The definition of a kite is a quadrilateral which has:
Two separate pairs of adjacent congruent sides
A kite is a four sided shape - the sides can be split into two separate pairs. Each pair consists of two sides which are next to each other(adjacent) and are the same length(congruent).
In the diagram
the pairs of (adjacent and congruent) sides are:
A kite also has the following properties:
The diagonals are perpendicular. The diagonals (blue lines in the diagram) cross each other at a 90° angle(right angles).
ONE diagonal bisects the other. As shown in the kite above, diagonal AB bisects(divides into two equal parts where it crosses) diagonal CD.
ONE diagonal bisects opposite angles. Diagonal AB bisects angles A and B.
ONE pair of opposite congruent angles. Angles C and D are the same size.
This diagram shows how a kite can be rearranged. Moving the triangle shapes creates a rectangle with the same area as the kite shape.
This tells us that we can use the rectangle area formula (area = height x base) to calculate the area of the kite.
What is the base and height of this rectangle? Look at the blue lines, these are the diagonals of the kite.
The diagonal d₂ is the same height as the rectangle and d₁ is twice the length of the base.
If we mutiply d₁ and d₂ together, the value will be twice the area of the kite. This is because d₁ is twice as long as the base, therefore in order to calculate the area of a kite we need to half this value:
Area of a kite = ½ d₁ d₂
If you want to calculate the perimeter(the distance around the outside) of a kite shape, all you need to do is add together the length of all the sides.
Because there are two pairs of congruent sides, you can multiply each congruent pair side length by 2 and add them together:
Perimeter = 2a + 2b
Where a is is the length of each side in one congruent pair and b is the length of each side in the other pair.
For example, in the diagram the two congruent sides are:
- AC and AD
- CB and DB
Perimeter = 2(8) + 2(12)
= 16 + 24
What is the area of this kite shape?
Area of a kite = ½ d₁ d₂
In order to calculate the area, first you need to calculate the lengths of the diagonals.
You can do this using some right triangle rules. The diagonals split the kite into four right triangles.
We know that the length of one side of triangle AED is 5. If you look at the size of the angles of this triangle you can see that it is a 45-45-90 triangle. Using the ratios of a 45-45-90 triangle we can calculate the length of line ED.
Line ED = 5
The length of diagonal CD is twice the length of line ED(AB bisects diagonal CD).
Diagonal CD = 10
Now we can use the
to calculate the length of diagonal AB. Look at triangle EDB. We
have the length of two sides of this right triangle: ED = 5, DB = 13 - we can use the pythagorean theorem to find the length of the third side. This triangle is also one of the
This gives us the length of the third side. Line EB = 12.
The length of the diagonal AB = AE + EB
AB = 5 + 12 = 17
Now we can put the diagonal lengths into the area formula:
Area = ½ x 5 x 17
= 42.5 units²