The area of a Trapezoid Formula is the number of unit squares that fit in the trapezoid, measured in square units (cm2, m2, in2, etc.). For example, if 15 unit squares of length 1 cm fit in a trapezoid, its area is 15 cm2. A Trapezoid Formula is a type of quadrilateral with a pair of parallel sides (called bases). This means that the other pair of sides are not parallel (called legs). It is not always possible to draw a unit square and measure the area of a trapezoid. Learners can learn about the formula for calculating the area of a trapezoid on the website of Extramarks.
What Is The Area Of Trapezoid?
The area of aTrapezoid Formula is the total space covered by its sides. An interesting point to note here is that if students know the lengths of all the sides, they can simply divide the Trapezoid Formula into smaller polygons, such as triangles or rectangles, find their areas, and sum them up to get the area of the trapezoid. However, there is a straightforward formula used to find the area of a Trapezoid Formula if students know the specific dimensions.
Area of Trapezoid Formula: The area of a trapezoid can be calculated from the lengths of its parallel sides and the distance (height) between them. The formula for the area of a trapezoid can be written as:
A = ½ (a + b) h
where (A) is the area of the trapezoid, (a) and (b) are the bases (parallel sides), and (h) is the height (perpendicular distance between a and b).
Area Of Trapezoid Formula
In geometry, students have objects of various shapes and sizes. Such a shape is a trapezoid. It has a slightly different shape compared to others. Since it has four sides, it belongs to the quadrilateral. A trapezoid is a two-dimensional geometric figure with four sides and at least one pair of opposite parallel sides. This article explains the trapezoidal area formula with an example.
A trapezoid is a special kind of quadrilateral, meaning it has four sides. Of these, a pair of sides are parallel but unequal in length. These two parallel sides are called the base. The other side is called the leg. The other two sides are not parallel, but may or may not be of equal length.
To make a trapezoid, students need a triangle. Any triangle will work: the right triangle, the obtuse triangle, the isosceles triangle, and the scalene triangle. Then cut the top of the triangle so that the cut plane is parallel to the bottom of the triangle. Now students have small triangles and trapezoids, and there are many types of trapezoids. These include isosceles trapezoids, straight trapezoids, and scalene trapezoids. A trapezoid with two non-parallel sides of equal length is called an isosceles trapezoid. A right-angled trapezoid has at least two right angles. A right-angled isosceles trapezoid is a Trapezoid Formula that has both a right-angled Trapezoid Formula and an isosceles trapezoid.
Area Of Trapezoid Without Height
A Trapezoid Formula or trapezoid, is a quadrilateral with at least one pair of parallel sides, and the parallel sides are called bases. If the other two sides are not parallel, they are called legs or sides. Otherwise, there are two base pairs. Real-life examples where students can see the range of trapezoids are handbags, popcorn cans, and dulcimers like guitars. The area of aTrapezoid Formula is the total space enclosed by its four sides, and there are two ways to find the area of a trapezoid.
The first method is a direct method that uses a direct formula to find the area of a Trapezoid Formula of known dimensions .
The second method first divides the Trapezoid Formula into smaller polygons, such as triangles or rectangles, taking all side lengths into account. Then find the areas of triangles and rectangles separately. Finally, add the area of the polygon to get the total area of the trapezoid. Refer to Extramarks for more information.
How To Derive Area Of Trapezoid Formula?
Formula for the area of a trapezoid:
The area can be calculated using the following simple steps to arrive at the area formula for a trapezoid.
Step-1: Add two parallel bases.
Step-2: Multiply the result of the above step by the height of the trapezoid.
Step-3: Dividing the result of Step-2 by 2.
Step-4: Get the area of the specified trapezoid.
Area Of Trapezoid Calculator
A trapezoidal prism is a 3D figure consisting of two congruent trapezoids connected by four rectangles. The trapezoid is up and down. They therefore form the base of the prism and have polygons forming the base. Four rectangles form the sides of a trapezoidal prism. So the trapezoidal prism is
- six faces
- 8 vertices
- 12 edges
Surface of a trapezoidal prism
The area of a trapezoidal prism is the sum of the areas of the prisms. This area equals the area of all faces of a trapezoidal prism. A trapezoidal prism has two trapezoidal faces and four rectangular faces, so the sum of their areas gives the surface area of the prism. However, there is a simple and straightforward formula for calculating the surface area of a trapezoidal prism. The formula is:
Surface area of trapezoidal prism = h(b + d) + l(a + b + c + d) square units. where h = height
b and d are base lengths
a + b + c + d is the perimeter
l is the side of the trapezoidal prism
Derivation of the surface of a trapezoidal prism
The base of the trapezoidal prism is trapezoidal
Also b and d are parallel sides of the trapezoid
H = distance between parallel sides
l = length of trapezoidal prism
So total area of trapezoidal prism (TSA) = 2 × base area + side area — (1)
Area of Trapezoid Formula = ½ (base 1 + base 2) height
Area of Trapezoid Formula base = h (b + d)/2 —— (2)
Trapezoidal Prism (LSA) Side Area = Sum of Areas of Each Rectangular Face
Therefore, LSA = (a × l) + (b × l) + (c × l) + (d × l) — (3)
Substituting the values of equations (2) and (3) into equation 1, i.e. the H. TSA equation, students get:
(TSA) = 2 × h(b + d)/2 + (a × l) + (b × l) + (c × l) + (d × l)
TSA = h(b + d) + [(a x l) + (b x l) + (c x l) + (d x l)]
Total area of trapezoidal prism = h (b + d) + l (a + b + c + d)
So TSA for trapezoidal prism = h(b + d) + l(a + b + c + d) unit square
Area Of Trapezoid Examples
Example: Find the area of a Trapezoid Formula with parallel sides of 32 cm and 12 cm respectively, and a height of 5 cm.
Solution: The base is given as a = 32 cm b = 12 cm; height h = 5 cm.
Area of Trapezoid Formula = A = ½ (a + b) h
A = ½ (32 + 12) × (5) = ½ (44) × (5) = 110 cm2.
area of Trapezoid Formula without height
If students know all the sides of theTrapezoid Formula but not the height, they can find the area of the trapezoid. In this case, students first need to calculate the height of the trapezoid. Let’s understand this with an example.
Example: How do students find the surface of a trapezoidal prism?
Solution: To find the surface area of a trapezoidal prism:
Step 1: Find the four sides (a, b, c, d) of the trapezoid. These represent the width of the four rectangles. Adding these four values gives the circumference P.
Step 2: Find the prism length h.
Step 3: Find the sides of the trapezoidal prism.
Step 4: Identify b1, b2, and h of the trapezoid. Now calculate the base area B using the formula (b1 + b2) h/2.
Step 5: Finally, substitute the value into the formula = 2B + lateral area to calculate the total area of the trapezoidal prism.
Practice Questions On Area Of Trapezoid
1: Find the area of a Trapezoid Formula whose bases (parallel sides) are 6 and 14 units, respectively, and whose non-parallel sides (legs) are 5 units each.
Solution: Try calculating the area of a Trapezoid Formula using the following steps.
Step 1: Area of Trapezoid Formula = ½ (a + b) h; h = height of trapezoid. Not specified in this case a = 6 units, b = 14 units, and non-parallel sides (legs) = 5 units each.
Step 2: Knowing the height of the trapezoid, students can calculate the area. Plotting the height of the Trapezoid Formula on both sides shows that the Trapezoid Formula is divided into a rectangle ABQP and two right triangles ADP and BQC.
Step 3: Because the rectangles have equal opposite sides, AP = BQ, and the sides are AD = BC = 5 units. As a result, the Pythagorean theorem can be used to calculate the heights AP and BQ.
Step 4: Find the length of DP and QC. ABQP is a rectangle, so AB = PQ and DC = 14 units. This means that PQ = 6 units, and the remaining combined length of DP + QC can be calculated as DC – PQ = 14 – 6 = 8. 8 ÷ 2 = 4 units. So DP = QC = 4 units.
Step 5: Now students can use the Pythagorean theorem to calculate the height of the trapezoid. Taking the right triangle ADP, students know that AD = 5 units and DP = 4 units, so AP = √(AD2 – DP2) = √(52 – 42) = √(25 – 16) = √9 = 3 units . ABQP is a rectangle with opposite sides of equal length, so AP = BQ = 3 units.
Step 6: Now that students know all the dimensions of the trapezoid, including its height, they can calculate its area using the following formula: Area of Trapezoid Formula = ½ (a + b) h; where h = 3 units, a = 6 units, b = 14 units. Substituting the values into the formula, students get the area of the trapezoid = ½ (a + b) h = ½ (6 + 14) × 3 = ½ × 20 × 3 = 30 unit2.
2: If one of the bases of a Trapezoid Formula is 8 units, the height is 12 units, and the area is 108 square units, find the length of the other base.
Solution: One of the bases ‘a’ = 8 units.
Let the other base will be ‘b’.
The area of the Trapezoid Formula will be A = 108 square units.
Its height will be ‘h’ = 12 units. Substituting all these values into the trapezoidal field, students get
A = ½ (a + b) h
108 = ½ (8 + b) × (12)
108 = 6 (8 + b)
Divide both sides by 6,
18 = 8 + b
b = 10
3: Find the area of an isosceles Trapezoid Formula with legs 8 units long and bases 13 and 17 units, respectively.
Solution: The base is a = 13 units and b = 17 units. Let h be its height.
A given Trapezoid Formula can be split into two congruent right triangles and a rectangle as follows.
Example of the area of a Trapezoid Formula with sides and no height
From the above figure,
x + x + 13 = 17
2x + 13 = 17
2x = 4
x = 2
Using the Pythagorean theorem,
x2 + h.h = 82
22 + h.h = 64
4 + h.h = 64
h = 2√15
The area of a givenTrapezoid Formula
A = ½ (a + b) h
A = ½ (13 + 17) × (2√15) = 30√15 = 116.18 square units
Answer: Area of given trapezoid = 116.18 square units. Example 3: Find the area of a trapezoid with bases of 7 and 9 units and a height of 5 units.
Solution: Area of trapezoid = ½(a + b)h; where a = 7, b = 9, h = 5.
Substituting these values into the formula gives:
A = ½ (a + b) h
A = ½ (7 + 9) × 5
A = ½ × 16 × 5 = 40 units 2
Therefore, the area of the trapezoid will be 40 square units.
FAQs (Frequently Asked Questions)
1. How will be the area of trapezoidal calculated?
The area of a trapezoid is the number of unit squares that fit inside it. Trapezoidal Area Calculator is an online tool that can be used to find the area of a trapezoid. If certain parameters, such as base value and height, are available, students can enter the inputs directly to calculate the area. Try Extramarks Trapezoid Area Calculator to calculate the area of a trapezoid in seconds. For more practice, look at the Trapezoid Worksheet section and use the calculator to solve the problem.
2. How to find the area of a trapezoidal formula?
Here, students can use triangles to prove the area of the trapezoidal formula. Taking a trapezoid with bases ‘a’ and ‘b’ and height ‘h’ proves the formula.
Step 1: Divide one of the legs into two equal parts and cut out the trapezoidal triangular part as shown.
Step 3: Install in a large triangle as shown below.
Step 4: In this way, the trapezoid is transformed into a triangle. Students can see that the areas of the trapezoid and the new large triangle remain the same even after being attached in this way. Students also know that the base of the new, larger triangle is (a + b) and the height of the triangle is h.
Step 5: Students can now state that the area of the trapezoid equals the area of the triangle.
Step 6: This can be written as area of trapezoid = ½ × base × height = ½ (a + b) h.
The formula for finding the area of a trapezoid has now been proven.