About the Kite Calculator
A kite is a quadrilateral with two pairs of equal adjacent sides — like the classic diamond-shaped flying kite. One diagonal (the axis of symmetry) cuts the kite into two mirror-image triangles, the two diagonals always cross at right angles, and the two angles between the unequal sides are equal.
This kite calculator works from either the two side lengths a and b plus the angle between them, or from the two diagonals and where the cross diagonal meets the symmetry axis. It returns the area, perimeter, both diagonals, all four interior angles and the radius of the inscribed circle (every convex kite has one).
It is useful for geometry homework, sizing fabric and spars for a real kite, laying out kite-shaped tiles or quilt pieces, and checking drawings. Angles are in degrees. If the sides and angle you enter produce a dart (a concave kite with one reflex angle), the calculator says so: area, perimeter and angles still apply, but a dart has no inscribed circle.
With the default inputs, the area is 39.3923. Change any value above to recalculate instantly.
How to use the kite calculator
- 1Choose whether you know the sides and angle or the diagonals.
- 2Enter sides a and b and the angle between them — or p, q and the crossing point.
- 3Read the area, perimeter and both diagonals.
- 4Check the four angles and the inradius for layout or proofs.
Formula and method
The diagonals of a kite are perpendicular, so the kite fills exactly half of the rectangle p × q around it: Area = pq/2. The symmetry diagonal p splits the kite into two congruent triangles with sides a, b and p, each with area ½·a·b·sin θ, so the area is also a·b·sin θ. The law of cosines in that triangle gives p from a, b and θ, and q follows from q = 2·Area/p.
In diagonal mode, the cross diagonal is bisected by p at a point x from the vertex where the two a sides meet, so a = √(x² + (q/2)²) and b = √((p − x)² + (q/2)²). The angles at the symmetry vertices come from the law of cosines and the four angles always total 360°. Every convex kite is tangential, so the inscribed circle has radius Area divided by the semiperimeter a + b.
- a, b
- The two pairs of equal adjacent sides
- θ
- Angle between sides a and b (the two equal angles)
- p
- Symmetry diagonal (joins the a–a and b–b vertices)
- q
- Cross diagonal (perpendicular to p, bisected by it)
- x
- Distance along p from the a–a vertex to the crossing point
- r
- Inradius
Worked examples
Sides 5 and 8 meeting at 100°
Area = 5 × 8 × sin 100° ≈ 39.39. The symmetry diagonal is √(25 + 64 − 80 cos 100°) ≈ 10.14 and the cross diagonal is 2 × 39.39 ÷ 10.14 ≈ 7.77. The other two angles are about 101.92° and 58.08°, so the angles total 360°.
Diagonals 12 and 8, crossing 3 units from the top
Area = 12 × 8 ÷ 2 = 48. The short sides are √(3² + 4²) = 5 and the long sides √(9² + 4²) ≈ 9.85, giving a perimeter of about 29.70. The inscribed circle has radius 48 ÷ 14.85 ≈ 3.23.
Flying kite with 60 cm and 90 cm spars, cross spar 20 cm from the top
The sail is half of 90 × 60, or 2,700 cm². The upper edges are √(20² + 30²) ≈ 36.06 cm and the lower edges √(70² + 30²) ≈ 76.16 cm, so you need about 224.4 cm of edging tape.
Frequently asked questions
How do you find the area of a kite?+
Multiply the two diagonals and divide by 2: A = pq/2. With diagonals of 12 and 8 the area is 48. If you know two unequal sides and the angle between them, use A = ab·sin θ instead.
What are the properties of a kite?+
A kite has two pairs of equal adjacent sides, perpendicular diagonals, one diagonal that bisects the other and is a line of symmetry, and one pair of equal opposite angles (the angles between unequal sides).
Is a rhombus a kite?+
Yes. A rhombus has all four sides equal, so it satisfies the kite definition in two ways. A square is both a rhombus and a kite. A general kite, however, is not a parallelogram.
How do I find the diagonals of a kite from its sides?+
You also need one angle. The symmetry diagonal is p = √(a² + b² − 2ab·cos θ) where θ is the angle between a and b, and the other diagonal is q = 2ab·sin θ ÷ p.
Does every kite have an inscribed circle?+
Every convex kite does, because the sums of opposite sides are equal (a + b = b + a). Its radius is the area divided by the semiperimeter, r = A ÷ (a + b).