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Areas Related to Circles Set-1 MCQ Online Practice Test | Class 10 | Quiz Tweek

Areas Related to Circles Set-1

Practice Important MCQs for Areas Related to Circles — Maths, Class 10.

“Areas Related to Circles” is a high-scoring chapter because most questions reduce to combining basic circle-area facts (sector and segment) with geometry of triangles, chords, and lenses. Mastery of these area relationships is essential for both CBSE board exams and competitive tests like JEE/NEET, since many problems are solved by identifying the correct angle, forming the right sector/triangle, and subtracting overlapping regions carefully.

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Q1. A circle is inscribed in a square of side $14\text{ cm}$ . What is the area (in cm $^2$ ) of the region inside the square but outside the circle?

(A)

$196-14\pi$

(B)

$196-98\pi$

(C)

$196-49\pi$

(D)

$49\pi$

Q2. In a circle of radius $14\text{ cm}$ , a chord subtends an angle of $120^\circ$ at the centre. What is the area (in cm $^2$ ) of the smaller segment cut off by this chord? Express your answer exactly.

(A)

$\dfrac{196\pi}{3}-49\sqrt{3}$

(B)

$\dfrac{196\pi}{3}-98\sqrt{3}$

(C)

$\dfrac{98\pi}{3}-49\sqrt{3}$

(D)

$98\pi-49\sqrt{3}$

Q3. A right-angled triangle with legs $6\text{ cm}$ and $8\text{ cm}$ has its hypotenuse as the diameter of a semicircle constructed outward on the hypotenuse. What is the area (in cm $^2$ ) of the region that lies inside the semicircle but outside the triangle?

(A)

$25\pi-24$

(B)

$12.5\pi+24$

(C)

$25\pi-48$

(D)

$12.5\pi-24$

Q4. Two circles, each of radius $10\text{ cm}$ , have their centres $10\text{ cm}$ apart. What is the area (in cm $^2$ ) common to both circles? Give the exact value.

(A)

$\dfrac{100\pi}{3}-25\sqrt{3}$

(B)

$\dfrac{200\pi}{3}-50\sqrt{3}$

(C)

$200\pi-50\sqrt{3}$

(D)

$\dfrac{300\pi}{3}-75\sqrt{3}$

Q5. A semicircle of radius $10\text{ cm}$ has a rectangle of width $12\text{ cm}$ placed inside it such that the rectangle’s base lies on the diameter and the top two corners touch the semicircular arc. What is the area (in cm $^2$ ) of the region inside the semicircle but outside the rectangle?

(A)

$50\pi-64$

(B)

$25\pi-96$

(C)

$50\pi-96$

(D)

$100\pi-96$

Q6. A chord of a circle of radius $7\text{ cm}$ subtends a central angle of $60^\circ$ . The area of the minor segment cut off by this chord is

(A)

$\displaystyle \frac{49\pi}{6}-\frac{49\sqrt3}{2}$

(B)

$\displaystyle \frac{49\pi}{12}-\frac{49\sqrt3}{4}$

(C)

$\displaystyle 49\!\left(\frac{\pi}{6}-\frac{\sqrt3}{4}\right)$

(D)

$\displaystyle \frac{49\pi}{3}-\frac{49\sqrt3}{4}$

Q7. In a circle of radius $13\text{ cm}$ , a chord is at a distance $5\text{ cm}$ from the centre. The area of the smaller segment determined by this chord equals

(A)

$\displaystyle 169\cos^{-1}\!\left(\frac{5}{13}\right)-60$

(B)

$\displaystyle 169\sin^{-1}\!\left(\frac{5}{13}\right)-60$

(C)

$\displaystyle 169\cos^{-1}\!\left(\frac{5}{13}\right)-120$

(D)

$\displaystyle 169\left(\cos^{-1}\!\left(\frac{5}{13}\right)-\frac{60}{169}\right)+60$

Q8. Triangle $ABC$ is right-angled at $C$ with $AC=6\text{ cm},\ BC=8\text{ cm}$ . A semicircle with diameter $AB$ (the hypotenuse) is drawn on the outside of the triangle. The area of the region inside this semicircle but outside $\triangle ABC$ is

(A)

$\displaystyle \frac{25\pi}{4}-24$

(B)

$\displaystyle 25\pi-24$

(C)

$\displaystyle \frac{25\pi}{2}-12$

(D)

$\displaystyle \frac{25\pi}{2}-24$

Q9. Two circles of equal radius $10\text{ cm}$ have their centres $10\text{ cm}$ apart. The area of their overlapping region (lens) is

(A)

$\displaystyle \frac{200\pi}{3}-25\sqrt{3}$

(B)

$\displaystyle 100\!\left(\frac{2\pi}{3}-\frac{\sqrt3}{2}\right)$

(C)

$\displaystyle \frac{200\pi}{3}-75\sqrt{3}$

(D)

$\displaystyle 100\left(\frac{\pi}{3}-\frac{\sqrt3}{4}\right)$

Q10. In a circle of radius $10\text{ cm}$ , two parallel chords on the same side of the centre are at distances $6\text{ cm}$ and $8\text{ cm}$ from the centre. The area of the region between these two chords (the part of the circle lying between the two chords) is

(A)

$\displaystyle 100\!\left(\cos^{-1}\!\frac{3}{5}-\cos^{-1}\!\frac{4}{5}\right)$

(B)

$\displaystyle 100\!\left(\cos^{-1}\!\frac{4}{5}-\cos^{-1}\!\frac{3}{5}\right)$

(C)

$\displaystyle 100\!\left(\cos^{-1}\!\frac{3}{5}-\cos^{-1}\!\frac{4}{5}\right)-48$

(D)

$\displaystyle 100\!\left(\cos^{-1}\!\frac{3}{5}+\cos^{-1}\!\frac{4}{5}\right)$

Q11. A sector of a circle has radius $14\text{ cm}$ and central angle $90^\circ$ . What is the area of the sector? (Give exact value in terms of $\pi$ .)

(A)

$196\pi\ \text{cm}^2$

(B)

$49\pi\ \text{cm}^2$

(C)

$98\pi\ \text{cm}^2$

(D)

$14\pi\ \text{cm}^2$

Q12. A semicircle has diameter $AB=10\text{ cm}$ . A square is inscribed in the semicircle with one side on $AB$ and the opposite two vertices on the semicircular arc. What is the area of the shaded region inside the semicircle but outside the square? (Give exact answer.)

(A)

$\dfrac{25\pi}{4}-10\ \text{cm}^2$

(B)

$10\pi-20\ \text{cm}^2$

(C)

$\dfrac{25\pi}{2}-20\ \text{cm}^2$

(D)

$\dfrac{25\pi}{6}-5\ \text{cm}^2$

Q13. In a circle of radius $10\text{ cm}$ , a chord is at a distance $5\text{ cm}$ from the centre. The smaller segment cut off by the chord has area equal to (use exact form).

(A)

$\dfrac{100\pi}{3}-25\sqrt{3}\ \text{cm}^2$

(B)

$50\pi-25\sqrt{3}\ \text{cm}^2$

(C)

$\dfrac{200\pi}{3}-50\sqrt{3}\ \text{cm}^2$

(D)

$100\pi-50\sqrt{3}\ \text{cm}^2$

Q14. Two circles of equal radius $5\text{ cm}$ have their centres at a distance $5\text{ cm}$ apart. What is the area of their overlapping region (intersection)? Give the exact value.

(A)

$25\pi-\dfrac{25\sqrt{3}}{2}\ \text{cm}^2$

(B)

$\dfrac{25\pi}{3}-\dfrac{25\sqrt{3}}{4}\ \text{cm}^2$

(C)

$50\pi-\dfrac{25\sqrt{3}}{2}\ \text{cm}^2$

(D)

$\dfrac{50\pi}{3}-\dfrac{25\sqrt{3}}{2}\ \text{cm}^2$

Q15. An isosceles right triangle has hypotenuse $10\text{ cm}$ . A circle is inscribed in the triangle. What is the area of the region inside the triangle but outside the inscribed circle? Give the exact value in simplest form.

(A)

$25-\dfrac{\pi}{2}\bigl(75-50\sqrt{2}\bigr)\ \text{cm}^2$

(B)

$25-25\pi\bigl(3-2\sqrt{2}\bigr)\ \text{cm}^2$

(C)

$25-\dfrac{\pi}{25}\bigl(75-50\sqrt{2}\bigr)\ \text{cm}^2$

(D)

$25-\pi\bigl(3-2\sqrt{2}\bigr)\ \text{cm}^2$

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