Question

An equilateral triangle and a regular hexagon have equal perimeters. If the area of triangle is 12dm2, then the difference in their area is

Answer 1

Answer:

6d

Step-by-step explanation:

Let a be the area of the triangle (so a = 12d) and p be its perimeter

A hexagon drawn around the triangle has area 2a and perimeter that is (2/√3)p (use Pythagoras' Theorem to get this).  So the hexagon in question is obtained by scaling this hexagon by a factor of √3/2.  But this scales the area by the square of the scale factor, so the area of the hexagon in question is 2a times (√3/2)^2; i.e. its area is 3a/2.

The difference between the areas is then 3a/2 - a = a/2.

When a = 12d, the difference then is a/2 = 6d.

Answer 2

Answer :-
1.6dm2
STEP BY STEP EXPLANATION:-

In an hexagona there are 6sides, while a triangle has three sides.

Side of Equilateral Triangle=a
Side of regular hexagon=b

By question,
Perimeter of Equilateral Triangle=Perimeter of Regular Hexagon
Therefore,
a+a+a= b+b+b+b+b+b
3a=6b
b=a/2

So ,the side of regular hexagon=a/2

Area of an equilateral triangle=√3/4a^2=12dm2
Therefore,a=4dm

Area of regular hexagon=3√3/2b^2
=3√3/2(a/2)^2 {b=a/2}
=3√3/2(2^2) {b=2}
=6√3dm2

Difference in there area,
Area of regular hexagon-Area of equilateral triangle=12dm2- 6√3dm2
=12-10.4 {√3=1.73}
=1.6dm2

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