Question

A square and a regular hexagon have equal perimeters. Then their areas are in the ratio:

Answer 1

Hint: Here from the given perimeters we will find the side of square and hexagon. And with the help of it we will find the ratio of their areas.

Formula used:

Let, the side of a square is a
The perimeter of a square is = 4a
The area of a square is $$a^2$$
Let, the side of a regular hexagon is b
The perimeter of the regular hexagon is = 6b
The area of a square is $${\displaystyle \frac{3\sqrt{3}}{2}}{b}^2$$

Complete step-by-step answer:
It is given that the square and a regular hexagon have equal perimeters.
If the side of a square is a
The perimeter of the square is = 4a
And if the side of the regular hexagon is b,
The perimeter of the regular hexagon is = 6b
Then by the given condition, we have
$$4a=6b$$
Let us solve the above equation to find “a”, we get,
$$a={\displaystyle \frac{6b}{4}}={\displaystyle \frac{3b}{2}}$$…..(1)
Let us mark it as equation (1)
The area of the square is $$a^2$$when the side of the square is a.
The area of the regular hexagon is $${\displaystyle \frac{3\sqrt{3}}{2}}{b}^2$$when the side of the regular hexagon is b.
Then the ratio of the area of the square and the regular hexagon is found by dividing both the areas,
$${\displaystyle \frac{a^2}{{\displaystyle \frac{3\sqrt{3}}{2}}{b}^2}}$$
Let us substitute equation (1) in the above equation we get,
$$={\displaystyle \frac{{({\displaystyle \frac{3b}{2}})}^2}{{\displaystyle \frac{3\sqrt{3}}{2}}{b}^2}}$$
Let us now solve the above equation we get,
$$={\displaystyle \frac{9b^2}{4}}\times {\displaystyle \frac{2}{3\sqrt{3}{b}^2}}$$
We have found that,
$${\displaystyle \frac{a^2}{{\displaystyle \frac{3\sqrt{3}}{2}}{b}^2}}$$$$={\displaystyle \frac{\sqrt{3}}{2}}$$

Hence the ratio of the area of the square and the regular hexagon is $$\sqrt{3}:2$$ .

Note:Square:
In geometry, a square is a regular quadrilateral that has four equal sides and four equal angles.
Hexagon:
A hexagon is a six-sided polygon or 6-gon.The total of the internal angles of a hexagon is 720°.
Ratio is found by dividing the values for which we have to find the ratio.