Table of Contents
Area of Trapezoid Formula
Introduction:
A trapezoid is a quadrilateral with one pair of parallel sides. It differs from other quadrilaterals because it has only one pair of parallel sides, while the other sides are non-parallel. The area of a trapezoid represents the amount of space enclosed by its four sides and can be found using this simple formula.
Formula for the Area of Trapezoid:
A trapezoid is a quadrilateral with one pair of parallel sides. The formula for the area (A) of a trapezoid is equal to half the sum of the lengths of its parallel sides (base1 and base2) multiplied by the height (h).
Mathematically, the formula can be written as:
A = (1/2) x (base1 + base2) x h
Derivation of Area of Trapezium Formula:
To understand how the area formula of trapezium is derived, we can break down the trapezoid into a rectangle and two triangles. The height of the trapezoid is the perpendicular distance between the parallel sides, and it determines how tall the trapezoid is.
The rectangle is formed by extending the shorter base (base1) to the longer base (base2). Its width is equal to the height of the trapezoid. Thus, the area of the rectangle is base1 x h.
Next, we have two triangles. Each triangle is formed by one of the bases and the height of the trapezoid. The area of a triangle is calculated as half the base multiplied by the height. Therefore, the combined area of both triangles is (1/2) x base2 x h.
Adding the area of the rectangle and the two triangles together, we get the total area of the trapezoid:
A = (base1 x h) + (1/2 x base2 x h) = (1/2) x (base1 + base2) x h
This formula allows us to calculate the area of a trapezoid by knowing the lengths of its parallel sides and the height. It is important to ensure that the bases are parallel for the formula to be valid.
Solved Examples on Area of Trapezoid Formula:
Example 1: A garden in the shape of a trapezoid has a longer base of 12 meters, a shorter base of 8 meters, and a height of 5 meters. What is the area of the garden?
Solution:
To find the area of the trapezoid, we use the formula:
Area = (a + b) x h / 2
Substituting the given values into the formula, we have:
Area = (12 + 8) x 5 / 2
Area = 20 x 5 / 2
Area = 100 / 2
Area = 50 square meters
Therefore, the area of the garden is 50 square meters.
Example 2: A road sign in the shape of a trapezoid has a bottom base of 6 feet, a top base of 4 feet, and a height of 3 feet. What is the area of the road sign?
Solution:
Using the formula for the area of a trapezoid:
Area = (a + b) x h / 2
Plugging in the given values, we get:
Area = (6 + 4) x 3 / 2
Area = 10 x 3 / 2
Area = 30 / 2
Area = 15 square feet
Therefore, the area of the road sign is 15 square feet.
Example 3: The area of a trapezoid is 72 square inches. The length of base1 is 10 inches, and the height is 8 inches. Find the length of base2.
Solution:
Using the trapezoid formula: A = (1/2) x (base1 + base2) x height
Substituting the given values: 72 = (1/2) x (10 + base2) x 8
Simplifying: 72 = 4 x (10 + base2)
Dividing both sides by 4: 18 = 10 + base2
Subtracting 10 from both sides: base2 = 8 inches
Therefore, the length of base2 is 8 inches.
Frequently Asked Questions on Area of Trapezoid Formula:
1: What is the formula for finding the area of a trapezoid?
Answer: The formula for finding the area of a trapezoid is (1/2) multiplied by the sum of the lengths of its parallel sides, known as the bases, multiplied by the height of the trapezoid. Mathematically, it can be expressed as Area = (1/2) x (base1 + base2) x height. The formula is derived from the concept that the area of a trapezoid can be calculated by dividing it into a rectangle and two triangles. The height represents the perpendicular distance between the parallel bases. By taking half of the sum of the bases and multiplying it by the height, we obtain the area of the trapezoid.
2: Can a trapezoid be a square?
Answer: The four attributes or characteristics of a trapezoid are:
- Bases: A trapezoid has two parallel sides, referred to as the bases. These bases can have different lengths.
- Legs: The legs of a trapezoid are the non-parallel sides. They connect the ends of the bases and can have different lengths.
- Height: The height of a trapezoid is the perpendicular distance between the two bases. It measures the shortest distance between the bases.
- Angles: A trapezoid has four angles. The two angles formed by each leg with the adjacent base are known as the base angles, while the two opposite angles are called the non-base angles or the top angles.
These attributes define the unique characteristics of a trapezoid and help in its identification and analysis.
3: Can a trapezoid have perpendicular diagonals?
Answer: Trapezoids do not necessarily have perpendicular diagonals. A trapezoid, however, can be drawn and oriented in such a way it has perpendicular diagonals.
4: Are the diagonals of a trapezoid equal?
Answer: No, the diagonals of a trapezoid are not necessarily equal in length. A trapezoid is a quadrilateral with one pair of parallel sides. The diagonals of a trapezoid are the line segments connecting non-adjacent vertices. In a trapezoid, the diagonals are usually of different lengths, except in the special case of an isosceles trapezoid. In an isosceles trapezoid, the diagonals are equal, as the non-parallel sides are congruent. However, in a general trapezoid, the diagonals can have different lengths, depending on the specific dimensions and shape of the trapezoid.
5: Can the trapezoid formula be used if the bases are not parallel?
Answer: No, the trapezoid formula is only applicable when the trapezoid has one pair of parallel sides. If the bases are not parallel, a different formula or approach is required to calculate the area.
6: What do you need to know to find the area of a trapezoid?
Answer: To find the area of a trapezoid, you need to know the lengths of the parallel sides (usually referred to as the base and top) and the perpendicular height between them. The base and top lengths are typically denoted as ‘b’ and ‘a,’ respectively, while the height is represented as ‘h.’ With these measurements, you can use the formula: A = (1/2) x (a + b) x h to calculate the area of the trapezoid.
7: How do you find the missing height of a trapezoid?
Answer: To find the missing height of a trapezoid, you can use the formula for the area of a trapezoid and rearrange it to solve for the height. The formula for the area of a trapezoid is A = (1/2) x (a + b) x h, where ‘a’ and ‘b’ are the lengths of the parallel sides, and ‘h’ is the height. Rearranging the formula gives us h = (2A) / (a + b), where ‘A’ is the known area of the trapezoid. By substituting the known values into the equation, you can find the missing height.
8: How do you find the area of a trapezoid with diagonals?
Answer: To find the area of a trapezoid when the diagonals are given, you can use the formula:
Area = (1/2) x d1 x d2
Where ‘d1’ and ‘d2’ are the lengths of the diagonals of the trapezoid. Simply substitute the values of the diagonals into the formula to calculate the area. Note that the diagonals must be within the trapezoid and intersect at a common point inside the trapezoid for this formula to be applicable.
