How to Calculate the Centroid of a Trapezoid with Examples
In this article, students will be able to learn about the topic of the centroid of a trapezoid. We will also look at the centroid of the trapezoid formula. But before we learn how to find the centroid of a trapezoid, students need to focus on the basics and start from the beginning.
The first thing that one needs to learn is the definition of a trapezoid. A trapezoid can be defined as a quadrilateral in which there are two parallel sides. A trapezoid is also known as a trapezium. So, if you see trapezium written in some other book, then don’t be confused. It means the same thing as a trapezoid.
A trapezoid can also be defined as a four-sided figure that is closed. It also covers some areas and has its perimeter. We will learn the formula for both area and perimeter of a trapezoid at a later point in this article.
It should be noted that a trapezoid is a two-dimensional figure and not a three-dimensional figure. The sides that are parallel to one another are known as the bases of the trapezoid. On the other hand, the sides that are not parallel to each other are known as lateral sides or legs. The distance between the two parallel sides is also known as the altitude.
Some readers might find it interesting to learn that there is also a disagreement over the exact definition of a trapezoid. There are different schools of mathematics that take up different definitions.
According to one of those schools of mathematics, a trapezoid can only have one pair of parallel sides. Another school of mathematics dictates that a trapezoid can have more than one pair of parallel sides.
This means that if we consider the first school of thought to be true, then a parallelogram is not a trapezoid. But according to the second school of thought, a parallelogram is a trapezoid. There are also different types of trapezoids. And those different types of trapezoids are:
Right Trapezoids
A right trapezoid contains a pair of right angles. We have also attached an image of a right trapezoid below.
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Isosceles Trapezoids
In an isosceles trapezoid, the non-parallel sides of the legs of the trapezoid are equal in length. An image depicting an isosceles trapezoid is attached below.
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Scalene Trapezoids
A scalene trapezoid is a figure in which neither the sides nor the angles of the trapezium are equal. For your better understanding, an image of a scalene trapezoid is attached below.
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The Formula for Area and Perimeter of a Trapezoid
Now, let’s look at the formula for calculating the area and perimeter of a trapezoid. According to experts, the area of a trapezoid can be calculated by taking the average of the two bases and multiplying the answer with the value for the altitude. This means that the formula for the area of a trapezoid can also be depicted by:
Area = ½(a + b) x h
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Moving on to the formula for the perimeter of a trapezoid, it can be described as the simple sum of all the sides. This means that if a trapezoid has four sides like a, b, c, and d, then the formula for the perimeter of a trapezoid can be represented by:
Perimeter = a + b + c + d.
The Properties of a Trapezoid
There are various important properties of a trapezoid. We have discussed those properties in the list that is mentioned below.
The diagonals and base angles of an isosceles trapezoid are equal in length.
If a median is drawn on a trapezoid, then the median will be parallel to the bases. And the length will also be the average of the length of the bases.
The intersection point of the diagonals is collinear to the midpoints of the two opposite sides.
If there is a trapezoid that has sides, including a, b, c, and d, and diagonals p and q, then the following equation stands true.
p2 + q2 = c2 + d2 + 2ab
In the next section, we will look at the centroid of a trapezoid formula.
The Formula for Centroid of a Trapezoid
In this section, we will look at the trapezoid centroid and the centroid formula for the trapezoid. As you must already know, a trapezoid is a quadrilateral that has two sides parallel. The centroid, as the name indicates, lies at the centre of a trapezoid. This means that for any trapezoid that has parallel sides a and b, the trapezoid centroid formula is:
X = {b + 2a / 3 (a + b)} x h
In this formula, h is the height of the trapezoid. Also, a and b are the lengths of the parallel sides.
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FAQs on Centroid of a Trapezoid: Definition, Formula & Method
1. What is the centroid of a trapezoid?
The centroid of a trapezoid is its geometric centre, often referred to as the center of mass or center of gravity for a uniform thin plate. It is the unique point where the trapezoid would perfectly balance if supported on a single point. This point always lies on the median, which is the line segment connecting the midpoints of the two parallel sides.
2. What is the formula used to calculate the centroid of a trapezoid?
To find the centroid's position, you typically calculate its height (y-coordinate) from one of the parallel bases. If a trapezoid with parallel sides 'a' and 'b' and height 'h' is placed with base 'a' on the x-axis, the height of the centroid (y_c) from this base is given by the formula:
y_c = (h/3) * [(2b + a) / (a + b)]
The x-coordinate depends on the trapezoid's placement on the plane. For an isosceles trapezoid symmetric about the y-axis, the x-coordinate is 0.
3. How do you calculate the centroid of a trapezoid with a worked example?
Let's calculate the centroid's height for a trapezoid with parallel sides a = 12 cm (the longer base) and b = 8 cm, and a height h = 6 cm. Assume base 'a' is on the x-axis.
Step 1: Use the centroid formula: y_c = (h/3) * [(2b + a) / (a + b)]
Step 2: Substitute the given values: y_c = (6/3) * [(2*8 + 12) / (12 + 8)]
Step 3: Simplify the expression: y_c = 2 * [(16 + 12) / 20] = 2 * [28 / 20]
Step 4: Calculate the final result: y_c = 2 * 1.4 = 2.8 cm.
Thus, the centroid is located at a height of 2.8 cm from the 12 cm base.
4. Is the centroid formula different for a right trapezoid?
No, the general formula for the height (y-coordinate) of the centroid is the same for all types of trapezoids, including a right trapezoid. However, finding the horizontal position (x-coordinate) can be simpler for a right trapezoid. You can easily decompose it into a rectangle and a triangle, find their individual centroids, and then calculate the weighted average to find the x-coordinate of the overall centroid.
5. How is the formula for the centroid of a trapezoid derived?
The formula for the centroid of a trapezoid is typically derived using one of two methods:
Method of Decomposition: The trapezoid is divided into simpler shapes with known centroids, such as a rectangle and a triangle (or two triangles). The centroid of the composite shape is then found by taking a weighted average of the centroids of the individual parts, where the 'weight' is the area of each part.
Method of Integration: This is a more formal calculus-based approach. The trapezoid is placed on a coordinate plane, and its boundaries are expressed as functions. The centroid's coordinates (x_c, y_c) are then calculated using integral formulas: x_c = (1/A) ∫x dA and y_c = (1/A) ∫y dA, where 'A' is the total area of the trapezoid. This method is precise and fundamental to mechanics.
6. How does the centroid of a trapezoid differ from the intersection of its diagonals?
These are two completely different points within a trapezoid. The centroid is the physical center of mass, located on the line connecting the midpoints of the parallel sides. The intersection of the diagonals is simply the point where the two diagonal lines cross each other. While both these points lie on the same vertical line (the median connecting the midpoints of parallel bases), they are located at different heights and only coincide in the special case where the trapezoid is a parallelogram.
7. What is the real-world importance of finding the centroid of a trapezoid?
Finding a trapezoid's centroid is essential in various practical fields:
Structural Engineering: To ensure the stability of structures with trapezoidal shapes, such as dams, bridge supports, and retaining walls, engineers must calculate the center of mass to balance loads and prevent tipping.
Physics and Mechanics: The centroid is used to predict an object's response to forces. It helps in analysing the rotational motion and equilibrium of trapezoidal components in mechanical systems.
Naval Architecture: The centroid of a ship's cross-section (often trapezoidal) is crucial for determining its stability in water.
