Rhombus and Kite Calculator

How it works

A rhombus is a four-sided figure with all sides equal: a square pushed over. A kite has two pairs of equal sides that sit next to each other, like a classic flying kite. Both have diagonals that cross at right angles, so for either shape the area is half the product of the diagonals. A rhombus is a special kite, so the rhombus modes are the kite modes with every side the same.

A rhombus from its side and an angle: the area is side² × sin θ, like any parallelogram, and the diagonals cut it into four right triangles, giving p = 2s cos(A/2) and q = 2s sin(A/2) where A is the acute angle. From the diagonals, each side is the hypotenuse of a right triangle with legs p/2 and q/2, and the acute angle is 2 atan(q/p). Either interior angle describes the same rhombus; results list the acute one as A and the obtuse one as B.

A kite from two sides and the angle between them: the axis of symmetry is the third side of the triangle those two sides make, by the law of cosines, and the cross diagonal follows from the area. The angle at the vertex where the two a sides meet is A, at the two b sides B, and A + B + 2θ = 360°. From the diagonals alone only the area is fixed, because the axis can cross the other diagonal anywhere along it, so the sides and angles are left blank.

Formula

area = p × q ÷ 2                      (rhombus or kite, from diagonals)
rhombus: area = s² sin θ,  perimeter = 4s,  p = 2s cos(A/2),  q = 2s sin(A/2)
         s = √((p/2)² + (q/2)²),  A = 2 atan(q ÷ p),  B = 180° − A
kite:    area = a b sin θ,  perimeter = 2(a + b),  p = √(a² + b² − 2ab cos θ),  q = 2 × area ÷ p
         A = 2 acos((a² + p² − b²) ÷ 2ap),  B = 2 acos((b² + p² − a²) ÷ 2bp)

Example

A rhombus with 5 ft sides and a 60° angle has an area of 25 × sin 60° ≈ 21.65 ft² and a perimeter of 20 ft. Its diagonals are 2 × 5 × cos 30° ≈ 8.66 ft and 2 × 5 × sin 30° = 5 ft: the short diagonal equals the side, because a 60° rhombus is two equilateral triangles. The angles are 60° and 120°.

A rhombus with diagonals of 8 ft and 6 ft has an area of 8 × 6 ÷ 2 = 24 ft², sides of √(4² + 3²) = 5 ft, a perimeter of 20 ft, and angles of 2 × atan(6 ÷ 8) ≈ 73.74° and 106.26°.

A kite with sides of 3 ft and 4 ft meeting at 90° has an area of 3 × 4 × sin 90° = 12 ft² and a perimeter of 14 ft. The axis is the 3-4-5 hypotenuse, 5 ft, the cross diagonal is 2 × 12 ÷ 5 = 4.8 ft, and the angles at the ends of the axis are about 106.26° and 73.74°.

Assumptions and limitations

• A rhombus has four equal sides; a kite has two pairs of equal adjacent sides. A rhombus is a kite, so either rhombus mode gives the same figure the kite modes would.
• From a kite's two diagonals alone only the area is fixed; for its sides and angles use the sides-and-angle method.
• Angles are in degrees. For a rhombus either interior angle may be entered; for a kite the angle is the one between the two unequal sides.
• All lengths are in the unit you chose. The calculator labels the results with it and does not convert between units.
• Results are rounded for display only.

Frequently asked questions