Area of Trapezoid
A trapezoid is a quadrilateral with one pair of parallel sides (which are known as bases). It means the other pair of sides can be nonparallel (which are known as legs). The area of a trapezoid is the number of unit squares that can be fit into it and it is measured in square units (like cm^{2}, m^{2}, in^{2}, etc). For example, if 15 unit squares each of length 1 cm can be fit inside a trapezoid, then its area is 15 cm^{2}.
It is not possible always to draw unit squares and measure the area of a trapezoid. There is a formula to find the area of a trapezoid which we are going to learn on this page.
1.  Area of Trapezoid Formula 
2.  How to Derive Area of Trapezoid Formula? 
3.  FAQs on Area of Trapezoid 
4.  Solved Examples 
5.  Practice Questions 
Area of Trapezoid Formula
To find the area of a trapezoid, it is enough to know the lengths of two of its parallel sides and the distance (height) between them. The area (A) of a trapezoid whose bases are a and b and whose height is h (the perpendicular distance between a and b) is,
A = ½ (a + b) h
Example
Find the area of a trapezoid whose bases are 32 cm and 12 cm and whose height is 5 cm.
Solution
The bases are a = 32 cm; b = 12 cm.
The height is h = 5 cm.
The area of the trapezoid is,
A = ½ (a + b) h
A = ½ (32 + 12) (5) = ½ (44) (5) = 110 cm^{2}.
How to Derive Area of Trapezoid Formula?
We can prove the formula to find the area of a trapezoid in two ways:
 Proof by using a parallelogram
 Proof by using a triangle
We will see the proof of the area of a trapezoid formula by using a triangle here. Consider the above trapezoid of bases a and b and height h. To prove the formula,
 Split one of the legs into two equal parts
 Cut a triangular portion of the trapezoid (as shown below in the top figure of the below diagram)
 Attach it at the bottom (as shown in the bottom figure of the below diagram)
This way, the trapezoid is rearranged as a triangle. We can easily see from the above diagram that the areas of both the trapezoid and the triangle are the same. Also, we can see that the base of the triangle is (a + b) and the height of the triangle is h.
The area of the trapezoid = The area of the triangle
The area of the trapezoid = ½ × base × height = ½ (a + b) h
Thus, we have proved the formula for finding the area of a trapezoid. Let's proceed to solve some examples based on the area of a trapezoid.
Solved Examples on Area of Trapezoid

Example 1: One of the bases of a trapezoid is of length 8 units. If its height is 12 units and its area is 108 square units. Then find the length of its other base.
Solution:
One of the bases is, a = 8 units.
Let the other base be 'b'.
Its area is, A = 108 square units.
Its height is, h = 12 units.
Substitute all these values in the area of trapezoid formula,
A = ½ (a + b) h
108 = ½ (8 + b) (12)
108 = 6 (8 + b)
Dividing both sides by 6,
18 = 8 + b
Subtracting 8 from both sides,
10 = b
Answer: The length of another base of the given trapezoid = 10 units.

Example 2: Find the area of an isosceles trapezoid with the length of each leg to be 8 units and the bases are of lengths 13 units and 17 units.
Solution:
The bases are a= 13 units and b = 17 units. Let us assume that its height to be h.
We can divide the given trapezoid into two congruent right triangles and a rectangle as follows:
From the above figure,
x + x + 13 = 17
2x + 13 = 17
2x = 4
x = 2
Using Pythagoras theorem,
x^{2} + h^{2} = 8^{2}
2^{2} + h^{2} = 64
4 + h^{2} = 64
h^{2} = 60
h = √60 = √4 × √15 = 2√15
The area of the given trapezoid is,
A = ½ (a + b) h
A = ½ (13 + 17) (2√15) = 30√15 square units
Answer: The area of the given trapezoid = 30√15 square units.
FAQs on Area of Trapezoid
How to Find the Area of a Trapezoid?
The area of a trapezoid is found using the formula, A = ½ (a + b) h, where 'a' and 'b' are the bases (parallel sides) and 'h' is the height (the perpendicular distance between the bases) of the trapezoid.
Why Is the Area of a Trapezoid ½ (a + b) h?
Consider a trapezoid of bases 'a' and 'b', and height 'h'. We can cut a triangularshaped portion from the trapezoid and attach it at the bottom so that the entire trapezoid is rearranged as a triangle. Then the triangle obtained has the base (a + b) and height h. By applying the area of a triangle formula, the area of the trapezoid (or triangle) = ½ (a + b) h. For more information, you can refer to How to Derive Area of Trapezoid Formula? section of this page.
How Do You Find the Missing Base of a Trapezoid if You Know the Area?
The area of a trapezoid whose bases are 'a' and 'b' and whose height is 'h' is A = ½ (a + b) h. If one of the bases (say 'a'), height, and area are given, then we will just substitute these values in the above formula and solve it for the missing base (a) as follows:
A = ½ (a + b) h
Multiplying both sides by 2,
2A = (a + b) h
Dividing both sides by h,
2A/h = a + b
Subtracting b from both sides,
a = (2A/h)  b
How Do You Find the Height of a Trapezoid With the Area and Bases?
The area of a trapezoid whose bases are 'a' and 'b' and whose height is 'h' is A = ½ (a + b) h. We can find the height of the trapezoid with area and bases by solving the above formula for h as follows:
A = ½ (a + b) h
Multiplying both sides by 2,
2A = (a + b) h
Dividing both sides by (a + b),
h = (2A) / (a + b).
What Is the Area of an Isosceles Trapezoid With Sides?
If the side lengths of an isosceles trapezoid are given, then we will divide it into two congruent right triangles and a rectangle. We find the areas of each of these shapes and add them which gives the area of the given trapezoid. You can see this in detail in Example 2 under the "solved examples" section of this page.
What Is the Area of a Trapezoid With Coordinates?
If the coordinates of vertices of a trapezoid are given, then we can find the lengths of its bases 'a' and 'b' using the distance formula. To find the height 'h' (the perpendicular distance between the bases), we can use the perpendicular distance of a point to a line formula (for this, we need to find the line equation of one of the bases).
The perpendicular distance from a point \((x_0,y_0)\) to a line ax + by + c =0 is \(\dfrac{ax_0+by_0+c}{\sqrt{a^2+b^2}}\).
Then we can apply the formula A = ½ (a + b) h to find the area of the trapezoid.
How Do You Find the Area of an Isosceles Trapezoid Without the Height?
If the height of the trapezoid is not given and all its sides are given instead, then we will divide it into two congruent right triangles and a rectangle. We find the areas of each of these shapes and add them which gives the area of the given trapezoid. You can see this in detail in Example 2 under the "solved examples" section of this page.