Centroid of a Trapezoid Formula
Last Updated :
28 Aug, 2024
A trapezoid is a type of quadrilateral with two parallel sides. A quadrilateral is a type of polygon with four sides. The sum of the four internal angles of a quadrilateral is 360°. Centroid refers to the center point of any figure. It is also known as the geometric center. In a 2D coordinate system, the centroid is always measured with respect to the x-axis and y-axis. In a trapezoid, the centroid is also measured with respect to these two axes.
Centroid of a Trapezoid Formula
A centroid point is the center point of the trapezoid. Centroid is represented in the form of coordinates of the center point of a trapezoid. A horizontal or vertical line through the centroid point divides it into two equal parts. It means, that for the height ‘h,’ a center point will be ‘h/2.’ Similarly, the other center point is calculated. These two points form the centroid.
The diagram of a trapezoid is shown below:
The Centroid of a figure is given by:
C = [XC, YC]
Where,
XC is the point on the x-axis
YC is the point on the y-axis
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2, [Tex]\frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h is the height
p is the base
q is the opposite parallel side
Comparing the values, we get:
XC = h/2
YC = [Tex]\frac{(2q + p)h}{3(p + q)}[/Tex]
Sample Problems
Question 1: Find the centroid of a trapezoid with a height of 4m and two parallel sides of 5m and 3m.
Solution:
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2, [Tex]\frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h = height
p and q are the two parallel sides
h = 4m
p = 5m
q = 3m
C = [4/2,[Tex] \frac{(2\times3+ 5)4}{3(5 + 3)}[/Tex]]
C = [2, 44/24]
C = [2, 1.83]
Thus, the centroid is [2, 1.83] or 2m with respect to the x-axis and 1.83 m with respect to the y-axis.
Question 2: Find the centroid of a trapezoid with a height of 2cm and two parallel sides of 6cm and 4cm.
Solution:
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2,[Tex] \frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h = height
p and q are the two parallel sides
h = 2cm
p = 6cm
q = 4cm
C = [2/2, [Tex]\frac{(2 \times 4 + 6)2}{3(6 + 4)}[/Tex]]
C = [1, 28/30]
C = [1, 0.93]
Thus, the centroid is [1, 0.93] or 1cm with respect to the x-axis and 0.93cm with respect to the y-axis.
Question 3: Find the centroid of a trapezoid with a height of 10m and two parallel sides of 7m and 4m.
Solution:
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2, [Tex]\frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h = height
p and q are the two parallel sides
h = 10m
p = 7m
q = 4m
C = [10/2, [Tex]\frac{(2 \times 4 + 7)10}{3(7 + 4)}[/Tex]]
C = [5, 150/33]
C = [5, 4.54]
Thus, the centroid is [5, 4.54] or 5m with respect to the x-axis and 4.54 m with respect to the y-axis.
Question 4: Find the centroid of a trapezoid with a height of 11cm and two parallel sides of 3cm and 2cm.
Solution:
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2,[Tex] \frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h = height
p and q are the two parallel sides
h = 1cm
p = 2cm
q = 3cm
C = [1/2, [Tex]\frac{(2 \times 3 + 2)1}{3(2 + 3)}[/Tex]]
C = [0.5, 8/15]
C = [0.5, 0.53]
Thus, the centroid is [0.5, 0.53] or 0.5cm with respect to x-axis and 0.53 cm with respect to y-axis.
Question 5: Find the centroid of a trapezoid with a height of 8m and two parallel sides of 5m and 3m.
Solution:
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2,[Tex] \frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h = height
p and q are the two parallel sides
h = 8m
p = 5m
q = 3m
C = [8/2, [Tex]\frac{(2 \times 3 + 5)8}{3(5 + 3)}[/Tex]]
C = [4, 88/24]
C = [4, 3.66]
Thus, the centroid is [4, 3.66] or 4m with respect to the x-axis and 3.66 m with respect to the y-axis.
Question 6: Find the centroid of a trapezoid with a height of 5m and two parallel sides of 7m and 9m.
Solution:
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2, [Tex]\frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h = height
p and q are the two parallel sides
h = 5m
p = 7m
q = 9m
C = [5/2, [Tex]\frac{(2 \times 9 + 7)5}{3(7 + 9)}[/Tex]]
C = [2.5, 125/48]
C = [2.5, 2.60]
Thus, the centroid is [2.5, 2.60] or 2.5m with respect to the x-axis and 2.60m with respect to the y-axis.
Question 7: Find the centroid of a trapezoid with a height of 20cm and two parallel sides of 15cm and 12cm.
Solution:
The formula to calculate the centroid of a trapezoid is given by:
C = [h/2, [Tex]\frac{(2q + p)h}{3(p + q)}[/Tex]]
Where,
h = height
p and q are the two parallel sides
h = 20cm
p = 15cm
q = 12cm
C = [20/2, [Tex]\frac{(2 \times 12 + 15)20}{3(15 + 12)}[/Tex]]
C = [10, 780/81]
C = [10, 9.62]
Thus, the centroid is [10, 9.62] or 10cm with respect to the x-axis and 9.62 cm with respect to the y-axis.
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Conclusion
The centroid of a trapezoid is a crucial point that represents the center of the mass or balance point of the trapezoid. It can be found using the straightforward formula based on the lengths of the bases and the height of the trapezoid. By understanding and applying this formula one can efficiently determine the centroid’s location which is essential in the various applications such as the engineering, design and architectural planning. The centroid formula helps in the simplifying complex geometric calculations and provides the geometric center that can be utilized for further analysis.
FAQs on Centroid of a Trapezoid Formula
What is the significance of the centroid in a trapezoid?
The centroid is significant as it represents the point where the trapezoid’s area is equally distributed. It is used in the various fields such as physics, engineering and architecture to the determine balance and stability.
Can the centroid of a trapezoid be outside the trapezoid?
No, the centroid of a trapezoid always lies within the boundaries of the trapezoid. It is the average location of the all points within the trapezoid’s shape.
How does the length of the bases affect the centroid’s position?
The centroid’s position is influenced by the lengths of the bases with centroid being closer to the longer base. The formula accounts for this by the weighting the longer base more heavily in the calculation.
Is the centroid formula applicable to all types of trapezoids?
Yes, the centroid formula applies to the all types of the trapezoids including the isosceles and right trapezoids as long as the trapezoid is defined by the two parallel sides and height.