
A circle is inscribed in a regular hexagon of side cm. Find the circumference of the inscribed circle.
Answer
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Hint: The circumference of the circle
In this question where a circle is inscribed in a regular hexagon we know the central angle of a hexagon , so we will find the other angles of the triangle and then we will find the radius of the circle which will be equal to the sides of the triangle and by using this radius we will find the circumference of the circle.
Complete step-by-step answer:
Side of the regular hexagon
Now since a regular hexagon has 6 sides, hence we can say the central angle of a hexagon
Now from the figure we can say (Central angle of regular hexagon is )
Now in the
Since the total sum of the angles of a triangle is equal to , hence we can write
Now since , so we can write
Therefore
Now since all the angles of the triangle are equal hence we can say the triangle is an equilateral triangle, so all the sides of the triangle will also be equal
Now this length will be the circum-radius of the circle which is inscribed in a regular hexagon
We know the Perimeter of a circle is given by the formula
Hence by substituting the value of the radius of the circle, we get
Therefore the circumference of the inscribed circle
Note: Students must note that there are six equilateral triangles in a hexagon so we can say the sides of the hexagon will also be equal to the radius of the hexagon.
In this question where a circle is inscribed in a regular hexagon we know the central angle of a hexagon
Complete step-by-step answer:
Side of the regular hexagon
Now since a regular hexagon has 6 sides, hence we can say the central angle of a hexagon

Now from the figure we can say
Now in the
Since the total sum of the angles of a triangle is equal to
Now since
Therefore
Now since all the angles of the triangle are equal hence we can say the triangle is an equilateral triangle, so all the sides of the triangle will also be equal
Now this length
We know the Perimeter of a circle is given by the formula
Hence by substituting the value of the radius of the circle, we get
Therefore the circumference of the inscribed circle
Note: Students must note that there are six equilateral triangles in a hexagon so we can say the sides of the hexagon will also be equal to the radius of the hexagon.
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