About Kite in Math
What is a Kite?
A kite is a quadrilateral consisting of two pairs of adjacent sides of same length and two pairs of opposite sides that are unequal in length. So the four interior angles formed in a kite include a pair of opposite angles formed by the unequal sides and a pair of opposite angles formed by the equal sides.
Based upon angle measurements, there are two types of kites:
- Convex kite: When all the interior angles are less than 180°
- Concave kite: When any one of the interior angles is greater than 180°
Diagonals of a Kite
The diagonals of a kite will always intersect each other at 90°. The intersecting diagonals are perpendicular to each other and thus divide the kite into four right angled triangles.
We notice that the shorter diagonal AC of the kite ABCD divides the kite into two isosceles triangles △ABC and △ADC. Whereas, the longer diagonal BD divides the kite into two congruent triangles △ABD and △BDC as shown in the image.
Area of a Kite
The area of a kite is the measure of space enclosed within the sides of the kite in a two-dimensional plane.
If \(d_1\) and \(d_2\) are the measures of the two diagonals of a kite then the formula to calculate its area is given by:
Area of kite, A = \(\frac{1}{2}\times d_1 \times d_2\)
The area is measured in square units such as square meters, square inches or square yards.
Solved Examples
Example 1: The area of a kite is 240 square units and one of its diagonals measures 20 units. Find the measure of the other diagonal of the kite.
Solution:
Area = 240 sq. units
\(d_2\) = 20 units
Use the formula for the area of a kite to find the other diagonal:
A = \(\frac{1}{2}\times d_1 \times d_2\) [Formula for the area of kite]
240 = \(\frac{1}{2}\times d_1 \times 20\) [Substitute the given values]
\(d_1\) = \(\frac{240 \times 2}{20}\) [Solve for \(d_1\)]
\(d_1\) = 24 [Simplify]
Therefore the measure of the other diagonal \(d_1\) is 24 units.
Example 2: Calculate the area of a kite if its diagonals measure 8 inches and 14 inches.
Solution:
Here, \(d_1\) = 8 inches, and \(d_2\) = 14 inches
We can find the area of the kite with the following formula:
Area, A = \(\frac{1}{2}\times d_1 \times d_2\) [Formula for the area of kite]
On substituting the values in the formula equation,
A = \(\frac{1}{2}\times 8 \times 14\)
= 56 square inches [Simplify]
Hence, the area of the kite is 56 square inches.
Example 3: The area of a kite-shaped garden is 240 square feet. If the distance between two opposite vertices of the garden is 6 feet then find the distance between its other two opposite vertices.
Solution:
The distance between the opposite vertices is the diagonal of the kite shaped garden.
According to the area of the kite formula,
Area, A = \(\frac{1}{2}\times d_1 \times d_2\) [Formula for the area of kite]
Here, A is given as 240 square feet and \(d_1\) is given as 6 feet.
On substituting the values in the formula equation,
240 = \(\frac{1}{2}\times 6 \times d_2\)
\(d_2\) = \(\frac{240 \times 2}{6}\) [Solve for \(d_2\)]
\(d_2\) = 80 [Simplify]
Hence, \(d_2\) is 80 feet.
Thus, the distance between the other two opposite vertices of the kite shaped garden is 80 feet.
Yes, a kite is a quadrilateral as it is a four-sided closed figure.
A square is considered as a special form of a kite.
Both the kite and trapezoid are quadrilaterals, but a kite has no parallel sides while trapezoids have one set of parallel sides.