Areas Related to Circles Set-2

Q1. A circle circumscribes a square of side $14\ \text{cm}$. Find the area of the region that lies inside the circle but outside the square.

Q2. In a circle of radius $10\ \text{cm}$, a chord is at a distance $6\ \text{cm}$ from the centre. The area of the smaller segment (rounded to one decimal place) is:

Q3. An equilateral triangle is inscribed in a circle of radius $10\ \text{cm}$. The area of the region inside the circle but outside the triangle is:

Q4. Two equal circles of radius $10\ \text{cm}$ have their centres $12\ \text{cm}$ apart. The area of their overlapping region (lens) is approximately:

Q5. $AC$ is the diameter of a semicircle of length $20\ \text{cm}$. A right triangle $ABC$ is drawn on the diameter $AC$ as hypotenuse, and the altitude from $B$ to $AC$ divides $AC$ into segments $8\ \text{cm}$ and $12\ \text{cm}$. The area of the part of the semicircle that lies outside triangle $ABC$ equals:

Q6. A sector of a circle of radius $14\text{ cm}$ has central angle $60^\circ$. The area of the corresponding minor segment (sector minus the isosceles triangle formed by the radii) is

Q7. A rectangle is inscribed in a semicircle of radius $10\text{ cm}$ with its base on the diameter. Let the rectangle's width be $2x$ and height be $y$ where $y=\sqrt{100-x^2}$. The maximum possible area of such a rectangle equals

Q8. An equilateral triangle is inscribed in a circle of radius $10\text{ cm}$. The area of the region inside the circle but outside the triangle is

Q9. Start with a circle of radius $6\text{ cm}$. Inscribe a square in it, then inscribe a circle in that square, then a square in that circle, and continue this process indefinitely. The sum of the areas of all circles in this infinite sequence equals

Q10. In a circle of radius $13\text{ units}$ a chord is at a distance $5$ units from the centre. If $\alpha=\arccos\!\left(\dfrac{5}{13}\right)$, the area of the smaller segment cut off by the chord can be written in closed form as

Q11. A circle is inscribed in a square of side $10\ \text{cm}$. What is the area (in cm$^2$) of the region inside the square but outside the circle?

Q12. Two equal circles of radius $10\ \text{cm}$ have their centers $12\ \text{cm}$ apart. The area common to both circles (the overlap) is given by which expression?

Q13. In a circle of radius $13\ \text{cm}$, a chord of length $10\ \text{cm}$ is drawn. The area (in cm$^2$) of the minor segment cut off by this chord equals:

Q14. Triangle $ABC$ is right-angled at $B$ with $AB=15\ \text{cm}$ and $BC=20\ \text{cm}$. A semicircle is drawn on the hypotenuse $AC$ (diameter $AC$) on the side opposite the triangle. What is the area (in cm$^2$) of the region inside this semicircle but outside the triangle?

Q15. Two concentric circles have radii $10\ \text{cm}$ and $6\ \text{cm}$. A chord of the larger circle is tangent to the smaller circle. What is the area (in cm$^2$) of the smaller segment of the larger circle cut off by this chord?