Kite

A kite is a quadrilateral where two pairs of adjacent sides are equal in length. The figure is called a kite because it resembles a classic kite that can fly in the wind.

A quadrilateral \( \large ABCD \) is a kite if \( \large AB = AD \) and \( \large BC = CD \).

 

Kite

 

Properties that characterize a kite

 

  • A kite has two pairs of adjacent sides of equal length.
  • The diagonals are perpendicular to each other.
  • One diagonal bisects the other.
  • The kite has one axis of symmetry (along the longest diagonal).
  • Two opposite angles are equal.

 

The fact that the diagonals are perpendicular means that the kite can be divided into right triangles.

Thus, Pythagoras' theorem and elementary trigonometry can be used in calculations.

 

 

Calculator

Formulas

Side a

$$ a = \frac{h}{\sin(\angle A)} $$

$$ a = \frac{d_2}{2 \sin\left(\frac{\angle A}{2}\right)} $$

$$ a = b \cdot \cos(\angle B) \pm \sqrt{d_1^2 - b^2 \cdot \sin^2(\angle B)} $$

Side b

$$ b = \sqrt{a^2 + d_1^2 - 2 \cdot a \cdot d_1 \cdot \cos(\frac{\angle A}{2})} $$

$$ b = \frac{d_1 \cdot \sin\left(\frac{\angle A}{2}\right)}{\sin(\angle B)} $$

Diagonal 1

$$ d_1 = \frac{2ab \cdot \sin(\angle B)}{d_2} $$

$$ d_1 = \sqrt{a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(\angle B)} $$

Diagonal 2

$$ d_2 = \sqrt{2 \cdot b^2 \cdot (1 - \cos(\angle C))} $$

$$ d_2 = \sqrt{2 \cdot a^2 \cdot (1 - \cos(\angle A))} $$

Height

$$ h = a \cdot \sin(\angle A) $$

Angle A

$$ \angle A = \cos^{-1}\!\left(\frac{2a^2 - d_2^2}{2a^2}\right) $$

$$ \angle A = \sin^{-1}\!\left(\frac{h}{a}\right) $$

Angle B

$$ \angle B = \cos^{-1}\!\left(\frac{a^2 + b^2 - d_1^2}{2ab}\right) $$

$$ \angle B = \sin^{-1}\!\left(\frac{d_1 \cdot d_2}{2ab}\right) $$

Angle C

$$ \angle C = \cos^{-1}\!\left(\frac{2b^2 - d_2^2}{2 \cdot b^2}\right) $$

$$ \angle C = \sin^{-1}\!\left(\frac{d_1 \cdot d_2}{2 \cdot b^2}\right) $$

Area

$$ A = \frac{1}{2} \cdot d_1 \cdot d_2 $$

$$ A = a \cdot b \cdot \sin(\angle B) $$

Perimeter

$$ P = 2 \cdot (a + b) $$