This chapter is crucial because it connects circle geometry to practical area problems commonly asked on board exams and in competitive tests. You learn standard results for sectors, segments, chords, and overlaps of circles/squares, which appear repeatedly in numericals and reasoning-based MCQs. Mastering these area relationships also strengthens your ability to convert a diagram into usable formulas quickly.
15
Minutes
15
Questions
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Marking
Q1. A circle circumscribes a square of side . Find the area of the region that lies inside the circle but outside the square.
Q2. In a circle of radius , a chord is at a distance 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 . The area of the region inside the circle but outside the triangle is:
Q4. Two equal circles of radius have their centres apart. The area of their overlapping region (lens) is approximately:
Q5. is the diameter of a semicircle of length . A right triangle is drawn on the diameter as hypotenuse, and the altitude from to divides into segments and . The area of the part of the semicircle that lies outside triangle equals:
Q6. A sector of a circle of radius has central angle . 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 with its base on the diameter. Let the rectangle's width be and height be where . The maximum possible area of such a rectangle equals
Q8. An equilateral triangle is inscribed in a circle of radius . The area of the region inside the circle but outside the triangle is
Q9. Start with a circle of radius . 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 a chord is at a distance units from the centre. If , 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 . What is the area (in cm) of the region inside the square but outside the circle?
Q12. Two equal circles of radius have their centers apart. The area common to both circles (the overlap) is given by which expression?
Q13. In a circle of radius , a chord of length is drawn. The area (in cm) of the minor segment cut off by this chord equals:
Q14. Triangle is right-angled at with and . A semicircle is drawn on the hypotenuse (diameter ) on the side opposite the triangle. What is the area (in cm) of the region inside this semicircle but outside the triangle?
Q15. Two concentric circles have radii and . A chord of the larger circle is tangent to the smaller circle. What is the area (in cm) of the smaller segment of the larger circle cut off by this chord?