Centroid of a Trapezoid Formula

Consider the following trapezium in which AB || CD. The trapezium OABC is placed such that the origin coincides with one of its vertices. G is its centroid. The lengths of its parallel sides are AB = a and OC = b and its height is h.

The coordinates of the centroid of the trapezium are given by the following formula.

• \[G\left ( \frac{h}{2},\frac{b+2a}{3(a+b)}h \right )\]

Let’s look at an example to see how to use this formula.

Some Solved Questions by Vedantu  

Question: Find the centroid of a trapezium of height 5 cm whose parallel sides are 6 cm and 8 cm.

Solution:

a=6cm,b=8cm,h=5cma=6cm,b=8cm,h=5cm

\[Gy=\frac{b+2a}{3(a+b)}h\]\[=\frac{8+12}{3(8+6)}\times5\]\[=\frac{20\times5}{42}\]=2.38cm

Hence, the centroid is 2.38 cm from the side whose length is 8 cm.

Why don’t you try to solve a problem for practice?

Question: Find the centroid of a trapezium of height 4.5 cm whose parallel sides are 4 cm and 8 cm.

Options:

(a) 2 cm from the side whose length is 4 cm

(b) 2 cm from the side whose length is 8 cm

(c) 3 cm from the side whose length is 4 cm

(d) 3 cm from the side whose length is 8 cm 

Answer: (b)

Solution:

a=4cm,b=8cm,h=4.5cma=4cm,b=8cm,h=4.5cm

\[Gy=\frac{b+2a}{3(a+b)}h\]\[=\frac{8+8}{3(8+4)}\times4.5\]\[=\frac{16\times4.5}{3\times 12}=\frac{72}{36}=2cm\]

Hence, the centroid is 2 cm from the side whose length is 8 cm.

What Exactly is a Centroid? Definition of a Centroid

A centroid, also known as a geometric center, is the center of mass of a uniformly dense object. To make it easier to understand, think of it as the point where you should place the tip of a pin in order to balance your geometric shape on it.

Formula for the Centroid of a Trapezoid

The centroid of a trapezoid formula can be used to determine the position of a trapezoid's centroid. A quadrilateral with two parallel sides is known as a trapezoid. A trapezoid's centroid is located halfway between the two bases. Let's look at a few examples of the centroid of a trapezoid formula.

A trapezoid is a four-sided quadrilateral with two parallel sides. A trapezoid's centroid is located halfway between the two bases. Use the formula below to find any trapezoid with parallel sides a and b.

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Find the formula for the centroid of a trapezoid at a distance of x in the table below.

\[x=\frac{b+2a}{3(a+b)}h\]

Where,

h = Trapezoid height

a, b = Parallel side lengths

Calculator for Centroid

Simply enter the vertices of your shape as Cartesian coordinates to utilize this centroid calculator. Let's look at how to find the trapezoid's centroid:

• Select the sort of form for which you want the centroid to be calculated. In this scenario, we'll go with an N-sided polygon.

• Fill in the N parameter (if required). For our example, we'll need to know how many sides a polygon has. We type 4 into the N box since the trapezoid is a quadrilateral.

• After that, the fields for entering coordinates will display. Enter the vertices of your shape's coordinates. Assume that the vertices of our trapezoid are:

• A = (1,1)

• B = (2,4)

• C = (5,4)

• D = (11,1)

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• The result will be displayed on our centroid calculator! The choice of a trapezoid's centroid is (4.974,2.231).

Example: Find the centroid of the trapezoid with the following dimensions: a = 12′, b = 5′, and h = 5′.

Solution: Given,

a = 12′; b = 5′; h = 5′

Using centroid of trapezoid formula,

x = \[\frac{b + 2a}{3(a + b)} × h\]

x = \[\frac{5 + 2 \times 12}{3(12 + 5)} × 5\]

x = 2.84

As a result, the trapezoid's centroid is at a distance of 2.84′.