Trapezoid Calculator
Calculate the area, midsegment, and perimeter of a trapezoid from its two parallel sides, height, and the two legs.
Comment utiliser cette calculatrice
- 1Enter the lengths of the two parallel sides.
- 2Enter the perpendicular height between them.
- 3Read the area and the midsegment.
- 4Add the two legs to get the perimeter.
Comment ça marche
Trapezoid
area = ½ × (a + b) × height midsegment = (a + b) ÷ 2 perimeter = a + b + c + d a and b are the parallel sides; height is between them
A trapezoid is a quadrilateral with one pair of parallel sides, called the bases, and two non-parallel sides, called the legs. Its area is found by averaging the lengths of the two parallel sides and multiplying by the perpendicular height between them. This works because the trapezoid can be thought of as a rectangle whose width is the average of the two bases — the short base is made up for by the long base, and the average is exactly the width of the equivalent rectangle. That average of the bases also equals the midsegment (or median), the segment joining the midpoints of the two legs, which always runs parallel to the bases at their average length. The height must be the perpendicular distance between the parallel sides, not the slanted length of a leg. The perimeter simply adds all four sides, which requires knowing the two legs as well as the two bases.
Exemple détaillé
A trapezoid with parallel sides of 5 and 8 and a height of 4 has an area of ½ × (5 + 8) × 4 = ½ × 13 × 4 = 26. Its midsegment is (5 + 8) ÷ 2 = 6.5, and if both legs are 5, its perimeter is 5 + 8 + 5 + 5 = 23.
Trapezoid Calculator : le guide complet
The averaging trick behind the area
The area formula for a trapezoid — average the two parallel sides, then multiply by the height — is one of the most intuitive in geometry once you see why it works. A trapezoid has a long base and a short base, and the region between them is more than a rectangle on the short base but less than a rectangle on the long base. Splitting the difference by using the average of the two bases gives exactly the right width for an equivalent rectangle of the same height and the same area. The overhang on the long side precisely fills the gap on the short side.
This averaging idea unifies the trapezoid with simpler shapes. A rectangle is just a trapezoid whose two parallel sides happen to be equal, and averaging two equal numbers gives that same number, so the formula collapses to length times width. A triangle can be seen as a trapezoid with one base shrunk to zero, and averaging a base with zero gives half the base, recovering the familiar one-half base times height. The trapezoid area formula is thus a kind of master formula, with the rectangle and triangle as special cases, which is part of why it is so worth understanding rather than merely memorising.
The midsegment and the perpendicular height
The midsegment of a trapezoid — the line connecting the midpoints of the two non-parallel legs — has an elegant property: its length is always exactly the average of the two parallel sides. This is the very quantity that appears in the area formula, which means the area can also be read as the midsegment times the height. The midsegment runs parallel to the bases and sits halfway between them, splitting the trapezoid into two smaller trapezoids of equal height. It is a useful line both in calculations and in construction and design, where the midline of a tapered shape often matters.
The one point of care in every trapezoid calculation is the height. It must be the perpendicular distance between the two parallel sides — the straight-across gap — not the length of a slanted leg. In a right trapezoid, where a leg meets the bases at a right angle, that leg is the height, but in a general or isosceles trapezoid the legs are longer than the perpendicular height because they slant. Using a leg length in place of the true height is the most common trapezoid mistake, inflating the area. When the height is not given directly, it often has to be found first, sometimes using the Pythagorean theorem on a leg and the horizontal offset of the bases.
Trapezoids in the real world
Trapezoids appear wherever a shape tapers from one width to another while keeping two sides parallel, which is surprisingly often in the built environment. Many bridges, dams, and retaining walls have trapezoidal cross-sections, wider at the base for stability and narrower at the top, and computing their cross-sectional area is the first step in finding their volume and the material needed. Roof trusses, lampshades, handbags, and countless architectural details take trapezoidal forms, and land parcels are frequently trapezoidal, making the area formula a staple of surveying.
The trapezoid also lends its name to a fundamental technique in calculus: the trapezoidal rule for estimating the area under a curve. By slicing the region beneath a curve into thin vertical strips and treating each as a trapezoid — with its top approximated by a straight line between two points on the curve — the total area can be estimated by summing the strips. This turns the simple trapezoid area formula into a powerful tool for numerical integration, used constantly in science and engineering when a curve is too complicated to integrate exactly. From dams and roofs to the mathematics of continuous change, the trapezoid's blend of a parallel pair and a taper makes it one of geometry's quietly essential shapes.
Questions fréquentes
How do I find the area of a trapezoid?
Average the two parallel sides and multiply by the perpendicular height: area = ½ × (a + b) × height. For parallel sides of 5 and 8 with a height of 4, that's ½ × 13 × 4 = 26. The height must be the straight-across distance, not a slanted leg.
What is the midsegment of a trapezoid?
The midsegment (or median) joins the midpoints of the two legs and runs parallel to the bases. Its length always equals the average of the two parallel sides, (a + b) ÷ 2 — the same value used in the area formula. So the area is also the midsegment times the height.
Do I need the legs to find the area?
No — the area needs only the two parallel sides and the perpendicular height between them. The legs (the slanted sides) are required only for the perimeter, which adds all four sides together.
What's the difference between the height and a leg?
The height is the perpendicular distance between the two parallel sides; a leg is a slanted non-parallel side, which is usually longer. Using a leg in place of the true height is a common mistake that overstates the area. Only in a right trapezoid is a leg also the height.