Area, perimeter, and diagonals for rectangles, parallelograms, and trapezoids — with step-by-step solutions.
A trapezoid has one pair of parallel sides (bases a and b). Height is the perpendicular distance between them.
If legs are equal (isosceles trapezoid) and Leg 1 is entered, the calculator uses it for both legs.
When calculating parallelogram or trapezoid area, the formula always requires the perpendicular height — the true vertical distance between the parallel sides — not the slant side length. These are different unless the shape has a right angle. If you only know the slant side (a) and the interior angle (θ), calculate the perpendicular height as h = a × sin(θ). For a parallelogram with base 10, slant side 8, and 30° angle: h = 8 × sin(30°) = 4, so area = 10 × 4 = 40 — not 80. For rectangles, the diagonal is always found using the Pythagorean theorem: d = √(l² + w²). A room that is 12 × 16 feet has a diagonal of √(144 + 256) = √400 = 20 feet exactly — a 3-4-5 triple scaled by 4.
A quadrilateral is any polygon with exactly four sides and four angles. The sum of interior angles in any quadrilateral is always 360° — a fact that holds whether the shape is a perfect square or a wildly irregular four-sided figure. This calculator covers three of the most common quadrilateral families: rectangles (including squares), parallelograms (including rhombuses), and trapezoids.
Each family has specific properties that simplify calculations. Rectangles have four right angles, making area simple: A = l × w. Parallelograms have two pairs of parallel sides, and area is base × perpendicular height (not slant height). Trapezoids have exactly one pair of parallel sides, and their area formula averages the two parallel bases before multiplying by height.
In construction and carpentry, these formulas appear constantly. A 12 × 16 foot room has area 192 sq ft. A parallelogram-shaped plot of land 100 feet along the base with a perpendicular depth of 60 feet has area 6,000 sq ft — the same as a rectangular plot with those dimensions, because only the perpendicular height matters, not the slant.
| Shape | Area | Perimeter | Diagonal |
|---|---|---|---|
| Rectangle | l × w | 2(l + w) | √(l² + w²) |
| Square | s² | 4s | s√2 |
| Parallelogram | b × h | 2(a + b) | — |
| Rhombus | (d1 × d2) / 2 | 4s where s = √((d1/2)² + (d2/2)²) | d1, d2 |
| Trapezoid | (a + b) × h / 2 | a + b + c + d | — |
The diagonal of a rectangle is found using the Pythagorean theorem: d = √(l² + w²). This makes intuitive sense because the diagonal divides the rectangle into two right triangles, where the diagonal is the hypotenuse. For an 8 × 6 rectangle: d = √(64 + 36) = √100 = 10. This is why the 6-8-10 (and 3-4-5) right triangles appear so often in construction — they create perfect right angles. Carpenters use the "3-4-5 check": if a corner truly measures 3 feet on one side, 4 feet on the other, and 5 feet diagonally, it's a perfect right angle.
For a square with side s, the diagonal is s√2 ≈ 1.414s. A square room that's 10 feet on each side has a diagonal of 10√2 ≈ 14.14 feet — the longest curtain rod that could fit diagonally across the room.
Calculates area, perimeter, and diagonal in feet-inches for any rectangular space.
View on Amazon →Measure interior angles of any quadrilateral or set precise mitre angles.
View on Amazon →Sketch and verify quadrilateral dimensions on a grid before calculating.
View on Amazon →The trapezoid area formula A = (a + b) / 2 × h is actually elegant: it multiplies the height by the average of the two parallel bases. This makes geometric sense — a trapezoid with parallel sides of 10 and 6 at height 4 has the same area as a rectangle that is 8 × 4 (the average base × the height): A = (10 + 6)/2 × 4 = 8 × 4 = 32 sq units.
The midsegment (or median) of a trapezoid connects the midpoints of the two non-parallel sides and equals exactly (a + b)/2 — the same average used in the area formula. This means area can also be expressed as A = midsegment × height, a useful shortcut when the midsegment length is known.
A rhombus is a parallelogram with all four sides equal. Its area can be calculated two ways:
A square is a special rhombus where both diagonals are equal (d1 = d2 = s√2). For a square with side 5: d = 5√2 ≈ 7.07, and area = (7.07 × 7.07)/2 ≈ 25 = s² ✓.