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Circle

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Every question from past board papers on Circle, newest paper first. Open any question for its full worked answer.

Maharashtra SSC Geometry (Mathematics Part II) Board Paper June 2026

Full paper →

  1. Q.1 (A)(ii) What is the measure of semicircular arc of a circle ?

    1 mark · See worked answer →

  2. Q.1 (B)(iii) In the following figure, ray PQ touches the circle at point Q, PQ = 12, PR = 9. Find PS.

    Figure: P is a point outside the circle; ray PQ is a tangent touching the circle at Q; a secant from P cuts the circle first at R and then at S, so P – R – S; the line through P, R, S carries an arrow beyond S.

    1 mark · See worked answer →

  3. Q.2 (A)(i) In the following figure, chord EN and chord FS intersect each other at point M in the exterior of the circle. If \(m(\text{arc } NS) = 125^\circ\), \(m(\text{arc } EF) = 37^\circ\), then complete the following activity to find \(\angle NMS\).

    Figure: M is a point outside the circle; one secant from M cuts the circle at E and then N (M – E – N) and the other cuts it at F and then S (M – F – S); both secants are extended with arrows beyond N and S.

    Activity : Chord EN and chord FS intersect each other at point M in the exterior of the circle.

    \(m\angle NMS = \dfrac{1}{2}\left[\text{____} - m(\text{arc } EF)\right]\)
    = \(\dfrac{1}{2}\left[\text{____} - 37\right]\)
    = \(\dfrac{1}{2} \times \text{____}\)
    \(\therefore m\angle NMS = \text{____}\)

    2 marks · See worked answer →

  4. Q.2 (A)(iii) In the following figure, point O is the centre of the circle, \(\angle AOB = 100^\circ\). Complete the following activity with the help of given figure.

    Figure: circle with centre O; X and Y are the points where the vertical diameter meets the circle, Y at the top and X at the bottom; A is on the lower-left of the circle and B on the lower-right, with \(\angle AOB = 100^\circ\) marked at O; A and B are joined to O.

    Activity :

    Type of arcName of the arcMeasure of the arc
    Minor arc________
    Major arc________

    2 marks · See worked answer →

  5. Q.2 (B)(iii) In the following figure, point G, D, E, F are concyclic points of a circle with centre C. \(\angle ECF = 70^\circ\), \(m(\text{arc } DGF) = 200^\circ\). Find \(m(\text{arc } DE)\).

    Figure: circle with centre C; D in the upper left, G in the upper right, E in the lower left and F in the lower right on the circle; segments CD, CE and CF are drawn, with \(\angle ECF\) marked at C.

    2 marks · See worked answer →

  6. Q.3 (B)(ii) Prove that, opposite angles of a cyclic quadrilateral are supplementary.

    3 marks · See worked answer →

  7. Q.3 (B)(iii) Draw a circle of radius 3.3 cm. Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at point P and Q. Write your observation about the tangents.

    3 marks · See worked answer →

  8. Q.5 (i) AB is a chord of a circle with centre O. AC is a diameter of a circle. Line AT is a tangent at A. Write the answers of the following questions :

    (a) Draw the figure using the given information.
    (b) Find the measure of \(\angle CAT\).
    (c) Find the measure of \(\angle ABC\).

    3 marks · See worked answer →

Maharashtra SSC Geometry (Mathematics Part II) Board Paper March 2026

Full paper →

  1. Q.1 (A)(2) Two circles having radii 4 cm and 5 cm touch each other internally. Then the distance between the centres is .................... .

    1 mark · See worked answer →

  2. Q.1 (B)(3) If the measure of minor arc is 70º, then find the measure of corresponding major arc.

    1 mark · See worked answer →

  3. Q.2 (A)(1) In the following figure, seg PS is a tangent segment, line PR is a secant. If PQ = 3.6, QR = 6.4, find PS.

    Figure: a circle with an external point P; the secant through P cuts the circle at Q (nearer to P) and R (farther), and the tangent from P touches the circle at S.

    Activity :

    PS2 = ____ × PR (Tangent secant segment theorem)
    = PQ × [PQ + QR]
    = ____ × [3.6 + 6.4]
    = 3.6 × ____
    ∴ PS2 = 36
    PS = ____

    2 marks · See worked answer →

  4. Q.2 (B)(3) In the following figure, chord MN and chord RS intersect at point D. If RD = 15, DS = 4, MD = 8, find DN.

    Figure: a circle with chords MN and RS intersecting at the point D inside the circle, so that R–D–S and M–D–N.

    2 marks · See worked answer →

  5. Q.3 (B)(1) Prove that : The tangent segments drawn from an external point to the circle are congruent.

    Figure: a circle with centre O and an external point P; tangent segments PA and PB touch the circle at A and B; seg OP, seg OA and seg OB are drawn.

    3 marks · See worked answer →

  6. Q.5 (1) Seg PQ is a chord of a circle. Arc PXQ is a major arc and Arc PYQ is a minor arc. ∠POQ is central angle of the circle :

    Figure: a circle with centre O and a chord PQ; the major arc PXQ passes through the point X on the larger part of the circle and the minor arc PYQ passes through the point Y on the smaller part; ∠POQ is marked at the centre and the inscribed angles ∠PXQ and ∠PYQ are shown.

    (a) Draw the figure from given information.

    (b) Find the measure of ∠PXQ, ∠PYQ and add m∠PXQ, m∠PYQ.

    (c) Find the measure of ∠POQ. Write the relation between ∠PXQ and ∠POQ.

    3 marks · See worked answer →

Maharashtra SSC Geometry (Mathematics Part II) Board Paper March 2025

Full paper →

  1. Q.1 (A)(2) ∠ACB is inscribed angle in a circle with centre O. If m∠ACB = 65º, then what is measure of its intercepted arc AXB ?

    1 mark · See worked answer →

  2. Q.1 (B)(4) In cyclic quadrilateral ABCD m∠A = 100º, then find m∠C.

    1 mark · See worked answer →

  3. Q.2 (A)(2) In the following figure chord MN and chord RS intersect at point D. If RD = 15, DS = 4, MD = 8, find DN by completing the following activity :

    Figure: circle with chords MN and RS intersecting at D inside the circle.

    Activity :

    ∴ MD × DN = ____ × DS .............. (Theorem of internal division of chords)
    ∴ ____ × DN = 15 × 4
    ∴ DN = ____ / 8
    ∴ DN = ____

    2 marks · See worked answer →

  4. Q.2 (B)(3) In the following figure, circle with centre D touches the sides of ∠ACB at A and B. If ∠ACB = 52º, find measure of ∠ADB.

    Figure: circle with centre D; CA and CB are tangents touching the circle at A and B; radii DA and DB drawn.

    2 marks · See worked answer →

  5. Q.3 (B)(2) Prove that tangent segments drawn from an external point to a circle are congruent.

    3 marks · See worked answer →

Maharashtra SSC Geometry (Mathematics Part II) Board Paper July 2025

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  1. Q.1 (A)(2) Two circles of radii 5.5 cm and 4.2 cm touch each other externally then distance between their centres is ....................

    1 mark · See worked answer →

  2. Q.2 (A)(2) In the following figure, seg PS is a tangent segment, line PR is a secant. If PQ = 3.6, QR = 6.4, then find PS by completing the following activity.

    Figure: circle with an external point P; the secant from P cuts the circle at Q (nearer) and R (farther); the tangent from P touches the circle at S.

    Activity :

    \(PS^2 = PQ \times ____\) ......... (tangent secant segments theorem)

    \(\therefore PS^2 = PQ \times (PQ + ____)\)

    \(\therefore PS^2 = 3.6 \times (3.6 + ____)\)

    \(\therefore PS^2 = 3.6 \times 10\)

    \(\therefore PS^2 = 36\)

    \(\therefore PS = ____\)

    2 marks · See worked answer →

  3. Q.2 (B)(3) In the following figure, \(m(\text{arc NS}) = 125^\circ\), \(m(\text{arc EF}) = 37^\circ\). Find \(m\angle NMS\).

    Figure: circle with chords NS and EF intersecting at the point M inside the circle; N, F on the upper part of the circle, E and S on the lower part.

    2 marks · See worked answer →

  4. Q.3 (A)(2) In the following figure, point O is the centre of the circle and length of chord AB is equal to the radius of the circle. Find the measures of :

    (i) ∠AOB
    (ii) arc AB
    (iii) ∠ACB

    Figure: circle with centre O; chord AB below the centre, C on the major arc AB above the centre; seg OA, seg OB, seg CA and seg CB drawn.

    by completing the activity.

    Activity :

    In \(\Delta AOB\),

    AO = OB = AB

    \(\therefore \Delta AOB\) is an ____ triangle.

    \(\therefore m\angle AOB = ____\)

    \(m\angle AOB = m(\text{arc AB}) = ____\) ......... (definition of measure of an arc)

    \(m\angle ACB = \dfrac{1}{2} \times ____\) ......... (inscribed angle theorem)

    \(= \dfrac{1}{2} \times 60^\circ\)

    \(m\angle ACB = ____\)

    3 marks · See worked answer →

  5. Q.4 (3) In the following figure Δ ABC is an isosceles triangle with perimeter 44 cm. The base BC is of length 12 cm. Side AB and AC are congruent. A circle touches the three sides of triangle as shown.

    Find the length of tangent segment from A to circle.

    Figure: isosceles Δ ABC with A at the top and base BC; the incircle touches AB at P, AC at Q and BC at R; centre of the circle marked inside.

    4 marks · See worked answer →

Maharashtra SSC Geometry (Mathematics Part II) Board Paper July 2024

Full paper →

  1. Q.1 (A)(2) Two circles of radii 5.5 cm and 3.3 cm respectively touch each other externally. Then the distance between their centres is .............

    1 mark · See worked answer →

  2. Q.1 (B)(2) In the above figure, \(m(\text{arc } MN) = 70^\circ\), find \(\angle MLN\).

    Figure: a circle with points L, M, N on it; chords LM and LN are drawn, so that \(\angle MLN\) is an inscribed angle standing on arc MN.

    1 mark · See worked answer →

  3. Q.2 (A)(1) In the above figure, chord PQ and chord RS intersect at point T. Then complete the following activity to prove, \(\angle STQ = \dfrac{1}{2}\left[m(\text{arc } SQ) + m(\text{arc } PR)\right]\)

    Figure: a circle with points P, S above and R, Q below on the circumference; chords PQ and RS intersect at T inside the circle; chord PS is also drawn.

    Activity :

    \(\angle STQ = \angle SPQ +\) ____ ............ (Remote interior angles theorem of a triangle)
    \(= \dfrac{1}{2}m(\text{arc } SQ) +\) ____ ........... (Inscribed angle theorem)
    \(= \dfrac{1}{2}\left[\text{____} + \text{____}\right]\)

    2 marks · See worked answer →

  4. Q.3 (B)(1) Prove that, ‘Opposite angles of a cyclic quadrilateral are supplementary’.

    Figure: a circle with centre O; points A, B, C, D on the circle in order, forming cyclic quadrilateral ABCD; diagonals AC and BD are drawn.

    3 marks · See worked answer →

  5. Q.5 (1) The diameter of circular garden is 13 meter. The distance between two gates of garden is 13 meter. An electric pole is to be erected on the circumference of the garden such that the difference between distance of the pole from each gate is 7 meter. Can such pole be erected ? If yes, find the distance of the pole from each gate.

    Figure: a circle of diameter 13 m with two gates A and B at the ends of a diameter; a pole P on the circumference with PA and PB drawn.

    3 marks · See worked answer →

Maharashtra SSC Geometry (Mathematics Part II) Board Paper March 2024

Full paper →

  1. Q.1 (A)(4) A circle having radius 3 cm, then the length of its largest chord is .................... .

    1 mark · See worked answer →

  2. Q.1 (B)(2) Two circles of radii 5 cm and 3 cm touch each other externally. Find the distance between their centres.

    1 mark · See worked answer →

  3. Q.2 (A)(1)

    Figure: a circle with centre O. Points A, B, C lie on the circle, B at the top and A, C lower down; chords AB and BC are drawn; X is a point on the arc AXC below, and the rays from A and from C through X are produced beyond X, outside the circle.

    In the above figure, ∠ABC is inscribed in arc ABC.

    If ∠ABC = 60º, find m∠AOC.

    Solution :

    ∠ABC = \(\dfrac{1}{2}\) m(arc AXC) ............ ____

    60º = \(\dfrac{1}{2}\) m(arc AXC)

    ____ = m(arc AXC)

    But m∠AOC = m(arc .....) ............ (Property of central angle)

    ∴ m∠AOC = ____

    2 marks · See worked answer →

  4. Q.2 (B)(3)

    Figure: a circle; two secants from an external point M cut the circle at E, N and at F, S respectively, so that M – E – N and M – F – S.

    In the above figure, m(arc NS) = 125º, m(arc EF) = 37º. Find the measure of ∠NMS.

    2 marks · See worked answer →

  5. Q.3 (A)(2)

    Figure: a circle with centre P; chords AB and CD intersect at E inside the circle; seg AC and seg BD are drawn (dashed).

    Given : Chords AB and CD of a circle with centre P intersect at point E.

    To prove : AE × EB = CE × ED

    Construction : Draw seg AC and seg BD.

    Fill in the blanks and complete the proof.

    Proof :

    In Δ CAE and Δ BDE

    ∠AEC ≅ ∠DEB ............ ____

    ____ ≅ ∠BDE (angles inscribed in the same arc)

    ∴ Δ CAE ~ Δ BDE ............ ____

    ∴ \(\dfrac{____}{DE} = \dfrac{CE}{____}\) ............ ____

    ∴ AE × EB = CE × ED.

    3 marks · See worked answer →

  6. Q.5 (2) AB is a chord of a circle with centre O. AOC is diameter of circle, AT is a tangent at A.

    Write answers of the following questions :

    (a) Draw the figure using given information.

    (b) Find the measures of ∠CAT and ∠ABC with reasons.

    (c) Whether ∠CAT and ∠ABC are congruent ? Justify your answer.

    3 marks · See worked answer →