Question

A regular Pentagon is drawn with all its vertices on a circle of radius 15 centimetres.Calculate the length of the sides of this Pentagon? Verified steps are required.​

Answer 1

From image :-

• ABCDE is a regular pentagon with its vertices on a circle.
• O is the centre of circle.
• OA and OB are radius of 15cm.
• OF is perpendicular drawn on AB side of pentagon.

Solution :-

→ ∠AOB = (360°/5) = 72° { Regular Pentagon divides the complete angle at centre in equal parts. }

Now, in ∆AOB ,

→ AO = OB = 15cm (Radius.)

So,

→ ∆AOB is an isosceles ∆.

Therefore,

→ Perpendicular OF, divides the Base and bisect the angle.

Hence,

→ AF = FB

→ ∠AOF = ∠BOF = (∠AOB/2) = (72°/2) = 36° .

Now, in Right ∆AFO,

→ sin 36° = Perpendicular / Hypotenuse

→ sin 36° = AF / OA

→ sin 36° = AF / 15

→ AF = (15 * sin 36°)

Therefore,

→ AB = 2 * AF { AF = FB .}

→ AB = 2 * (15 * sin 36°)

→ AB = 30 * sin 36° .

Putting value of sin 36° = 0.587 ,

→ AB = 30 * 0.587

→ AB ≈ 17.63 cm. (Ans.)

Hence, Each sides of regular Pentagon is equal to 17.73 cm.(Approx) .

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Shortcut :-

• Circum - radius of a regular Polygon with total N - sides of length L = L / [2 * sin(180°/N)]

in question we have given that :-

• N = 5
• Circum - radius = 15cm.

Putting both values in formula we get :-

→ 15 = L / [ 2 * sin(180°/5)]

→ 15 = L / [2 * sin36°]

→ L = 15 * 2 * sin36°

→ L = 30 * sin36°

→ L = 30 * 0.587

→ L ≈ 17.63 cm. (Ans.)

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Answer 2

Answer:

From image :-

ABCDE is a regular pentagon with its vertices on a circle.

O is the centre of circle.

OA and OB are radius of 15cm.

OF is perpendicular drawn on AB side of pentagon.

Solution :-

→ ∠AOB = (360°/5) = 72° { Regular Pentagon divides the complete angle at centre in equal parts. }

Now, in ∆AOB ,

→ AO = OB = 15cm (Radius.)

So,

→ ∆AOB is an isosceles ∆.

Therefore,

→ Perpendicular OF, divides the Base and bisect the angle.

Hence,

→ AF = FB

→ ∠AOF = ∠BOF = (∠AOB/2) = (72°/2) = 36° .

Now, in Right ∆AFO,

→ sin 36° = Perpendicular / Hypotenuse

→ sin 36° = AF / OA

→ sin 36° = AF / 15

→ AF = (15 * sin 36°)

Therefore,

→ AB = 2 * AF { AF = FB .}

→ AB = 2 * (15 * sin 36°)

→ AB = 30 * sin 36° .

Putting value of sin 36° = 0.587 ,

→ AB = 30 * 0.587

→ AB ≈ 17.63 cm. (Ans.)

Hence, Each sides of regular Pentagon is equal to 17.73 cm.(Approx) .

_______________

Shortcut :-

Circum - radius of a regular Polygon with total N - sides of length L = L / [2 * sin(180°/N)]

in question we have given that :-

N = 5

Circum - radius = 15cm.

Putting both values in formula we get :-

→ 15 = L / [ 2 * sin(180°/5)]

→ 15 = L / [2 * sin36°]

→ L = 15 * 2 * sin36°

→ L = 30 * sin36°

→ L = 30 * 0.587

→ L ≈ 17.63 cm. (Ans.)

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