A circle is inscribed in a regular hexagon of side cm. Find (i) the circumference of the inscribed circle (ii) the area of the inscribed circle

(i) Side of the regular hexagon (a) = cm. Now since a regular hexagon has 6 sides, hence we can say the central angle of a hexagon = = 60°. From figure, ∠AOB = 60°. Now in the ΔAOB, Since the total sum of the angles of a triangle is equal to 180° ∴ ∠AOB + ∠OAB + ∠OBA = 180° .......(1) Since, OA = OB = radius of circle, So, ∠OAB = ∠OBA = x (let) (Angles opposite to equal sides are equal). Substituting value in equation 1 we get, ⇒ 60° + x + x = 180° ⇒ 2x + 60° = 180° ⇒ 2x = 120° ⇒ x = 60°. 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. ∴ AO = BO = AB = cm. Draw perpendicular from O to AB. From figure, AT = BT = cm. (As altitude and median are same in equilateral triangle.) From figure, In right angle triangle OAT, Hence, radius of circle = OT = 3 cm. We know the circumference of a circle is given by the formula, Circumference = 2πr = = cm. Hence, circumference of inscribed circle = cm. (ii) Area of inscribed circle = πr 2 = = cm 2 . Hence, area of inscribed circle = cm 2 .

In the figure (ii) given below, ABC is an equilateral triangle of side 8 cm. A, B and C are the centers of circular arcs of equal radius. Find the area of the shaded region correct upto 2 decimal places.

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In the adjoining figure, a chord AB of a circle of radius 10 cm subtends a right angle at the centre O. Find the area of the sector OACB and of the major segment. Take π = 3.14.

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In the figure (ii) given below, a piece of cardboard, in the shape of a trapezium ABCD and AB || CD and ∠BCD = 90°, quarter circle BFEC is removed. Given AB = BC = 3.5 cm and DE = 2 cm. Calculate the area of the remaining piece of the cardboard.

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In the figure (i) given below, ABC is a right angled triangle, ∠B = 90°, AB = 28 cm and BC = 21 cm. With AC as diameter a semi-circle is drawn and with BC as radius a quarter circle is drawn. Find the area of the shaded region correct to two decimal places.

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Mathematics

A circle is inscribed in a regular hexagon of side cm. Find (i) the circumference of the inscribed circle (ii) the area of the inscribed circle

Mensuration

Related Questions

In the figure (ii) given below, ABC is an equilateral triangle of side 8 cm. A, B and C are the centers of circular arcs of equal radius. Find the area of the shaded region correct upto 2 decimal places.

In the adjoining figure, a chord AB of a circle of radius 10 cm subtends a right angle at the centre O. Find the area of the sector OACB and of the major segment. Take π = 3.14.

In the figure (ii) given below, a piece of cardboard, in the shape of a trapezium ABCD and AB || CD and ∠BCD = 90°, quarter circle BFEC is removed. Given AB = BC = 3.5 cm and DE = 2 cm. Calculate the area of the remaining piece of the cardboard.

In the figure (i) given below, ABC is a right angled triangle, ∠B = 90°, AB = 28 cm and BC = 21 cm. With AC as diameter a semi-circle is drawn and with BC as radius a quarter circle is drawn. Find the area of the shaded region correct to two decimal places.