Exploring the Centroid and Area of Trapezoids: Formulas and Concepts
Comprehensive Definition, Description, Examples & Rules
Introduction
The trapezoids are essential subjects of study in mathematics and geometrical calculations as they help the students understand the fundamental concepts of geometry, such as angles and parallel lines, and they directly relate to quadrilaterals.
Importance in Geometry and Practical Application
The primary significance of a trapezoid includes:
- You can also effectively understand the properties like diagonals and bases while following the trapezoid.
- Engineers need to understand the trapezoid shape very well, as in engineering, the trapezoid has a significant role, and if the building has a particular shape of a trapezoid, then understanding it is crucial for civil engineers.
Definition of a Trapezoid
The trapezoid is a quadrilateral with two parallel and two non-parallel sides. A trapezoid is also known as a trapezium and is a four-sided close shape that covers a particular area and has its perimeter.
Characteristics
The primary characteristics of the trapezoid are:
- The top and the bottom bases of the trapezoid are parallel to each other.
- Angles that are next to each other have a sum of 180 degrees.
- The total sum of the trapezoid is 360 degrees.
- The opposite sides of the trapezoids are of the same length in the case of an isosceles trapezoid.
- The median of a trapezoid is always parallel to both the bases of the trapezoid.
There is a basic difference between the other quadrilaterals and a trapezoid, as a quadrilateral is any four-sided shape, and one of the most recognized four-sided shapes is a square, rectangle, parallelogram, or any different non-recognized shape.
- A rectangle has two equal sides opposite each other and is highly different from a trapezoid.
- A square has all equal sides, which is also not equal to the trapezoid.
- The trapezoid has many similarities with a parallelogram, as its opposite sides are parallel and equal in length. In contrast, the trapezoid has opposite sides parallel, but they are not equal in length.
Centroid of a Trapezoid
The centroid of a trapezoid is the center of the trapezoid and is represented in the form of coordinates of the center point. It can be represented through a vertical or a horizontal line through the central point that divides the center into two equal parts.
The step-by-step centroid of a trapezoid formula is:
- G = [ h/2, {(b + 2a)/ 3 (a + b)}× h ]
Here,
h is the distance among the parallel sides of the trapezoid.
a and b, the exact length of the parallel sides of the trapezoid.
Example:
Find the centroid of a trapezoid with a height of 10 cm and parallel sides lengths of 8 cm and 12 cm.
Solution:
Using the formula: G = [ h/2, {(b + 2a)/ 3 (a + b)}× h ]
Here,
h is 10 cm
a is 8 cm
b is 12 cm
G = [ 10/2, {(12 + 2×8)/ 3 (8 + 12)}× 10 ]
= (5,4.67)
The coordinates for the center of the trapezoid are (5, 4.67)
or the center is 4.67 cm from the side with a length of 8 cm.
Centroid of a Trapezium
A trapezoid is also known as a trapezium; there is no difference between a trapezoid and a trapezium, and it is an entirely similar thing. For various reasons, a trapezoid is known to be a trapezium, and people define the shape by that name. A trapezium is a similar shape with four sides, and the opposite bases of the sides are parallel to each other.
As a trapezoid and trapezium are similar, there is no difference between the formula to calculate the centroid of trapezium as it will be:
- G = [ h/2, {(b + 2a)/ 3 (a + b)}× h ]
Where,
h = the distance among the parallel sides of the trapezium.
a and b = the length of the parallel sides of the trapezium.
Example:
Find the centroid of a trapezoid with a height of 12 cm and parallel sides lengths of 5 cm and 4 cm.
Solution:
Using the formula: G = [ h/2, {(b + 2a)/ 3 (a + b)}× h ]
Here,
h is 12 cm
a is 5 cm
b is 4 cm
G = [ 12/2, {(4 + 2×5)/ 3 (5 + 4)}× 12 ]
= (6,6.22)
The coordinates for the center of the trapezoid are (6,6.22)
or the center is 6.22 cm from the side with a length of 5 cm.
Centroid Formula
The centroid formula that you will have to follow for all types of different geometric shapes is:
Triangle
The triangle is one of the most common two-dimensional shapes, and there is a particular formula that you have to use to find the centroid of a triangle through the intersection of all its medians. The formula is:
- ((x1+x2+x3)/3, (y1+y2+y3)/3)
Square
It is the easiest shape you can calculate because all four sides of the square are equal, so the square’s diagonals are very easy to calculate. There is no particular formula to find the centroid of a square because the point at which the square’s diagonals intersect is known to be the centroid of the square.
Rectangle
The calculation of the centroid of a rectangle is also similar, but calculating the center point is a bit different. The exact point where the rectangle’s diagonals meet is the rectangle’s center, but you need to find the width from the x-axis and the height from the y-axis.
Visual Representation
Here in this blog, we are focusing on the trapezoid, and we will follow how the centroid of the trapezoid is calculated through a visual representation.
Find the centroid of a trapezoid with a height of 8 cm and parallel sides lengths of 8 cm and 10 cm.
Solution:
Using the formula: G = [ h/2, {(b + 2a)/ 3 (a + b)}× h ]
Here,
h is 8 cm
a is 8 cm
b is 10 cm
G = [ 8/2, {(10 + 2×8)/ 3 (8 + 10)}× 8]
= (4,3.85)
The coordinates for the center of the trapezoid are (6,3.85)
or the center is 3.85 cm from the side with a length of 8 cm.
Area of Trapezoid when all Sides are Given
The area of trapezium when all sides are given is calculated by measuring the average of all the parallel sides and then multiplying it by its height. If all sides of the trapezoid are provided, then it is very easy for you to calculate the area of a trapezoid. One of the primary requirements for calculating the area of a trapezoid is the height of the trapezoid. If the height of the trapezoid is not provided, then you need to divide the trapezoid into a triangle and then calculate the height with the Pythagoras.
The formula that you should use for calculating the area of a trapezoid is:
- A = {(a + b)/2}× h
Example:
Find the area of a trapezoid with sides 12 cm, 8 cm, 10 cm, and 8cm.
Parallel sides are 12 cm and 10 cm
Assuming 12 cm is the base
Dividing the trapezoid into a triangle:
So, 10cm + x + x = 12cm
So, x = 1 cm
Now applying the Pythagoras theorem:
8² = 1² + h²
h² = 64 – 1
h = √63 = 7.94
Using the formula: A = {(a + b)/2}× h
A = {(12 + 10)/2}× 7.94
= 87.34
So, the area of the trapezoid is 87.34 cm²
Trapezoid Formula
There are various formulas that you can use for your trapezoid, and the formulas are:
Area
For calculating the area of a trapezoid you will use the formula:
- A = {(a + b)/2}× h
Perimeter
The formula can calculate the perimeter of the trapezoid:
- a + b + c + d
This is basically when you add up all the trapezoid’s sides and get the perimeter.
Centroid
To calculate the centroid of the trapezoid you have to use the formula:
- G = [ h/2, {(b + 2a)/ 3 (a + b)}× h ]
Suppose you want to use these formulas effectively. In that case, the primary lookout for you is to calculate the height of the trapezoid, mainly while calculating the area and centroid of the trapezoid. If you calculate the height, it will be easy to make further calculations, and understanding the Pythagoras Theorem is a primary part for you.
The real-world scenario for using the trapezoid formula is:
- Civil engineers primarily can use the trapezoid formula and do effective calculations while building.
Art and design use a lot of sectors of the trapezoid formula.
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Key Takeaways
- A trapezoid or a trapezium is the same thing with opposite sides of the shape parallel to each other.
- The central point of the trapezoid is the centroid of the trapezoid.
- Civil engineers and art designers use the trapezoid very effectively.
Quiz
Question comes here
Frequently Asked Questions
The centroid of the trapezoid is the central point of the trapezoid when all the medians intersect. You might have to determine the main point of the trapezoid.
The common formulas that you can use for the trapezoid are:
- Area: {(a + b)/2}× h\
- Perimeter: a + b + c + d
- Centroid: [ h/2, {(b + 2a)/ 3 (a + b)}× h ]
Yes, there are real-life scenarios where you can use the trapezoid:
- Civil Engineering
- Art and Design
- Mathematics and Geometry
To learn about geometry and geometric formulas, you can quickly check our Edulyte maths page, as we have effective details about geometry and geometric formulas that will help you use them in your daily life.
