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Mathematics

Grade9

Easy

Question

The area of a regular hexagon is 54√3 and the perimeter is 36. Find the length of the sides and the apothem.

s = 6, a = $3 \sqrt{3}$
s = 8, a = $4 \sqrt{3}$
s = 8, a = $3 \sqrt{3}$
s = 6, a = $4 \sqrt{3}$

hint Hint:

Polygon is a two- dimensional closed figure which is made up of three or more line segments. Each polygon has its own properties based on the number of sides. Area of a regular polygon can be determined using the formula $A r e a space o f space t h e space p o l y g o n equals 1 half cross times P e r i m e t e r space o f space t h e space p o l y g o n cross times A p o t h e m$ . Here, we have to find the apothem and length of each side of the given polygon using the area formula.

The correct answer is: s = 6, a = $3 square root of 3$

In the question it is given that
Area of a regular hexagon= 54√3
Perimeter of the hexagon= 36
Here, we have to find the length of the sides and the apothem of the given hexagon.
Firstly, find the length of the regular polygon using the perimeter formula.
Perimeter of the polygon= Length of the polygon(s) $cross times$ No. of sides
$rightwards double arrow$ 36= s $cross times$ 6
$rightwards double arrow$ s= $36 over 6$
$rightwards double arrow$ s= 6
Now, find the apothem of the given hexagon using the area formula.
Area of the polygon= $1 half cross times P e r i m e t e r space o f space t h e space p o l y g o n cross times A p o t h e m left parenthesis a right parenthesis$
$rightwards double arrow$ 54√3= $1 half cross times 36 cross times a$
$rightwards double arrow$ a= $fraction numerator 54 square root of 3 cross times 2 over denominator 36 end fraction$
$rightwards double arrow$ a= $3 square root of 3$
So, the length of the regular polygon is 6 and apothem of the given hexagon is $3 square root of 3$ .
Therefore, the correct option is a, i.e., s= 6, a= $3 square root of 3$ .

A perpendicular line segment joining the centre and one of the sides of the regular polygon is called apothem of the polygon.

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