Area of Trapezoid

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 cm², m², in², etc). For example, if 15 unit squares each of length 1 cm can be fit inside a trapezoid, then its area is 15 cm². A trapezoid is a type of quadrilateral with one pair of parallel sides (which are known as bases). It means the other pair of sides can be non-parallel (which are known as legs). It is not always possible to draw unit squares and measure the area of a trapezoid. So, let us learn about the trapezoid area formula and learn how to find the area of a trapezoid without the height in this article.

The area of a trapezoid is the total space covered by its sides. An interesting point to be noted here is that if we know the length of all the sides we can simply split the trapezoid into smaller polygons like triangles and rectangles, find their area, and add them up to get the area of the trapezoid. However, there is a direct formula that is used to find the area of a trapezoid if we know certain dimensions.

The area of a trapezoid can be calculated if the length of its parallel sides and the distance (height) between them is given. The formula for the area of a trapezoid is expressed as,

A = ½ (a + b) h

where (A) is the area of a trapezoid, 'a' and 'b' are the bases (parallel sides), and 'h' is the height (the perpendicular distance between a and b)

Example:

Find the area of a trapezoid whose parallel sides are 32 cm and 12 cm, respectively, and whose height is 5 cm.

Solution:

The bases are given as, a = 32 cm; b = 12 cm; the height is h = 5 cm.

The area of the trapezoid = A = ½ (a + b) h

A = ½ (32 + 12) × (5) = ½ (44) × (5) = 110 cm².

When all the sides of the trapezoid are known, and we do not know the height we can find the area of the trapezoid. In this case, we first need to calculate the height of the trapezoid. Let us understand this with the help of an example. It is to be noted that this method can be used in the case of an isosceles trapezoid.

Example: Find the area of a trapezoid in which the bases (parallel sides) are given as 6 and 14 units respectively, and the non-parallel sides (legs) are given as, 5 units each.

Solution: Let us calculate the area of the trapezoid using the following steps.

• Step 1: We know that the area of a trapezoid = ½ (a + b) h; where h = height of the trapezoid which is not given in this case; a = 6 units, b = 14 units, non parallel sides (legs) = 5 units each. This means that it is an isosceles trapezoid because the non-parallel sides are equal in length.
• Step 2: So, if we find the height of the trapezoid, we can calculate the area. If we draw the height of the trapezoid on both sides we can see that the trapezoid is split into a rectangle ABQP and 2 right-angled triangles, ADP and BQC.
• Step 3: Since a rectangle has equal opposite sides, this means AP = BQ and it is given that the sides AD = BC = 5 units. So, the height AP and BQ can be calculated using the Pythagoras theorem.
• Step 4: Now, let us find the length of DP and QC. Since ABQP is a rectangle, AB = PQ and DC = 14 units. This means PQ = 6 units, and the remaining combined length of DP + QC can be calculated as follows. DC - PQ = 14 - 6 = 8. So, 8 ÷ 2 = 4 units. Therefore, DP = QC = 4 units.
• Step 5: Now, the height of the trapezoid can be calculated using the Pythagoras theorem. Taking the right-angled triangle ADP, we know that AD = 5 units, DP = 4 units, so AP = √(AD² - DP²) = √(5² - 4²) = √(25 - 16) = √9 = 3 units. Since ABQP is a rectangle, in which the opposite sides are equal, AP = BQ = 3 units.
• Step 6: Now, that we know all the dimensions of the trapezoid including the height, we can calculate its area using the formula, area of a trapezoid = ½ (a + b) h; where h = 3 units, a = 6 units, b = 14 units. After substituting the values in the formula, we get, area of a trapezoid = ½ (a + b) h = ½ (6 + 14) × 3 = ½ × 20 × 3 = 30 unit².

We can prove the area of a trapezoid formula by using a triangle here. Taking a trapezoid of bases 'a' and 'b' and height 'h', let us prove the formula.

• Step 1: Split one of the legs into two equal parts and cut a triangular portion of the trapezoid as shown below.
• Step 3: Attach it at the bottom as shown, such that it forms a big triangle.

• Step 4: This way, the trapezoid is rearranged as a triangle. Even after we attach it this way, we know that the area of the trapezoid and the new big triangle remains the same. We can also see that the base of the new big triangle is (a + b) and the height of the triangle is h.
• Step 5: So, it can be said that the area of the trapezoid = the area of the triangle
• Step 6: This can be written as, area of the trapezoid = ½ × base × height = ½ (a + b) h

Thus, we have proved the formula for finding the area of a trapezoid.

The area of a trapezoid is the number of unit squares that can fit into it. Area of trapezoid calculator is an online tool that helps to find the area of a trapezoid. If certain parameters such as the value of base or height is available we can directly give the inputs and calculate the area. Try Cuemath's Area of a Trapezoid Calculator and calculate the area of a trapezoid within a few seconds. For more practice check out the area of trapezoid worksheets and solve the problems with the help of the calculator.

☛ Related Articles

• Area of Equilateral Triangle
• Area of Square
• Area of Parallelogram
• Area of Rectangle
• Area of Rhombus
• Area of Pentagon
• Area of Circle

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What is Area of Trapezoid in Math?

The area of a trapezoid is the number of unit squares that can fit into it. We know that a trapezoid is a four-sided quadrilateral in which one pair of opposite sides are parallel. The area of a trapezoid is calculated with the help of the formula, Area of trapezoid = ½ (a + b) h, where 'a' and 'b' are the bases (parallel sides) and 'h' is the perpendicular height. It is represented in terms of square units.

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?

The formula for the area of a trapezoid can be proved easily. Consider a trapezoid of bases 'a' and 'b', and height 'h'. We can cut a triangular-shaped 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 to Find the Missing Base of a Trapezoid if you Know the Area?

We know that 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 to Find the Height of a Trapezoid With the Area and Bases?

If the area and the bases of a trapezoid is known, then we can calculate its height using the formula, Area of trapezoid = ½ (a + b) h; where 'a' and 'b' are the bases and 'h' is the height. In other words, we can find the height of the trapezoid by substituting the given values of the area and the two bases.

How to Find the Area of a Trapezoid Without the Height?

If the height of the trapezoid is not given and all its sides are given, then we can divide the trapezoid into two congruent right triangles and a rectangle. It is to be noted that this method can be applied only if it is an isosceles trapezoid in which the non-parallel sides are of equal length. Using the Pythagoras theorem in the right-angled triangles, we can calculate the height. After we get the height, we can use the formula, A = ½ (a + b) h, to get the area of the trapezoid. A detailed explanation about this method is given above on this page along with a solved example, when we need to find the area of a trapezoid without the height.

What is the Formula for Area of Trapezoid?

The formula that is used to find the area of a trapezoid is expressed as, Area of trapezoid = ½ (a + b) h; where a' and 'b' are the bases (parallel sides) and 'h' is the height of the trapezoid.