b_1)^2 : (b_2)^2 \).
Angle Relationships
In a trapezoid, each pair of co-interior (same-side) angles (one from each base) is supplementary — they sum to 180°:
Relationship to Other Shapes
A trapezoid is related to:
- Parallelogram: when \(b_1 = b_2\) — both formulas give \(A = b \times h\)
- Triangle: when \(b_2 = 0\) — formula gives \(A = \frac{1}{2}b_1 \times h\)
- Rectangle: when \(b_1 = b_2\) and all angles are 90°
Real-World Applications of the Trapezoid Area Formula
Trapezoids appear far more frequently in the real world than their relatively modest coverage in school curricula might suggest. Understanding their area calculation is directly applicable to a wide range of professional and everyday contexts.
- Architecture and construction: Trapezoidal cross-sections appear in retaining walls, embankments, and sloped roofs. Engineers calculate trapezoidal cross-sectional areas to determine material volumes in cut-and-fill earthworks — \( \text{Volume} = A_{\text{trapezoid}} \times \text{length} \).
- Land surveying: Irregular land parcels are often approximated as trapezoids for quick area estimation. GPS survey data frequently yields trapezoidal subdivisions for area calculation.
- Civil engineering — channel design: Trapezoidal irrigation channels and drainage canals are standard in civil engineering because the trapezoidal cross-section is the most hydraulically efficient open-channel shape for earth-lined canals.
- Numerical integration (Trapezoid Rule): In calculus, the trapezoid rule approximates the definite integral of a function by summing trapezoidal areas under the curve: \( \int_a^b f(x)\,dx \approx \sum \frac{(f(x_i)+f(x_{i+1}))}{2} \cdot \Delta x \). Each term is exactly the trapezoid area formula.
- Graphic design and art: Trapezoidal perspective shapes create forced-perspective effects. Understanding how to calculate their area helps with material estimation and scaling.
- Urban planning: Trapezoidal building footprints and plots are common at street intersections. Planners use the formula to calculate floor-to-area ratios and zoning compliance.
Key Formulas Reference
| Property | Formula | Notes |
|---|---|---|
| Area | \( A = \frac{1}{2}(b_1+b_2) \cdot h \) | h = perpendicular height |
| Midsegment length | \( m = \frac{b_1+b_2}{2} \) | Parallel to both bases |
| Area via midsegment | \( A = m \times h \) | Equivalent to main formula |
| Perimeter (general) | \( P = b_1 + b_2 + \ell_1 + \ell_2 \) | Sum of all four sides |
| Perimeter (isosceles) | \( P = b_1 + b_2 + 2a \) | a = equal leg length |
| Isosceles height | \( h = \sqrt{a^2 - \left(\frac{b_1-b_2}{2}\right)^2} \) | Pythagoras on half-overhang |
| Missing base from area | \( b_2 = \frac{2A}{h} - b_1 \) | Rearranged area formula |
| Missing height from area | \( h = \frac{2A}{b_1+b_2} \) | Rearranged area formula |
| Co-interior angles | \( \angle A + \angle D = 180° \) | Same-side angle pair |
Frequently Asked Questions
What is the formula for the area of a trapezoid?
The area of a trapezoid is \( A = \frac{1}{2}(b_1 + b_2) \times h \), where \(b_1\) and \(b_2\) are the parallel bases and \(h\) is the perpendicular height between them. It can equally be written as \( A = \frac{(b_1+b_2)h}{2} \) or \( A = m \times h \) where m is the midsegment (average of the two bases).
What is the difference between height and leg in a trapezoid?
The height (h) is the perpendicular distance between the two parallel bases. The legs are the two non-parallel sides connecting the bases. In a general or isosceles trapezoid, the legs are slanted and their length is greater than the height. In a right trapezoid, one leg is perpendicular to the bases and equals the height; the other leg is still slanted.
How do you find the area of a trapezoid without the height?
If you know the four side lengths but not the height, you can compute the height geometrically. For an isosceles trapezoid with leg length a: \( h = \sqrt{a^2 - \left(\frac{b_1-b_2}{2}\right)^2} \). For a right trapezoid, one leg is the height. For a general scalene trapezoid with all four sides known, use coordinate geometry to reconstruct the shape and compute h.
Can both bases of a trapezoid be the same length?
If \(b_1 = b_2\), the shape is technically a parallelogram, not a trapezoid (under the exclusive definition). However, the area formula still gives the correct answer: \( A = \frac{1}{2}(b+b)h = b \times h \), which is the area of a parallelogram. Under the inclusive definition, a parallelogram is a special case of a trapezoid.
What is an isosceles trapezoid and what are its properties?
An isosceles trapezoid has two equal legs. Its key properties: (1) it is symmetric about the perpendicular bisector of the bases; (2) the two pairs of base angles are equal — \( \angle A = \angle B \) and \( \angle C = \angle D \); (3) its diagonals are equal in length; (4) it is always a cyclic quadrilateral (can be inscribed in a circle).
How is the trapezoid formula related to the triangle and parallelogram formulas?
All three formulas are unified: if \(b_2 = 0\), the trapezoid becomes a triangle and the formula gives \( A = \frac{1}{2}b_1 h \) — the triangle area formula. If \(b_1 = b_2 = b\), the trapezoid becomes a parallelogram and the formula gives \( A = \frac{1}{2}(2b)h = bh \) — the parallelogram formula. The trapezoid formula is the general case that contains both as special instances.
What is the trapezoid rule in calculus?
The trapezoid rule is a numerical integration method. To approximate \( \int_a^b f(x)\,dx \), divide the interval into n subintervals of width \( \Delta x \) and sum the areas of the trapezoids formed under the curve: \( \approx \sum_{i=0}^{n-1} \frac{f(x_i)+f(x_{i+1})}{2} \cdot \Delta x \). Each term is exactly the trapezoid area formula \( \frac{1}{2}(b_1+b_2)h \).
What does the midsegment of a trapezoid equal?
The midsegment (also called the median) of a trapezoid connects the midpoints of the two non-parallel legs. Its length is always exactly the arithmetic mean of the two bases: \( m = \frac{b_1+b_2}{2} \). The midsegment is parallel to both bases. The area of the trapezoid can be written as \( A = m \times h \) — the midsegment length times the height.
