66
sq units
36.5
units
11
units
10.1832
units
13.0636
units
66
sq units
36.5
units
11
units
10.1832
units
13.0636
units
A trapezoid (known as 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 or lateral sides. The Trapezoid Calculator computes the area, perimeter, median (midsegment), and diagonal lengths of any trapezoid given both parallel sides, the height, and both legs.
The trapezoid is one of the most commonly encountered quadrilaterals in practical applications. Bridge cross-sections, retaining walls, dam profiles, and architectural elements frequently take trapezoidal form. In irrigation engineering, trapezoidal channels are preferred because they provide efficient water flow with minimal erosion. The shape balances structural strength with material efficiency.
The area formula \( A = \frac{(a + b) \cdot h}{2} \) is perhaps the most well-known result associated with trapezoids. It states that the area equals the product of the height and the average of the two parallel sides. Geometrically, this can be understood by noting that a trapezoid can be decomposed into a rectangle with width equal to the shorter base and two right triangles, or alternatively, two copies of the trapezoid can be arranged to form a parallelogram with base \( (a + b) \) and height \( h \).
The median (also called the midsegment) is the line segment connecting the midpoints of the two legs. Its length equals the average of the two bases: \( m = (a + b)/2 \). The median is parallel to both bases and has the remarkable property that the area of the trapezoid equals the product of the median and the height: \( A = m \cdot h \). This makes the median a powerful computational tool.
The perimeter is simply the sum of all four sides: \( P = a + b + c + d \), where \( c \) and \( d \) are the leg lengths. For an isosceles trapezoid (where the legs are equal), additional symmetry properties apply: the diagonals are equal, the base angles are equal, and the shape has a line of symmetry perpendicular to the bases.
The diagonal lengths depend on all five parameters (both bases, height, and both legs). They can be computed by establishing a coordinate system with one base along the x-axis and using the distance formula. The offset of the top base relative to the bottom determines the diagonal lengths, which are generally unequal unless the trapezoid is isosceles.
Trapezoids play an important role in numerical analysis through the trapezoidal rule, which approximates definite integrals by dividing the area under a curve into trapezoidal strips. This method, one of the simplest numerical integration techniques, directly applies the trapezoid area formula to approximate curved regions. The error analysis of the trapezoidal rule connects to deeper results in calculus and approximation theory.
The calculator uses the following formulas:
Area:
$$A = \frac{(a + b) \cdot h}{2}$$
Perimeter:
$$P = a + b + c + d$$
Median (Midsegment):
$$m = \frac{a + b}{2}$$
Diagonals: Computed by placing the trapezoid in a coordinate system with the bottom base \( b \) along the x-axis. The horizontal offset of the top-left vertex is determined from the leg lengths and height using the relation \( x_1 = \frac{b - a}{2} + \frac{c^2 - d^2}{2(b - a)} \) for the general case, then the distance formula gives each diagonal length.
The area measures the total surface enclosed by the trapezoid. The perimeter is the total boundary length. The median gives the length of the midsegment, which is useful for determining the center line of trapezoidal sections (e.g., in channel design). The diagonals connect opposite vertices; they are equal for isosceles trapezoids but differ for general trapezoids. All measurements use the same linear unit, with area in the corresponding square unit.
Inputs
Results
A trapezoid with parallel sides 8 and 14, height 6, and legs 7 and 7.5. The area is 66 sq units and the median is 11 units.
Inputs
Results
An isosceles trapezoid with equal legs of approximately 5.831 (satisfying h² + 3² = leg² with h=5 and half the base difference = 3).
In American English, a trapezoid has exactly one pair of parallel sides. In British English, the same shape is called a trapezium. Confusingly, in British usage, "trapezoid" refers to a quadrilateral with no parallel sides. This calculator uses the American definition.
For a trapezoid with sides \( a, b, c, d \) where \( a \) and \( b \) are parallel, the height can be found using \( h = \frac{2}{|b-a|}\sqrt{s_1(s_1 - |b-a|)(s_1 - c)(s_1 - d)} \), where \( s_1 = \frac{|b-a| + c + d}{2} \). This comes from applying Heron's formula to the triangle formed by collapsing the shorter base to a point.
A right trapezoid has one leg perpendicular to both bases, creating two right angles. This makes the perpendicular leg equal to the height. Right trapezoids are common in architecture and structural design, where one vertical wall meets a horizontal base.
The trapezoidal rule approximates a definite integral by dividing the area under a curve into strips shaped like trapezoids. Each strip has parallel sides equal to the function values at consecutive points and height equal to the step size. The area formula \( A = (a+b)h/2 \) is applied to each strip.
This depends on the definition. Under the exclusive definition, a trapezoid has exactly one pair of parallel sides, so a parallelogram is not a trapezoid. Under the inclusive definition (used in some textbooks), a trapezoid has at least one pair of parallel sides, making parallelograms a special case.
The median (midsegment) connects the midpoints of the legs and is parallel to both bases. Its length equals the average of the two bases: \( m = (a+b)/2 \). The trapezoid's area can be expressed as \( A = m \cdot h \), and the median divides the trapezoid into two smaller trapezoids of equal height.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
