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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.

What is the Area of Trapezoid?

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.

Area of Trapezoid Formula

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.

Area of Trapezoid without Height

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.

How to Derive Area of Trapezoid Formula?

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.

Area of a Trapezoid Calculator

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.

How to Calculate the Area of a Trapezoid

A trapezoid, also known as a trapezium, is a 4-sided shape with two parallel bases that are different lengths. The formula for the area of a trapezoid is A = ½(b₁+b₂)h, where b₁ and b₂ are the lengths of the bases and h is the height.If you only know the side lengths of a regular trapezoid, you can break the trapezoid into simple shapes to find the height and finish your calculation. When you’re finished, just label your units!

Finding the Area Using Height and Base Lengths

Add together the lengths of the bases. The bases are the 2 sides of the trapezoid that are parallel with one another. If you aren’t given the values for the base lengths, then use a ruler to measure each one. Add the 2 lengths together so you have 1 value.

For example, if you find that the top base (b₁) is 8 cm and the bottom base (b₂) is 13 cm, the total length of the bases is 21 (8 cm + 13 cm = 21 cm, which reflects the “b = b₁ + b₂” part of the equation).

2. Measure the height of the trapezoid. The height of the trapezoid is the distance between the parallel bases. Draw a line between the bases, and use a ruler or other measuring device to find the distance. Write the height down so you don’t forget it later in your calculation.

The length of the angled sides, or the legs of the trapezoid, is not the same as the height. The leg length is only the same as the height if the leg is perpendicular to the bases.

3. Multiply the total base length and height together. Take the sum of the base lengths you found (b) and the height (h) and multiply them together. Write the product in the appropriate square units for your problem.

In this example, 21 cm x 7 cm = 147 cm² which reflects the “(b)h” part of the equation.

4. Multiply the product by ½ to find the area of the trapezoid. You can either multiply the product by ½ or divide the product by 2 to get the final area of the trapezoid since the result will be the same. Make sure you label your final answer in square units.

For this example, 147 cm² / 2 = 73.5 cm², which is the area (A).

Calculating Area of a Trapezoid If You Know the Sides

Break the trapezoid into 1 rectangle and 2 right triangles. Draw straight lines down from the corners of the top base so they intersect and form 90-degree angles with the bottom base. The inside of the trapezoid will have 1 rectangle in the middle and 2 triangles on either side that are the same size and have 90-degree angles. Drawing the shapes helps you visualize the area better and helps you find the height of the trapezoid.

This method only works for regular trapezoids.

2. Find the length of one of the triangle’s bases. Subtract the length of the top base from the length of the bottom base to find the amount that’s left over. Divide the amount by 2 to find the length of the triangle’s base. You should now have the length of the base and the hypotenuse of the triangle.

For example, if the top base (b₁) is 6 cm and the bottom base (b₂) is 12 cm, then the base of the triangle is 3 cm (because b = (b₂ – b₁)/2 and (12 cm – 6 cm)/2 = 6 cm which can be simplified to 6 cm/2 = 3 cm).

3. Use the Pythagorean theorem to find the height of the trapezoid. Plug the values for the length of the base and the hypotenuse, or the longest side of the triangle, into A² + B² = C², where A is the base and C is the hypotenuse. Solve the equation for B to find the height of the trapezoid. If the length of the base you found is 3 cm and the length of the hypotenuse is 5 cm, then in this example:

Fill in the variables: (3 cm)² + B² = (5 cm)²
Simplify the squares: 9 cm +B² = 25 cm
Subtract 9 cm from each side: B² = 16 cm
Take the square root of each side: B = 4 cm

Tip: If you don’t have a perfect square in your equation, then simplify it as much as possible and leave a value with a square root. For example, √32 = √(16)(2) = 4√2.

4. Plug the base lengths and height into the area formula and simplify it. Put the base lengths and the height into the formula A = ½(b₁ +b₂)h to find the area of the trapezoid. Simplify the number as much as you can and label it with square units.

Write the formula: A = ½(b₁+b₂)h
Fill in the variables: A = ½(6 cm +12 cm)(4 cm)
Simplify the terms: A = ½(18 cm)(4 cm)
Multiply the numbers together: A = 36 cm².

Area of Trapezoid Examples

Example 1: If one of the bases of a trapezoid is equal to 8 units, its height is 12 units and its area is 108 square units, find the length of the other base.

Solution:One of the bases is ‘a’ = 8 units.Let the other base be ‘b’.The area of the trapezoid 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) h108 = ½ (8 + b) × (12)108 = 6 (8 + b)Dividing both sides by 6,18 = 8 + bb = 10

Answer: The length of the other base of the given trapezoid = 10 units.

Example 2: Find the area of an isosceles trapezoid in which the length of each leg is 8 units and the bases are equal to 13 units and 17 units respectively.

Solution:

The bases are a = 13 units and b = 17 units. Let us assume that its height is 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,

x² + h² = 8²

2² + h² = 64

4 + h² = 64

h² = 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 = 116.18 square units

Answer: The area of the given trapezoid = 116.18 square units.

Example 3: Find the area of a trapezoid in which the bases are given as 7 units and 9 units and the height is given as 5 units.

Solution: The area of a trapezoid = ½ (a + b) h; where a = 7, b = 9, h = 5.Substituting these values in the formula, we get:A = ½ (a + b) hA = ½ (7 + 9) × 5A = ½ × 16 × 5 = 40 unit²Therefore, the area of the trapezoid is 40 square units.

Examples Using Trapezoid Formula

Example 1: If the perimeter of a trapezoid is 60 units and three of its sides are 15 units, 20 units, and 16 units respectively, find the measure of the fourth side using the Trapezoid Formula.
Solution:

Given: Perimeter = 60 units, a = 15 units, b = 20 units, c = 16 units, d = ?

We know that, according to Trapezoid Formula,

Perimeter of a trapezoid = Sum of all the sides

⇒ a + b + c + d = 60

⇒ 15 + 20 + 16 + d = 60

⇒ d = 9 units

Answer: Thus, the fourth side measures 9 units.

Example 2: Using area of trapezoid formula, find the area of a trapezoid whose bases are 19 units and 15 units and height is 8 units.
Solution:

Given:

a = 17 units

b = 19 units

h = 8 units

We know that, according to trapezoid formula,

Area of a trapezoid = h(a + b) / 2

= 8 (15 + 19) / 2

= 4 × 34

= 136 units²

Answer: Thus, the area of the trapezoid is 136 units²

Example 3: If the area of the trapezoid is 120 inches and the lengths of the bases are 12 inches and 20 inches, find the height of the trapezoid using the trapezoid formula?

Solution:

Let us assume that the bases are a and b, and the height of the trapezoid is h. Using the given information, we have to find height which is the distance between the bases. Let us substitute all these values in the area of a trapezoid formula:

Area of trapezoid formula, A = [(a + b)/2] × h

120 = [(20 + 12)/2] × h

120 = 16 × h

h = 7.5 inches

Therefore, the height of the trapezoid is 7.5 inches.

Answer: The height of the trapezoid is 7.5 inches.

What is Trapezoid Formula?

We will be learning the following Trapezoid Formula

Perimeter of a Trapezoid
Area of a Trapezoid

Formula to Calculate Perimeter of a Trapezoid

The perimeter of a trapezoid is defined as the sum of all its sides or the complete boundary of the trapezoid. Consider a trapezoid ABCD as shown below with side measures a,b,c, and d. Let’s look into the Trapezoid Formula

The perimeter of the trapezoid formula is calculated by finding the sum of all the sides i.e, AB + BC + CD + DA

Perimeter of a Trapezoid = Sum of all the sides = a + b + c + d

where, a, b,c, and d are the sides of the trapezoid.

FAQs on Area of Trapezoid

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.

What Is Perimeter of Trapezoid Formula?

The perimeter of a Trapezoid is Sum of all the sides. It is expressed as P = a + b + c + d. Where, a, b,c, and d are the sides of the trapezoid.

What Is the Area of Trapezoid Formula?

The area of a trapezoid formula is expressed as, A = (1/2) × h × (a + b). Where ‘a’ is the shorter base, ‘b’ is the longer base, and ‘h’ is the distance between the two bases

How To Calculate Height of Trapezoid Using Trapezoid Formula?

Area of trapezoid formula, A = [(a + b)/2] × h
To calculate the height of the trapezoid we can mold the area of the trapezoid formula as
h = 2A/(a+b). Where ‘a’ is the shorter base, ‘b’ is the longer base, and ‘h’ is the distance between the two bases, and A is the area of a trapezoid.

What Are the Two Basic Trapezoids Formulas?

The two basic trapezoid formulas are:
The perimeter of a Trapezoid is Sum of all the sides. Expressed as P = a + b + c + d. Where, a, b,c, and d are the sides of the trapezoid.
Area of trapezoid formula, A = [(a + b)/2] × h.

Area of a Trapezoid