How do you find the area of a kite?

The most common formula for the area of a kite is A = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals. If you only know the side lengths a and b and the angle between them θ, you can also use A = a × b × sin(θ).

What inputs do I need for this kite calculator?

You can use several input sets: (1) both diagonals, (2) both distinct side lengths and the angle between them, or (3) one side and both diagonals. The calculator automatically detects a valid combination and computes all remaining properties.

Are the diagonals of a kite always perpendicular?

Yes. In a geometric kite, the diagonals are perpendicular. One diagonal is the axis of symmetry: it bisects the other diagonal and splits the kite into two congruent triangles.

Can this kite calculator handle invalid dimensions?

The calculator checks for basic geometric consistency, such as positive lengths and triangle inequality in the right triangles formed by the diagonals. If the inputs cannot form a valid kite, it shows a clear error message instead of a misleading result.

Kite Calculator

Compute the area, perimeter, diagonals, side lengths, inradius, circumradius, and angles of a geometric kite from different input combinations.

Kite dimensions

Supported inputs (fill any valid combination):

Both diagonals d₁, d₂
Both side lengths and included angle a, b, θ
One side and both diagonals a, d₁, d₂ or b, d₁, d₂

Units:

Side a (short pair)

Side b (long pair)

Diagonal d₁ (axis of symmetry)

Diagonal d₂ (bisected)

Angle between a and b (θ)

deg

Results

Area

–

unit²

Perimeter

–

Diagonal d₁

–

Diagonal d₂

–

Side a

–

Side b

–

Inradius (incircle)

–

Circumradius (approx.)

–

Angle between equal sides (θ)

–

deg

Other vertex angle (φ)

–

deg

Kite diagram

The sketch below shows the naming convention used by this calculator. It is not to scale.

How the kite calculator works

This kite calculator is bidirectional: it accepts several different input sets and automatically computes all remaining properties that are geometrically determined. It also validates your inputs so you do not accidentally compute a kite that cannot exist.

1. Basic properties of a kite

A kite is a quadrilateral with:

Two pairs of adjacent sides equal: \(AB = AD = a\) and \(BC = CD = b\).
Perpendicular diagonals: \(d_1 \perp d_2\).
One diagonal (here \(d_1\)) is an axis of symmetry and bisects the other diagonal.

Key formulas used:

Area from diagonals: \[ A = \frac{d_1 \cdot d_2}{2} \]
Area from sides and included angle: \[ A = a \cdot b \cdot \sin(\theta) \]
Perimeter: \[ P = 2(a + b) \]
Right triangles formed by diagonals: \[ a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \] for the shorter side if the vertex lies on the bisected diagonal, and similarly for \(b\).

2. What this calculator can find

Depending on the inputs you provide, the tool can compute:

Area \(A\)
Perimeter \(P\)
Both diagonals \(d_1, d_2\)
Both side lengths \(a, b\)
Inradius (if the kite is tangential / has an incircle)
An approximate circumradius (radius of a circle through the vertices, when possible)
Vertex angles \(\theta\) (between equal sides) and \(\varphi\) (between unequal sides)

3. Valid input combinations

The calculator checks for these combinations, in this order:

Both diagonals known (\(d_1, d_2\)):
- Area: \(A = d_1 d_2 / 2\)
- Side lengths from right triangles: \[ a = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} \] (for the pair attached to the bisected diagonal) and similarly for \(b\) if geometry is consistent.
Both sides and included angle (\(a, b, \theta\)):
- Area: \(A = a b \sin\theta\)
- Diagonals from triangle geometry and perpendicularity constraints.
One side and both diagonals (e.g. \(a, d_1, d_2\)):
- Use Pythagoras in the right triangles formed by the diagonals to solve for the missing side and angles.

If the numbers violate basic geometric constraints (for example, negative lengths or impossible right triangles), the calculator shows an error instead of returning nonsense.

4. Units and precision

Choose any length unit (mm, cm, m, in, ft). The formulas are unit-agnostic.
All lengths are displayed with up to 4 decimal places; angles with 2 decimal places.
Area is reported in squared units (e.g. m², in²) based on your unit selection.

Worked example

Suppose a kite has diagonals \(d_1 = 10\ \text{cm}\) and \(d_2 = 6\ \text{cm}\).

Area: \[ A = \frac{10 \cdot 6}{2} = 30\ \text{cm}^2 \]
Half-diagonals: \[ \frac{d_1}{2} = 5,\quad \frac{d_2}{2} = 3 \]
Side length (for the triangles that share these halves): \[ a = \sqrt{5^2 + 3^2} = \sqrt{34} \approx 5.83\ \text{cm} \]
If the other pair of sides is the same (a symmetric kite), then \(b = a\) and the perimeter is: \[ P = 2(a + b) = 4a \approx 23.32\ \text{cm} \]

Frequently asked questions

Can a kite be a rhombus?

Yes. A rhombus is a special case of a kite where all four sides are equal (\(a = b\)). In that case, the formulas in this calculator reduce to the standard rhombus formulas.

What is the difference between a kite and a deltoid?

In many geometry texts, “kite” and “deltoid” are synonyms. Some authors use “deltoid” for the pure geometric shape and “kite” for the flying object, but the properties and formulas are the same.

Does every kite have an incircle?

No. Only tangential kites (where all sides are tangent to a single circle) have an incircle. This requires a specific relationship between the side lengths. When that condition is not met, the inradius shown by the calculator is left undefined.

Kite calculator FAQ

What is a kite in geometry?

In geometry, a kite is a quadrilateral with two pairs of adjacent sides equal in length. The equal sides meet at two vertices, and the diagonals are perpendicular. One diagonal is an axis of symmetry and bisects the other diagonal and the angles between equal sides.

How do I calculate the area of a kite?

The most convenient formula is \(A = \dfrac{d_1 d_2}{2}\), where \(d_1\) and \(d_2\) are the diagonal lengths. If you know the side lengths and the included angle instead, you can use \(A = a b \sin(\theta)\). The calculator supports both approaches.

Which inputs should I use?

Use the measurements you actually have:

For textbook problems, diagonals are often given – enter \(d_1\) and \(d_2\).
For construction or design, you may know side lengths and the angle – enter \(a\), \(b\), and \(\theta\).
If you measured one side and both diagonals, enter those three and the calculator will infer the rest.

Why do I get an “invalid kite” error?

This happens when the numbers cannot form a valid kite. Common causes include: negative or zero lengths, an angle outside 0–180°, or diagonals and sides that violate the Pythagorean relationships of the right triangles formed by the diagonals. Double‑check your measurements and units.