Trapezoids and kites are quadrilaterals. The word trapezoid is common in the US and Canadian syllabus, however, the word trapezoid, taken from the word trapezium, is derived from the Greek word “trapeza”. The meaning of “trapeza” is table. In our daily lives, we can see different objects that are shaped like a trapezium like briefcases, a kite and so on.
When talking about trapezoids, kite, which is a really familiar shape, is also a type of quadrilateral. We can see people flying different types of kites. But the general shape of a kite is the same as that of a geometric trapezoid.
The trapezoid is a quadrilateral that has one pair of opposite sides parallel to each other. It is also a two-dimensional figure. Each of the opposite sides that are parallel is called the base of the trapezium, and the other two sides are known as the legs of the trapezium. As you have already seen, a trapezoid has four sides, four vertices and four angles. The sum of the adjacent interior angles of a trapezoid is \(180^{\circ }\) i.e. the adjacent interior angles of a trapezoid are supplementary to each other.
In the trapezoid ABCD, A and D, B and C are adjacent interior angles forming pairs of supplementary angles. Sides AB and CD are parallel sides.
A kite is a quadrilateral in which the two adjacent pairs of sides are equal. It is a two dimensional figure. The angles formed by unequal sides are also equal. There are two types of kites.
If all the interior angles of the kite are less than \(180^{\circ }\) then the kite is called a convex kite. If any one of the interior angles of a kite is greater than \(180^{\circ }\) the kite is called a concave kite.
The diagonals of a kite always intersect each other at \(90^{\circ }\). The two perpendicular intersecting diagonals divide the kite into four right angled triangles.
The shorter diagonal of a kite divides the kite into two isosceles triangles.
The longer diagonal of a kite divides the kite into two congruent triangles.
The area of a trapezoid is the area enclosed by the sides of the same trapezoid in a two-dimensional plane. It is measured in square units such as square meters, square inches or square yards.
The area ‘A’ of a trapezoid is one-half of its height “h” and the sum of its bases \(“b_{1}”\) and \(“b_{2}”\)
As seen in the figure, the diagonal of this trapezoid divides the trapezoid into two triangles. Thus, the area of the trapezoid is the sum of the area of two triangles, ADC and ABC.
Algebraically, the formula for the area of a trapezoid is written as
\(\text{Area of trapezoid}(A)=\frac{1}{2}\times \text{h}\left( b_{1}+b_{2} \right)\)
The area of a kite is the area enclosed by the sides of a kite in a two-dimensional plane. It is measured in square units such as square meters, square inches or square yards.
The area “A” of a kite is one-half of the product of the length of its two diagonals \(“d_{1}”\) and \(“d_{2}”\)
Algebraically, the formula for the area of a kite is written as,
\(\text{Area of kite}\text{(A)}= \frac{1}{2}\times \text{d}_{1}\times \text{d}_{2}\)
Example1: Find the area of the trapezoid shown in the image.
Solution:
\(\text{Area of trapezoid} = \frac{1}{2}\times \text{h}\left( b_{1}+b_{2} \right)\)
= \(\frac{1}{2}\times 7\times \left( 6+9 \right)\) [Substitute]
= \(\frac{1}{2}\times 7\times \left( 15 \right)\) [Add]
= 52.5 [Multiply]
The area of the trapezoid is 52.5 square feet
Example 2: Find the area of the trapezoid.
Solution:
\(\text{Area of trapezoid} = \frac{1}{2}\times \text{h}\left( b_{1}+b_{2} \right)\)
= \(\frac{1}{2}\times 6\times \left( 9+12 \right)\) [Substitute]
= \(\frac{1}{2}\times 6\times \left( 21 \right)\) [Substitute]
= 63 [Multiply]
The area of the trapezoid is 63 square inches.
Example 3: The shape of a trapezoid can be applied to find the area of a county in Illinois. Given that the population of this county is about 28750 people, what will be the number of people per square mile of the same county?
Solution:
You are given the population and the dimensions of a country shaped like a trapezoid. You are asked to find the number of people per square mile.
Use the formula for the area of a trapezoid to find the area of the county Then divide the population by the area to find the number of people per square mile.
\(\text{Area of trapezoid} = \frac{1}{2}\times \text{h}\left( b_{1}+b_{2} \right)\)
= \(\frac{1}{2}\times 25\times \left( 20+30 \right)\) [Substitute]
= \(\frac{1}{2}\times 25\times \left( 50 \right)\) [Substitute]
= 625 [Multiply]
The area of the county is 625 square miles.
\(\frac{28750 \text{ people}}{625\text{ mile}^{2}}= 46\) [Divide]
So, there are about 46 people per square mile of this county in Illinois.
Example 4: Find the area of the kite shown:
Solution:
First decompose the kite into two triangles. Then the area of the kite will be given by the sum of the areas of the two triangles. We have two triangles with a base 10 ft. and heights 4ft. and 6ft. respectively.
\(\text{Area of kite} = \frac{1}{2}\times 10\times 4 + \frac{1}{2}\times 10\times 6\)
= 20 + 30 [Multiply]
= 50 [Add]
The area of the kite is about 50 square feet.
Example 5: Find the base area of a kite shaped box that contains a new kite. The length of two diagonals of the kite are 20 inches and 15 inches respectively.
Solution:
The box that contains the kite is in the shape of a kite. The base area of the kite box is calculated using the formula for the area of a kite
\(\text{Area of kite}\text{(A)}= \frac{1}{2}\times \text{d}_{1}\times \text{d}_{2}\) [Area of kite formula]
= \(\frac{1}{2}\times 15\times 20\) [Replace d1 with 15 and d2 with 20]
= 150 [Simplify]
The base area of the kite box is 150 square inches.
Yes, a square is also a kite. If two adjacent sides of a quadrilateral are equal then the figure can be considered a kite. For a square all four sides are equal so automatically its adjacent sides are equal i.e. the condition for the kite is satisfied.
Yes, a parallelogram is also a trapezoid. The opposite sides of a parallelogram are parallel to each other. This shows us that the condition for a shape to be a trapezoid i.e. at least one pair opposite sides are parallel, is automatically satisfied.
Yes, in a rectangle the opposite sides are parallel and equal in length. Therefore, the minimum condition for a trapezoid i.e. one pair of opposite sides that are parallel, is satisfied.
Yes, a square is also a trapezoid. All the four sides of a square are equal and opposite sides are parallel to each other. Therefore, the opposite pair of parallel sides makes the square a trapezoid.