Latest TCS Aptitude Question SOLUTION: An equilateral triangle and a regular hexagon have equal parameters then what is the ration of their areas? A) 6:1 b) 1:6 c) 3:2 d) 2:3

Regular hexagon=6 equilateral triangles
=> area of equilateral triangle =root(3)/4*a^2
=> area of hexagon= 6 *(root(3)/4*a^2)=3*(root(3)/2*a^2)
So, ratio of their areas is 1:6

area of triangle=root(3)a^2/4
area of hexagon=root(3)*3b^2/2
perimeter are same so hexagon b =a/2
(root(3)a^2/4)/(root(3)*3(a/2)^2/2)
so,result will be 2:3

area of equilateral=root(3)/4*a^2
area of hexagon=6*a^2*root(3)/4
solving two equations
ans=1:6

area of equilateral triangle=(sqrt3/4)s^2
area of hexagon=(3sqrt3/2)s^2
ans= 1:6

FOR A HEXAGON
Perimeter = 6 * radius
Area = (3√3/2) * (radius)^2
FOR AN EQUILATERAL TRIANGLE
Perimeter = 3 * side
Area = (side^2)√3/4
Since Hexagon perimeter = Equilateral triangle perimeter
then 6 * radius = 3 * side
Divide both sides of the equal sign by 3.
6 * radius / 3 = 3 * side / 3
6 * radius / 3 = side
Flip the equation around.
side = 6 * radius / 3
side = 2 * radius
This equation above gives me the relationship I needed between the radius and the side variables.

Next we deal with the ratio of the areas by using their area formulas.

. . . Area of hexagon . . . . . . (3√3/2)(radius)^2
-------------------------------------- = --------------------------
Area of equilateral triangle . . . (side^2)√3/4

Expand the exponent variables

. . . Area of hexagon . . . . . . (3√3/2)(radius)(radius)
-------------------------------------- = --------------------------------
Area of equilateral triangle . . (side)(side)√3/4

Since we calculated that side = 2 * radius, then substitute 2radius for side.

. . . Area of hexagon . . . . . . (3√3/2)(radius)(radius)
-------------------------------------- = --------------------------------
Area of equilateral triangle . . (2radius)(2radius)√3/4

cancel out common terms radius and radius

. . . Area of hexagon . . . . . . (3√3/2)
-------------------------------------- = ---------------
Area of equilateral triangle . . (2)(2)√3/4

. . . Area of hexagon . . . . . . (3√3)/2
-------------------------------------- = ------------
Area of equilateral triangle . . 4(√3)/4

cancel out common terms 4 and √3

. . . Area of hexagon . . . . . . 3/2
-------------------------------------- = -----
Area of equilateral triangle. . . 1

. . . Area of hexagon. . . . . . . 3
-------------------------------------- = -----
Area of equilateral triangle. . . 2