Trapezoid Calculator
Calculate the area, perimeter, height, angles, midsegment, diagonals, and 3D volume of any trapezoid with real-time SVG visual feedback and step-by-step formula breakdowns.
What is a trapezoid?
A trapezoid is a four-sided convex polygon (quadrilateral) with at least one pair of parallel opposite sides. In geometry and mathematics, the two parallel sides are designated as the bases (Base 1 / Base a and Base 2 / Base b), while the two non-parallel opposite sides are called the legs (Side c and Side d).
The perpendicular distance between the two parallel bases is defined as the height (also referred to as the altitude, represented by h). The line segment connecting the midpoints of the two non-parallel legs is called the midsegment (or median, represented by m). The midsegment is always parallel to both bases, and its length equals the average of the two base lengths: m = (a + b) / 2.
There is a regional naming difference in English terminology. In American English (US), this geometric shape is called a trapezoid, whereas a quadrilateral with no parallel sides is called a trapezium. In British English (UK, Australia, India, South Africa), the shape with parallel sides is called a trapezium, and a quadrilateral with no parallel sides is called a trapezoid. Computational tools such as Wolfram Alpha, GeoGebra, Symbolab, Omni Calculator, Cuemath, and Math is Fun recognize both terms interchangeably for the same four-sided geometric figure.
Click or hover over any geometric element to inspect its mathematical role, properties, and formula:
Parallel Bases (a & b)
The two parallel sides of a trapezoid are designated as Base 1 (top base, a) and Base 2 (bottom base, b). They never intersect and dictate the overall orientation and width of the polygon.
How to Use the Trapezoid Calculator
To use the trapezoid calculator, follow these five straightforward steps to compute area, perimeter, and missing geometric dimensions:
- Select your calculation mode: Choose from six available modes — Bases & Height (Standard), 4 Sides (Without Height), Isosceles Trapezoid, Right Trapezoid, Find Height (from Area & Bases), or 3D Volume (Footing / Trench).
- Enter the known dimensions: Input the values for Top Base (a), Bottom Base (b), and either Height (h), Left Leg (c), Right Leg (d), or Known Area (A).
- Select your measurement units: Choose individual units for each input field (feet, meters, yards, inches, centimeters, millimeters, kilometers, or miles), or use the unit synchronization bar to set all inputs to a uniform unit in one click.
- Review the instant output properties: The calculator updates immediately, displaying the total area in square feet (sq ft), square meters (m²), acres (ac), hectares (ha), and square yards (yd²), along with total perimeter, midsegment, diagonals, and interior angles.
- Inspect the diagram and step-by-step breakdown: Examine the real-time SVG diagram to verify shape proportions and review the mathematical steps to copy or export your results.
Trapezoid Formulas
There are four primary mathematical formulas used to determine the geometric properties of a trapezoid: the area formula, the perimeter formula, the height formula, and the interior angles equations.
Area of a Trapezoid
The area of a trapezoid equals the average length of the two parallel bases multiplied by the perpendicular height.
When the height is unknown but all four side lengths (a, b, c, d) are known, the area of a trapezoid is computed using the four-sides trapezoid area formula:
Perimeter of a Trapezoid
The perimeter of a trapezoid equals the sum of all four boundary sides.
For an isosceles trapezoid where the two non-parallel legs are identical (c = d), the perimeter formula is P = a + b + 2c. For a right trapezoid with perpendicular height h and slant leg d = √((b − a)² + h²), the perimeter formula is P = a + b + h + √((b − a)² + h²).
Height of a Trapezoid
The height of a trapezoid is the perpendicular distance measured at a 90-degree angle between the two parallel bases.
Depending on the available measurements, the height (h) is derived using one of four distinct mathematical methods:
- From Known Area and Bases:
h = 2A / (a + b) - From 4 Side Lengths (a, b, c, d):
h = √((-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)) / (2|b - a|) - In an Isosceles Trapezoid (with Leg c):
h = √(c² − ((b − a) / 2)²) - Using Trigonometry (with Leg c and Base Angle θ):
h = c × sin(θ)
Angles of a Trapezoid
The interior angles of any trapezoid sum to exactly 360 degrees (2π radians).
Because the two bases are parallel line segments cut by the transversal legs, the consecutive interior angles between the bases are supplementary:
- Left side angles: ∠A + ∠D = 180° (Bottom-left angle and Top-left angle sum to 180°).
- Right side angles: ∠B + ∠C = 180° (Bottom-right angle and Top-right angle sum to 180°).
- Total quadrilateral sum: ∠A + ∠B + ∠C + ∠D = 360°.
Adjust the dimension sliders below to see how geometric transformation visually proves the trapezoid area formula:
Types of Trapezoids
There are three main types of trapezoids categorized by side lengths and interior angle relationships: scalene trapezoids, isosceles trapezoids, and right trapezoids.
Scalene Trapezoid
A scalene trapezoid has four sides of unequal lengths and four unequal interior angles. Neither the parallel bases nor the non-parallel lateral legs share equal measurements (a ≠ b and c ≠ d).
Scalene trapezoids possess no lines of symmetry. Real estate land parcels, property boundaries bordering curved roads or waterways, and agricultural fields frequently take the form of scalene trapezoids. To compute the area of a scalene trapezoid, determine the perpendicular altitude between the two bases and apply the formula Area = ((a + b) / 2) × h.
Isosceles Trapezoid
An isosceles trapezoid has two non-parallel legs of equal length (c = d) and congruent base angles.
An isosceles trapezoid exhibits three specific geometric properties:
- Congruent base angles: The bottom base angles are equal (∠A = ∠B), and the top base angles are equal (∠C = ∠D).
- Equal diagonal lengths: Both diagonals have identical length, calculated by the formula d₁ = d₂ = √(a × b + c²).
- Reflectional symmetry: The shape possesses one vertical axis of symmetry passing through the midpoints of both parallel bases.
Right Trapezoid
A right trapezoid (also called a right-angled trapezoid) has two adjacent right angles of 90 degrees.
In a right trapezoid, one of the non-parallel legs is perpendicular to both parallel bases. This perpendicular leg serves directly as the height (c = h). The opposite leg is slanted, with its length calculated using the Pythagorean theorem: d = √((b − a)² + h²). Right trapezoids are common in architectural floor plans, building setback calculations, civil drainage channels, and roof trusses.
Select a trapezoid classification to view its unique geometric anatomy, symmetry axes, and formulas:
How to Calculate the Area of a Trapezoid
To calculate the area of a trapezoid, complete the following six calculation steps:
- Measure the top base (a): Measure the length of the shorter parallel side in linear units (e.g., 20 feet or 6 meters).
- Measure the bottom base (b): Measure the length of the longer parallel side using the same linear units (e.g., 36 feet or 11 meters).
- Measure the perpendicular height (h): Measure the straight-line vertical distance between the two parallel bases at a 90-degree angle (e.g., 14 feet or 4.25 meters).
- Add the two bases together: Calculate the sum of the parallel bases (a + b). For 20 ft and 36 ft, the sum is 20 + 36 = 56 ft.
- Divide the sum by 2: Calculate the average base length (midsegment m). For 56 ft, the average is 56 ÷ 2 = 28 ft.
- Multiply by the height: Multiply the average base length by the perpendicular height (m × h). For 28 ft and 14 ft, the area is 28 × 14 = 392 sq ft (36.42 m² or 0.009 acres).
Example: Calculate the Area of a Trapezoid
Consider a real-world land surveying example involving a residential building lot shaped like a trapezoid.
Step 2 (Average base / Midsegment): 160 ft ÷ 2 = 80 ft
Step 3 (Multiply by height): 80 ft × 75 ft = 6,000 sq ft
Step 4 (Acreage conversion): 6,000 sq ft ÷ 43,560 sq ft/acre = 0.1377 acres
Step 5 (Metric conversion): 6,000 sq ft × 0.092903 = 557.42 m² (0.0557 hectares)
If the lot owner plans to pour a concrete foundation footing shaped as a trapezoidal trench measuring 60 ft long, 2 ft wide at the top, 3 ft wide at the bottom, and 1.5 ft deep:
- Cross-section area: ((2 + 3) / 2) × 1.5 = 2.5 × 1.5 = 3.75 sq ft.
- 3D Trench Volume: 3.75 sq ft × 60 ft = 225 cubic feet.
- Concrete needed in cubic yards: 225 ÷ 27 = 8.33 cubic yards of concrete.
Select a preset or customize dimensions to estimate land parcel acreage and 3D concrete / earthwork volume:
How to Find the Height of a Trapezoid
To find the height of a trapezoid, use one of the following four calculation scenarios based on your known measurements:
- When Area (A) and both bases (a, b) are known:
Rearrange the area equation to solve for height: h = 2A / (a + b). For example, if a trapezoid has an area of 500 sq ft, a top base of 20 ft, and a bottom base of 30 ft: h = (2 × 500) / (20 + 30) = 1,000 / 50 = 20 ft. - When only four side lengths (a, b, c, d) are known:
Apply the altitude formula: h = √((-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)) / (2|b - a|). This algebraic equation calculates the exact altitude without requiring interior angle measurements. - In an isosceles trapezoid with equal legs (c):
Drop a perpendicular line from the top vertex to the bottom base to form a right triangle. The base of this right triangle is (b − a) / 2. Apply the Pythagorean theorem: h = √(c² − ((b − a) / 2)²). - In a right trapezoid with slant leg (d):
The perpendicular height is solved via the Pythagorean theorem using the base difference: h = √(d² − (b − a)²).
Select the data you have available to instantly compute the perpendicular height (h):
Trapezoid Calculations:
The following reference table presents calculated areas, perimeters, and midsegments across ten standard trapezoid dimensions in both Imperial (feet, square feet, acres) and Metric (meters, square meters, hectares) units:
| Top Base (a) | Bottom Base (b) | Height (h) | Midsegment (m) | Area (sq ft / m²) | Acreage / Hectares | Perimeter (P) |
|---|---|---|---|---|---|---|
| 10 ft (3.05 m) | 18 ft (5.49 m) | 8 ft (2.44 m) | 14 ft (4.27 m) | 112.00 sq ft (10.41 m²) | 0.00257 ac (0.00104 ha) | 45.89 ft (13.99 m) |
| 15 ft (4.57 m) | 25 ft (7.62 m) | 12 ft (3.66 m) | 20 ft (6.10 m) | 240.00 sq ft (22.30 m²) | 0.00551 ac (0.00223 ha) | 66.00 ft (20.12 m) |
| 20 ft (6.10 m) | 40 ft (12.19 m) | 15 ft (4.57 m) | 30 ft (9.14 m) | 450.00 sq ft (41.81 m²) | 0.01033 ac (0.00418 ha) | 96.06 ft (29.28 m) |
| 30 ft (9.14 m) | 50 ft (15.24 m) | 25 ft (7.62 m) | 40 ft (12.19 m) | 1,000.00 sq ft (92.90 m²) | 0.02296 ac (0.00929 ha) | 133.85 ft (40.80 m) |
| 50 ft (15.24 m) | 90 ft (27.43 m) | 40 ft (12.19 m) | 70 ft (21.34 m) | 2,800.00 sq ft (260.13 m²) | 0.06428 ac (0.02601 ha) | 229.44 ft (69.93 m) |
| 66 ft (20.12 m) | 132 ft (40.23 m) | 66 ft (20.12 m) | 99 ft (30.18 m) | 6,534.00 sq ft (607.03 m²) | 0.15000 ac (0.06070 ha) | 345.86 ft (105.42 m) |
| 100 ft (30.48 m) | 200 ft (60.96 m) | 100 ft (30.48 m) | 150 ft (45.72 m) | 15,000.00 sq ft (1,393.55 m²) | 0.34435 ac (0.13935 ha) | 523.61 ft (159.60 m) |
| 150 ft (45.72 m) | 250 ft (76.20 m) | 120 ft (36.58 m) | 200 ft (60.96 m) | 24,000.00 sq ft (2,229.67 m²) | 0.55096 ac (0.22297 ha) | 660.00 ft (201.17 m) |
| 200 ft (60.96 m) | 400 ft (121.92 m) | 200 ft (60.96 m) | 300 ft (91.44 m) | 60,000.00 sq ft (5,574.18 m²) | 1.37741 ac (0.55742 ha) | 1,047.21 ft (319.19 m) |
| 300 ft (91.44 m) | 600 ft (182.88 m) | 250 ft (76.20 m) | 450 ft (137.16 m) | 112,500.00 sq ft (10,451.59 m²) | 2.58264 ac (1.04516 ha) | 1,483.10 ft (452.05 m) |
Frequently Asked Questions
A trapezoid differs from other quadrilaterals because it has exactly one pair of parallel opposite sides (in standard inclusive geometry, at least one pair), whereas parallelograms, rectangles, and rhombuses have two pairs of parallel opposite sides, and general irregular quadrilaterals have zero parallel sides. The two parallel sides of a trapezoid are called bases (Base 1 and Base 2), while the non-parallel sides are called legs. In a trapezoid, the consecutive interior angles between the two parallel bases are supplementary, summing to 180 degrees.
The area of a trapezoid with height 5 meters and bases 8 meters and 1 meter is 22.5 square meters (242.19 square feet). The calculation uses the standard trapezoid area formula: Area = ((a + b) ÷ 2) × h = ((8 m + 1 m) ÷ 2) × 5 m = (9 m ÷ 2) × 5 m = 4.5 m × 5 m = 22.5 m². In alternative land measurement units, 22.5 square meters equals 0.00225 hectares (0.00556 acres or 26.91 square yards).
To find the perimeter of a trapezoid, add the lengths of all four outer boundary sides: Perimeter = a + b + c + d, where a is the top base, b is the bottom base, c is the left leg, and d is the right leg. For an isosceles trapezoid where both legs are equal (c = d), the perimeter formula simplifies to Perimeter = a + b + 2c. For a right trapezoid with perpendicular height h and slant leg c, the perimeter formula is Perimeter = a + b + h + √(h² + (b − a)²).
To find the height of a trapezoid when the area and both parallel bases are known, divide twice the area by the sum of the two bases: Height (h) = 2A ÷ (a + b). When only the four side lengths (a, b, c, d) are known without the area, calculate the height using the four-sides altitude formula: h = √((-a + b + c + d)(a - b + c + d)(a - b + c - d)(a - b - c + d)) ÷ (2|b - a|). For an isosceles trapezoid with leg c, use the Pythagorean theorem: h = √(c² - ((b - a) ÷ 2)²).
Yes, you can calculate trapezoid area without height if all four side lengths (top base a, bottom base b, left leg c, right leg d) are known. First, calculate the height h using the algebraic formula: h = √((-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)) ÷ (2|b - a|). Once the height is determined, multiply the average of the two bases by the calculated height: Area = ((a + b) ÷ 2) × h. The trapezoid calculator above automates this entire four-sided computation instantly.
The midsegment (also called the median) of a trapezoid is the line segment connecting the midpoints of the two non-parallel legs. The length of the midsegment equals the arithmetic average of the two parallel bases: m = (a + b) ÷ 2. The midsegment is always parallel to both bases, and the area of any trapezoid equals the midsegment length multiplied by the height: Area = m × h.
The difference between a trapezoid and a trapezium is purely regional terminology between American English and British English. In American English (US and Canada), a trapezoid is a quadrilateral with one pair of parallel sides, and a trapezium has zero parallel sides. In British English (UK, Australia, New Zealand, India, and South Africa), a trapezium is a quadrilateral with one pair of parallel sides, and a trapezoid has no parallel sides. Both terms refer to the exact same geometric shape across mathematical software and geometry calculators.
To calculate the volume of a 3D trapezoidal footing, prism, trench, drainage channel, or pond, multiply the two-dimensional cross-sectional trapezoid area by the length or depth of the structure: Volume = Area × Length = [((a + b) ÷ 2) × h] × L. To convert cubic feet (cu ft) of concrete or earthwork to cubic yards (cu yd), divide the cubic feet by 27. For metric volume, calculate the area in square meters and multiply by length in meters to get cubic meters (m³).