How this instrument works
A trapezoid has exactly one pair of parallel sides, called the bases, of lengths a and b, separated by a perpendicular height h. A rectangle built on the shorter base and a rectangle built on the longer base bracket the true shape from below and above, so the sensible area to expect is somewhere between a·h and b·h — and it lands exactly at their arithmetic mean: A = (a + b) ⁄ 2 × h. No approximation is involved; the average is exact.
The reason is a fact about the trapezoid's slanted sides, called legs: the segment joining the midpoints of the two legs — the midsegment — is parallel to both bases and has length exactly (a + b) ⁄ 2, regardless of how steeply the legs lean. Slide the small triangular sliver above that midline down to fill the matching gap on the opposite side, and the trapezoid rearranges, without changing its area, into a plain rectangle of width (a + b) ⁄ 2 and height h. The formula is simply that rectangle's area.
Two limits are worth knowing. Shrink base b toward zero and the shape narrows into a triangle; the formula quietly becomes A = a·h ⁄ 2, the familiar half-base-times-height rule. Make a equal to b and the legs straighten into a rectangle, or a parallelogram if they lean; the formula becomes A = a·h. One caution earns its place here: h is the perpendicular gap between the two bases, not the length of a slanted leg — measuring along a leg instead of straight across is the single most common way to get this wrong.
- Enter the length of one parallel side into Base a — either parallel side works, since the formula treats both the same way.
- Enter the length of the other parallel side into Base b.
- Enter the perpendicular distance between the two bases, not the slanted leg length, into Height.
- Read the computed area in the Area field, in whatever squared unit your bases and height share.
Worked example — a 3 m and 5 m garden bed
A trapezoidal garden bed has parallel edges of 3 m and 5 m, set 4 m apart at their perpendicular distance. Area: A = (3 + 5) ⁄ 2 × 4 = 8 ⁄ 2 × 4 = 4 × 4 = 16 m² — the figure to hand a turf supplier who sells sod by the square metre.
The midsegment shortcut gives the same number faster: average 3 and 5 to get 4, and the bed behaves like a plain 4 m by 4 m square for area purposes, even though none of its actual sides measures 4 m. Sixteen square metres, the same answer both ways.
Questions
What is the formula for the area of a trapezoid?
A = (a + b) ⁄ 2 × h, where a and b are the two parallel sides and h is the perpendicular height between them. The division by two comes from averaging the bases: a trapezoid's area sits exactly at the midpoint between a rectangle built on its short side and one built on its long side. With a = 3, b = 5, h = 4, that gives (3 + 5) ⁄ 2 × 4 = 16.
Is the height the same as the length of the slanted side?
No — height is the perpendicular distance between the two parallel bases, measured straight across, not along a leg. Only in a right trapezoid, where one leg meets the bases at 90°, does a side's length equal the height. Using a slanted leg's length in place of h is the most common arithmetic mistake this formula invites.
How does the trapezoid area formula relate to the triangle and rectangle formulas?
It generalizes both. Let base b shrink toward zero and A = (a + b) ⁄ 2 × h reduces to A = a·h ⁄ 2, the triangle rule. Let b equal a and it reduces to A = a·h, the rectangle rule. A trapezoid sits between a triangle and a rectangle, and its formula sits between theirs the same way.
What is a trapezoid's midsegment, and why does it equal (a + b) ⁄ 2?
The midsegment is the line joining the midpoints of the two legs. It is always parallel to both bases and, by a classical theorem, always exactly (a + b) ⁄ 2 long — the average of the two bases — no matter how the legs lean. Multiplying that average width by the height h gives the same figure as the standard area formula, because sliding a triangular sliver across the midline turns the trapezoid into a rectangle of that width without changing its area.
Does this formula only apply to isosceles or right trapezoids?
No — it holds for every trapezoid: right, isosceles, or scalene-legged. The formula only needs the two parallel base lengths and the perpendicular height between them; it has no opinion about how the legs are angled or whether they are equal in length. That is also why it reappears, applied repeatedly to thin strips, as the trapezoidal rule for approximating the area under a curve in calculus.