Introduction to solve centroid formula:
In general, Centroid formula is a point on a given body or shape at which the entire mass of the body acts (center of gravity of the mass), it might also be the center of area for certain shapes. For a triangle, solving centroid is the point at which the medians of the triangle intersect; they intersect at the ratio 2:1. In the case of polygons the Centroid is found using the boundary co-ordinates solving.
Solving Centroid Formulae:
In this case the Centroid of the triangle is taken and the formula used to find out solving the centroid of a triangle is,
G (x1+x2+x3)/3 , (y1+y2+y3)/3
Where,
(x1, y1)
(x2, y2)
(x3, y3)
are the co-ordinates of the triangle.
In general, for any shape in the x-y plane the Centroid formulae can be generalized to,
G (x1+x2+x3+….+xn)/3n , (y1+y2+y3+….+yn))/n
Where,
(x1, y1)
(x2, y2)
(x3, y3)
(........)
(........)
(xn, yn) are the co-ordinates of the given shape.
Example Problems on Solving Centroid Formula:
1. Calculate the Centroid of a triangle whose co-ordinates are (3, 6) (4, 2) (3, -4)
Sol:
The given points are (3, 6) (4, 2) (3, -4),
Therefore solving,
(x1, y1) is (3, 6)
(x2, y2) is (4, 2)
(x3, y3) is (3, -4)
Formulae for the Centroid of triangle is,
G (x1+x2+x3)/3 , (y1+y2+y3)/3
(3+4+3)/3 , (6+2-4)/3
(10)/3 , (8-4)/3
10/3 , 4/3
3.33 , 1.33
Therefore the Centroid is (3.33, 1.33)
2. Calculate the Centroid of a triangle whose co-ordinates are (4, 8) (3, 2) (5, -4)
Sol:
The given points are (4, 8) (3, 2) (5, -4),
Therefore solving,
(x1, y1) is (4, 8)
(x2, y2) is (3, 2)
(x3, y3) is (5, -4)
Formulae for the Centroid of triangle is,
G (x1+x2+x3)/3 , (y1+y2+y3)/3
(4+3+5)/3 , (8+2-4)/3
(12)/3 , (10-4)/3
12/3 , 6/3
4 , 2
Therefore the Centroid is (4, 2).
3. Calculate the Centroid of the quadrilateral, whose co- ordinates are (3, 2) (5, -4) (4, 2) (3, -4)
Sol:
The given points are (3, 2) (5, -4) (4, 2) (3, -4),
Therefore,
(x1, y1) is (3, 2)
(x2, y2) is (5, -4)
(x3, y3) is (4, 2)
(x4, y4) is (3, -4)
Formulae for the Centroid is
G (x1+x2+x3+….+xn)/3n , (y1+y2+y3+….+yn))/n,
(3+5+4+3)/4 , (2-4+2-4)/4,
(15)/4 , (4-8)/4,
3.75 , -4/4
3.75 , -1
Therefore the Centroid is (3.75, -1)
In general, Centroid formula is a point on a given body or shape at which the entire mass of the body acts (center of gravity of the mass), it might also be the center of area for certain shapes. For a triangle, solving centroid is the point at which the medians of the triangle intersect; they intersect at the ratio 2:1. In the case of polygons the Centroid is found using the boundary co-ordinates solving.
Solving Centroid Formulae:
In this case the Centroid of the triangle is taken and the formula used to find out solving the centroid of a triangle is,
G (x1+x2+x3)/3 , (y1+y2+y3)/3
Where,
(x1, y1)
(x2, y2)
(x3, y3)
are the co-ordinates of the triangle.
In general, for any shape in the x-y plane the Centroid formulae can be generalized to,
G (x1+x2+x3+….+xn)/3n , (y1+y2+y3+….+yn))/n
Where,
(x1, y1)
(x2, y2)
(x3, y3)
(........)
(........)
(xn, yn) are the co-ordinates of the given shape.
Example Problems on Solving Centroid Formula:
1. Calculate the Centroid of a triangle whose co-ordinates are (3, 6) (4, 2) (3, -4)
Sol:
The given points are (3, 6) (4, 2) (3, -4),
Therefore solving,
(x1, y1) is (3, 6)
(x2, y2) is (4, 2)
(x3, y3) is (3, -4)
Formulae for the Centroid of triangle is,
G (x1+x2+x3)/3 , (y1+y2+y3)/3
(3+4+3)/3 , (6+2-4)/3
(10)/3 , (8-4)/3
10/3 , 4/3
3.33 , 1.33
Therefore the Centroid is (3.33, 1.33)
2. Calculate the Centroid of a triangle whose co-ordinates are (4, 8) (3, 2) (5, -4)
Sol:
The given points are (4, 8) (3, 2) (5, -4),
Therefore solving,
(x1, y1) is (4, 8)
(x2, y2) is (3, 2)
(x3, y3) is (5, -4)
Formulae for the Centroid of triangle is,
G (x1+x2+x3)/3 , (y1+y2+y3)/3
(4+3+5)/3 , (8+2-4)/3
(12)/3 , (10-4)/3
12/3 , 6/3
4 , 2
Therefore the Centroid is (4, 2).
3. Calculate the Centroid of the quadrilateral, whose co- ordinates are (3, 2) (5, -4) (4, 2) (3, -4)
Sol:
The given points are (3, 2) (5, -4) (4, 2) (3, -4),
Therefore,
(x1, y1) is (3, 2)
(x2, y2) is (5, -4)
(x3, y3) is (4, 2)
(x4, y4) is (3, -4)
Formulae for the Centroid is
G (x1+x2+x3+….+xn)/3n , (y1+y2+y3+….+yn))/n,
(3+5+4+3)/4 , (2-4+2-4)/4,
(15)/4 , (4-8)/4,
3.75 , -4/4
3.75 , -1
Therefore the Centroid is (3.75, -1)
