How this instrument works
A right trapezoid has one leg standing perpendicular to both parallel bases — the height — and one leg leaning, the slant leg, that closes the gap between the shorter base and the longer one. Drop a vertical line from the top of the short base down to the long one and a right triangle appears, with the height as one leg and the horizontal overhang, the difference between the two bases, as the other. The slant leg is that triangle's hypotenuse, so rearranging the Pythagorean theorem recovers the overhang directly: overhang = √(slant² − h²), and adding that back onto the shorter base gives the longer one, b₂ = b₁ + √(slant² − h²).
This sheet starts from a different set of measurements than the site's combined Right Trapezoid calculator, which asks for both bases directly and derives the slant leg from them. Here the longer base is the unknown quantity, recovered instead from the shorter base, the height, and the slant leg — useful whenever the far edge of a trapezoid-shaped panel, roof section, or plot of land is awkward to reach directly but a diagonal board, cable, or fence line along the slant can be read off easily. Once the longer base is recovered, area follows the ordinary trapezoid formula, A = ½(b₁+b₂)h, exactly as it would had both bases been measured from the start.
One boundary case is worth flagging: the slant leg can never be shorter than the height, since it is the hypotenuse of a right triangle that includes the height as one leg. When slant equals height exactly, the recovered overhang is zero, the longer base equals the shorter one, and the shape has quietly become a rectangle rather than a true trapezoid. Entering a slant shorter than the height describes a triangle that cannot exist, which is why the sheet flags that combination instead of returning a number.
- Enter the known shorter side into Base 1 (shorter, known).
- Enter the perpendicular side's length into Height (the perpendicular leg).
- Enter the leaning fourth side's length into Slant leg length.
- Read Base 2 (longer, solved): the longer base recovered via the Pythagorean theorem.
- Read Area for the enclosed area, computed from both bases and the height.
Worked example — base 6, height 4, slant 5
A right trapezoid has a shorter base of 6, a perpendicular height of 4, and a measured slant leg of 5 — a classic 3-4-5 triangle hiding in the offset, since √(5²−4²) = √(25−16) = √9 = 3. Adding that 3 onto the shorter base gives the longer base: b₂ = 6 + 3 = 9.
With both bases now known, the area follows directly: A = ½(6+9) × 4 = ½ × 15 × 4 = 30. The slant leg alone was enough to pin down a base that was never measured directly — only the height and the shorter base needed a tape measure, and the geometry supplied the number 9 in between them.
Questions
How is the longer base recovered from the slant leg?
By rearranging the Pythagorean theorem. The slant leg is the hypotenuse of a hidden right triangle whose other two legs are the height and the horizontal overhang between the two bases, so overhang = √(slant² − h²), and adding that overhang to the shorter base gives b₂ = b₁ + √(slant² − h²). With a shorter base of 6, height of 4, and slant of 5, that overhang works out to 3, so b₂ = 9.
How is this different from the site's Right Trapezoid calculator?
That page takes both bases directly as inputs and computes the slant leg from them. This one works the other direction: it assumes the longer base hasn't been measured yet and recovers it from the shorter base, the height, and the slant leg instead, useful whenever the far edge of the shape is harder to reach than its slanted side.
What if the slant leg is shorter than the height?
That combination is impossible for a real right trapezoid, since the slant leg is the hypotenuse of a triangle that includes the height as one leg, and a hypotenuse can never be shorter than either leg it belongs to. The sheet checks for this and flags it rather than returning a negative value under the square root.
What happens when the slant leg exactly equals the height?
The recovered overhang becomes zero, so the longer base equals the shorter base and the shape becomes a rectangle. Both legs then stand perpendicular to the bases, and what was a leaning slant leg is now just a second height running parallel to the first.
Does the area formula change because the longer base is derived rather than measured?
No — once b₂ is recovered, area still comes from the standard trapezoid formula, A = ½(b₁+b₂)h, the same identity used regardless of whether both bases were measured directly or one of them was worked out from the slant leg first.