The perimeter of a square circumscribed about a circle is 6 cm larger
| April 23, 2021 | education
The perimeter of a square circumscribed about a circle is 6 cm larger than the perimeter of a pentagon inscribed in the same circle. Find the radius of this circle.
The perimeter of the square is equal to: Ravsd = 4 * AB.
The radius of the inscribed circle in the square is equal to half the length of its side, then Ravsd = 4 * 2 * R = 8 * R cm.
Since a regular pentagon is inscribed in a circle, the length of its side is:
KН = 2 * R * Sin (180/5) = 2 * R * Sin36.
Then P5 = 5 * 2 * R * Sin36 = 10 * R * Sin36.
By condition, Ravsd – P5 = 6 cm.
8 * R – 10 * R * Sin36 = 6.
R * (8 – 10 * Sin36) = 6.
R = 6 / (8 – 10 * Sin36) cm
Answer: The radius of the circle is 6 / (8 – 10 * Sin36) cm.
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