Centroid of a Trapezoid Formula (original) (raw)

Last Updated : 23 Jul, 2025

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.

The **centroid of a trapezoid is its center point, represented by coordinates. It lies along the vertical line connecting the midpoints of the bases. For a height h, the vertical centroid position is **not always h/2—it depends on the lengths of the bases. The centroid divides the trapezoid into two equal-area parts.

The diagram of a trapezoid is shown below:

centroid_of_a_trapezoid

Centroid of a Trapezoid

Where,

Cx is the point on the x-axis,
Cy is the point on the y-axis.

The formula to calculate the centroid of a trapezoid is given by:

G = [h/2, \frac{(2a + b)h}{3(a + b)}]

Where,

h is the height
p is the base
q is the opposite parallel side
The first coordinate, h/2, gives the horizontal center (assuming the bases are aligned horizontally).
The second coordinate, [(2a + b).h / 3(a+b)] gives the vertical distance of the centroid from the base of length b

**Example: Let's try to find the centroid of a trapezoid with dimensions: p = 14, q = 6, h = 7

**Solution:

We know, p = 14, q = 6, h = 7

Applying the centroid of trapezoid formula,
Comparing the values, we get:
h/2 = 7/2
G = [h/2, \frac{(2a + b)h}{3(a + b)}]
G=\frac{(2(14) +6)7}{3(14 + 6)} = \frac{(28+6)7}{3.20}= \frac{17.7}{3.20}=\frac{119}{30}=3.97

Therefore, the centroid of the trapezoid is at a distance of 3.97.

Solved Questions on Centroid of a Trapezoid

**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:

G = [h/2, \frac{(2a + b)h}{3(a + b)}]

Where,
h = height, p and q are the two parallel sides
h = 4m, p = 5m, q = 3m
C = [4/2,\frac{(2\times5+ 3)4}{3(5 + 3)}= \frac{(10+ 3)4}{3(5 + 3)}=\frac{52}{24}=2.17]
C = [2, 2.17]

Thus, the centroid is [2, 2.17] or 2m with respect to the x-axis and 2.17 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:

G = [h/2, \frac{(2a + b)h}{3(a + b)}]

Where, h = height p and q are the two parallel sides
h = 2cm, p = 6cm, q = 4cm
G = [2/2, \frac{(2\times6+ 4)2}{3(6 + 4)}= \frac{(12+ 4)2}{3(10)}=\frac{16.2}{3.10}=\frac{16}{15}=1.07 cm]
G = [1, 0.1.07 cm]

Thus, the centroid is [1, 1.07] cm 1cm with respect to the x-axis and 1.07cm 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:

G = [h/2, \frac{(2a + b)h}{3(a + b)}]

Where, h = height p and q are the two parallel sides
h = 10m, p = 7m, q = 4m
G = [10/2, \frac{(2\times7+ 4)10}{3(7 + 4)}= \frac{(14+ 4)10}{3(11)}=\frac{18.10}{3.11}=\frac{180}{33}= 5.45 m]
G = [5, 5.45]

Thus, the centroid is [5, 5.45] or 5m with respect to the x-axis and 5.45 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:

G = [h/2, \frac{(2a + b)h}{3(a + b)}]

Where, h = height p and q are the two parallel sides
h = 11cm, p = 2cm, q = 3cm
G = [11/2, \frac{(2\times3+ 2)11}{3(3 + 2)}= \frac{(6+ 2)11}{3(5)}=\frac{11.8}{3.5}=\frac{88}{15}= 5.87 cm]
G = [5.5, 5.87]

Thus, the centroid is [5.5, 5.87] or 5.5cm with respect to x-axis and 5.87 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:

G = [h/2, \frac{(2a + b)h}{3(a + b)}]

Where, h = height p and q are the two parallel sides .
h = 8m, p = 5m, q = 3m
G = [8/2, \frac{(2\times5+ 3)8}{3(5 + 3)}= \frac{(10+ 3)8}{3(8)}=\frac{8.13}{3.8}=\frac{104}{24}= 4.33 m]
G = [4, 4.33 ]

Thus, the centroid is [4, 4.33 ] or 4m with respect to the x-axis and 4.33 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:
G = [h/2, \frac{(2a + b)h}{3(a + b)}]
Where, h = height p and q are the two parallel sides .
h = 5m, p = 7m, q = 9m
G = [5/2, \frac{(2\times7+ 9)5}{3(7 + 9)}= \frac{(14+ 9)5}{3(16)}=\frac{5.23}{3.16}=\frac{115}{48}= 2.40m
G = [2.5, 2.40]

Thus, the centroid is [2.5, 2.40] or 2.5m with respect to the x-axis and 2.40m 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:
G = [h/2, \frac{(2a + b)h}{3(a + b)}]
Where, h = height p and q are the two parallel sides .
h = 20cm, p = 15cm, q = 12cm
G = [20/2, \frac{(2\times15+ 12)20}{3(15 + 12)}= \frac{(30+ 12)20}{3(27)}=\frac{20.42}{3.27}=\frac{840}{81}= 10.37m]
G = [10, 10.37]

Thus, the centroid is [10, 10.37] or 10cm with respect to the x-axis and 10.37 cm with respect to the y-axis.

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 various applications such as engineering, design, and architectural planning. The centroid formula helps in simplifying complex geometric calculations and provides the geometric center that can be utilized for further analysis.