Centroid of a Trapezoid Formula

A trapezoid is a type of quadrilateral with two parallel sides. A quadrilateral is a type of polygon with four sides. The sum of 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, 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:

Centroid of a figure is given by:

C = [X_C, Y_C]

Where,

X_C is the point on the x-axis

Y_C 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:

X_C = h/2

Y_C = [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.