Quadrilateral Calculator - Area and Perimeter
Choose your quadrilateral type, enter the required measurements, and instantly get the area, perimeter, and key properties. Supports rectangles, squares, parallelograms, rhombuses, trapezoids, kites, and general irregular quadrilaterals. The show-your-work panel explains every formula step so you can follow the math.
What is a quadrilateral?
A quadrilateral is any polygon with exactly four sides and four interior angles. The sum of the interior angles is always 360 degrees, regardless of the shape. The family includes the most familiar plane figures in everyday life: floors, screens, fields, and table tops are usually rectangles or squares. More general quadrilaterals such as parallelograms, trapezoids, kites, and irregular four-sided polygons also appear throughout geometry, engineering, and architecture. This calculator covers all seven common types and applies the exact formula for each one, so you get the most accurate result possible.
Formulas for every quadrilateral type
The formula used depends on the specific shape. A rectangle uses area = base x width, and a square simplifies that to side^2. A parallelogram replaces width with the perpendicular height between the two parallel sides: area = base x height. A rhombus and a kite share the same formula: area = (d1 x d2) / 2, where d1 and d2 are the two diagonals - for a rhombus the diagonals bisect each other at right angles, and for a kite one diagonal lies along the axis of symmetry. A trapezoid uses the average of its two parallel sides times the perpendicular height: area = (b1 + b2) / 2 x h. For a general irregular quadrilateral, Bretschneider's formula gives the area from four side lengths (a, b, c, d) and two opposite interior angles: area = sqrt((s - a)(s - b)(s - c)(s - d) - abcd x cos^2((alpha + gamma) / 2)), where s = (a + b + c + d) / 2 is the semi-perimeter. All perimeters are simply the sum of the four side lengths.
Diagonals and their significance
A diagonal connects two non-adjacent vertices of a quadrilateral. Every quadrilateral has exactly two diagonals. In a rectangle or square, both diagonals are equal in length and bisect each other. In a parallelogram, the diagonals bisect each other but are generally unequal, and their lengths can be found using the law of cosines: d^2 = a^2 + b^2 - 2ab x cos(angle). In a rhombus, the diagonals are perpendicular and bisect each other, so each diagonal splits the rhombus into four right triangles. In a kite, the main diagonal lies along the axis of symmetry and bisects the cross diagonal at right angles. For a trapezoid, the diagonals are generally unequal and do not bisect each other unless the trapezoid is isosceles. Knowing a diagonal can be useful for cutting materials, laying out foundations, or checking whether an angle is truly square.
How to use this calculator
Select the shape type from the dropdown. The input fields update to show only the measurements relevant to that shape. Enter positive values for all required fields. The area, perimeter, and diagonals appear instantly. The show-your-work panel at the bottom traces every formula step with your actual numbers so you can follow the calculation or check it by hand. The reference table lists all seven shapes and their formulas at a glance. For irregular quadrilaterals, you need four side lengths and the two interior angles at opposite vertices (alpha and gamma). The sum alpha + gamma must be less than 360 degrees for a valid convex or concave shape.
Quadrilateral area and perimeter formulas
| Shape | Area formula | Perimeter formula | Key property |
|---|---|---|---|
| Square | a^2 | 4a | All sides equal, all angles 90 deg |
| Rectangle | a x b | 2(a + b) | Opposite sides equal, all angles 90 deg |
| Parallelogram | b x h | 2(b + a) | Opposite sides equal and parallel |
| Rhombus | (d1 x d2) / 2 | 4a | All sides equal, diagonals bisect at 90 deg |
| Trapezoid | (b1 + b2) x h / 2 | b1 + b2 + legs | Exactly one pair of parallel sides |
| Kite | (d1 x d2) / 2 | 2(a + b) | Two pairs of adjacent equal sides |
| Irregular | Bretschneider formula | a + b + c + d | No special symmetry required |
Quick reference for the six most common quadrilateral types, where a, b are sides, h is perpendicular height, and d1, d2 are diagonals.
Frequently asked questions
What is the difference between a rhombus and a square?
Both have four equal sides, but a square also has four right angles (90 degrees each), while a rhombus only requires opposite angles to be equal. A square is a special case of a rhombus where all angles happen to be 90 degrees. Equivalently, a rhombus whose diagonals are equal in length is a square.
Why does a trapezoid area formula use the average of the two bases?
Imagine slicing a trapezoid horizontally at its mid-height. The cross-section at that point has the average length of the two bases. Multiplying this average width by the full height gives the area, which is the same as the sum of the top and bottom bases divided by two, then multiplied by the height. This is why area = (b1 + b2) / 2 x h.
What is Bretschneider's formula and when do I need it?
Bretschneider's formula calculates the area of any quadrilateral from its four side lengths and two opposite interior angles. You need it when your shape does not fit a special type like a rectangle or parallelogram and you do not have the diagonal lengths or perpendicular heights. It generalises Heron's formula for triangles to four-sided figures and works for both cyclic and non-cyclic quadrilaterals.
Can a quadrilateral have all four sides equal but still not be a square?
Yes - that is exactly what a rhombus is. A rhombus has four equal sides, but unless all its angles are also 90 degrees it is not a square. A rhombus can look like a tilted or flattened square. Similarly, a rectangle has four right angles but its four sides are generally not all equal.
How do I find the area of a quadrilateral if I only know the four side lengths?
Four side lengths alone do not uniquely determine the area of a quadrilateral, because the shape can flex. You also need at least one angle or one diagonal to fix the shape. If you additionally know two opposite angles (alpha and gamma), use Bretschneider's formula. If the quadrilateral is cyclic (all four vertices lie on a circle), alpha + gamma = 180 degrees, and the formula reduces to Brahmagupta's formula: area = sqrt((s-a)(s-b)(s-c)(s-d)).
What is the interior angle sum of any quadrilateral?
The four interior angles of any simple (non-self-intersecting) quadrilateral always add up to exactly 360 degrees. This follows because any quadrilateral can be split into two triangles by a diagonal, and each triangle contributes 180 degrees. For a convex quadrilateral this rule is straightforward; for a concave one, one angle is a reflex angle (greater than 180 degrees) while the others compensate.