Quantitative Aptitude

Geometry

1,950 Questions
All Circle Tangents and Chords 772 Coordinate Geometry 360 Circle and Arc Measures 216 Triangle Centers and Medians 175 Composite and Inscribed Shapes 174 Similarity, Congruence, and Proportionality 118 Right Triangles and Pythagorean Theorem 100 Standard 2D Mensuration 22 Polygon and Angle Sums 10 3D Solid Mensuration 3

Geometry Questions

Multiple choice

Find the area of sector of circle whose radius is 5.6 cm and angle on the center of circle by arc is 45o. उस वृत्त के वृत्तखंड का क्षेत्रफल ज्ञात कीजिये जिसकी त्रिज्या 5.6 सेमी. है तथा चाप द्वारा केंद्र पर 45o का कोण बनता है।

  1. 18.36 cm2

  2. 13.56 cm2

  3. 14.32 cm2

  4. 12.32 cm2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of sector = (θ/360) × πr². Given r = 5.6 cm, θ = 45°. Area = (45/360) × π × (5.6)² = (1/8) × π × 31.36 = 3.92π ≈ 3.92 × 3.14 ≈ 12.32 cm². Option D is correct.

Multiple choice

Find the area of sector of circle whose radius is 8.4 cm and angle on the center of circle by arc is 60o. उस वृत्त के वृत्तखंड का क्षेत्रफल ज्ञात कीजिये जिसकी त्रिज्या 8.4 सेमी. है तथा चाप द्वारा केंद्र पर 60o का कोण बनता है।

  1. 18.36 cm2

  2. 13.56 cm2

  3. 14.32 cm2

  4. 36.96 cm2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of sector = (θ/360°) × πr². With θ = 60°, r = 8.4 cm: Area = (60/360) × π × 8.4² = (1/6) × 3.14 × 70.56 = 36.96 cm². Option A (18.36) is exactly half - perhaps if θ was 30° instead of 60°.

Multiple choice

A triangle has sides of lengths 6 cm, 8 cm and 10 cm. The area of the circle inscribed in the triangle is एक त्रिभुज की भुजाएं 6 सेमी, 8 सेमी और 10 सेमी लंबी हैं। त्रिभुज में अन्तर्निहित वृत्त का क्षेत्रफल कितना होगा?

  1. 4π cm.2 / सेमी.2

  2. 6π cm.2 / सेमी.2

  3. 8π cm.2 / सेमी.2

  4. 2π cm.2 / सेमी.2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The 6-8-10 triangle is a right triangle (3-4-5 scaled by 2). Inradius r = Area/semiperimeter for any triangle. Area = (6×8)/2 = 24 cm², semiperimeter = 12 cm, so r = 24/12 = 2 cm. Area of incircle = πr² = π(2)² = 4π cm².

Multiple choice

An arc of 30o in one circle is double an arc in a second circle, the radius of which is three times the radius of the first. Then the angle subtended by the arc of the second circle at its center is एक वृत्त में 30o के कोण वाला एक चाप दूसरे वृत्त के एक चाप का दोगुना है, जिसकी त्रिज्या पहले वाले की त्रिज्या का तीन गुना है | तो दूसरे वृत्त द्वारा इसके केंद्र पर अन्तरित किया जाने वाला कोण है

  1. 3o

  2. 4o

  3. 5o

  4. 6o
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For circle 1: arc length L1 = (30/360) × 2πr = πr/6. For circle 2: radius R=3r, let angle be θ. Given L1 = 2×L2, so πr/6 = 2 × (θ/360) × 2π(3r). Solving: πr/6 = 2 × (θ/360) × 6πr. Canceling πr: 1/6 = 12θ/360, so θ = 360/(72) = 5°. The key is relating arc lengths through radius and angle.

Multiple choice

Find length of the arc whose central angle is 90° and radius of the circle is 3.5 cm? वृत्त का एक चाप जिसका केंद्रीय कोण 90° और वृत्त की त्रिज्या 3.5 सेमी है, चाप की लम्बाई क्या होगी?

  1. 11 cm / सेमी

  2. 5.5 cm / सेमी

  3. 16.5 cm / सेमी

  4. 22 cm / सेमी
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Arc length formula: L = (θ/360) × 2πr, where θ = 90° and r = 3.5 cm. L = (90/360) × 2π × 3.5 = (1/4) × 7π = 1.75π ≈ 5.5 cm. The arc length is 5.5 cm.

Multiple choice

PA is a tangent from an external point P to a circle with centre O. If ?POB =1150 , find ?APO. (where AB is diameter)

  1. 300

  2. 350

  3. 250

  4. 200
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

AB is diameter, so ∠ABO = 90° (angle in semicircle). In ΔAOB, ∠AOB = 115° (given as POB, same angle), so ∠OAB = 180° - 90° - 115° = 25°. PA is tangent, so ∠OAP = 90° (radius ⟂ tangent). In ΔOAP, ∠APO = 180° - 90° - 65° = 25°. The key is recognizing tangent-radius perpendicularity and angle in semicircle property.

Multiple choice

A piece of wire in the form of an arc of a circle of radius 6 cm, subtends 1500 at the centre. The wire is bent into a complete circle.Find the radius of the circle.

  1. 2 cm.

  2. 3 cm.

  3. 2.5 cm

  4. 1.8 cm.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Arc length of a sector = (θ/360) × 2πr, where θ is the central angle. Here arc length = (150/360) × 2π × 6 = 5π cm. When bent into a full circle, circumference = 2πR = 5π, giving R = 2.5 cm. Options A, B, and D are calculation errors.

Multiple choice

If PT is a tangent and AB is a chord of circle and they intersect at the point P externally and 2PT = AP and AB = 18 units, then PT is equal to- यदि PT एक स्पर्शरेखा है और AB वृत्त की एक जीवा है और वे बाह्य बिंदु P पर एक दूसरे को प्रतिच्छेदित करती हैं और 2PT = AP और AB = 18 इकाई, तो PT बराबर है

  1. 6 units/इकाई

  2. 9 units/इकाई

  3. 12 units/इकाई

  4. 15 units/इकाई
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By tangent-secant theorem: PT² = PA × PB. Given 2PT = AP (so AP = 2PT) and AB = 18. Then PB = AP - AB = 2PT - 18. Substituting: PT² = 2PT(2PT - 18). PT² = 4PT² - 36PT. Therefore PT = 12 units.

Multiple choice

Find the length of the chord that is at a distance of 21 cm from the center of a circle of radius 29 cm. उस जीवा की लम्बाई ज्ञात कीजिए जो 29 सेमी त्रिज्या वाले वृत्त के केन्द्र से 21 सेमी की दूरी पर है।

  1. 36 cm/सेमी

  2. 42 cm/सेमी

  3. 40 cm/सेमी

  4. 20 cm/सेमी
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a circle, chord length L at distance d from center with radius r is L = 2√(r² - d²). Here: 2√(29² - 21²) = 2√(841-441) = 2√400 = 2 × 20 = 40 cm.

Multiple choice

In a triangle, length of all the three medians are 9 cm, 12 cm and 15 cm respectively. Then what will be the area of the triangle ? एक त्रिभुज में सभी तीनों माध्यिकाओं क्रमशः 9 सेमी., 12 सेमी. और 15 सेमी. हैं तो त्रिभुज का क्षेत्रफल क्या होगा ?

  1. 36 cm2/सेमी2

  2. 72 cm2/सेमी2

  3. 54 cm2/सेमी2

  4. 64 cm2/सेमी2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a triangle with medians of lengths 9 cm, 12 cm, and 15 cm, these form a right triangle (9² + 12² = 81 + 144 = 225 = 15²). The area of a triangle with medians ma, mb, mc is (4/3) times the area of the triangle formed by the medians. Area of triangle with sides 9, 12, 15 = (1/2) * 9 * 12 = 54 cm². Therefore, area of original triangle = (4/3) * 54 = 72 cm². Option A (36 cm²) is incorrectly calculated without the 4/3 factor. Option C (54 cm²) is the area of the median triangle, not the original triangle. Option D (64 cm²) has no mathematical basis in this problem.

Multiple choice

In a circle with radius of 5 cm two chords AB and AC are such that AB = AC = 6 cm, then the length of chord BC is- एक 5 सेमी. त्रिज्या वाले वृत्त में दो जीवायें AB और AC इस प्रकार हैं कि AB = AC = 6 सेमी., तो जीवा BC की लम्बाई है-

  1. 9.6 cm/सेमी.

  2. 4.8 cm/सेमी.

  3. 1.4 cm/सेमी.

  4. 23.04 cm/सेमी.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

दिए गए वृत्त में (त्रिज्या = 5 cm), AB = AC = 6 cm। PAB और PAC समद्विबाहु त्रिभुजे हैं। बिंदु P से AB और AC पर लंब D और E हैं। AD = sqrt(6^2 - 5^2) = sqrt(11)। बिंदु A से BC पर लंब की लंबाई = AE = sqrt(36 - 11) = 5 cm। BC = 2 × BD = 2 × sqrt(36 - 25) = 2 × sqrt(11) ≈ 9.6 cm।

Multiple choice

Two equal circle touch each other externally at C and AB is a common tangent to the circles. Then ?ACB =?

  1. 30o

  2. 60o

  3. 45o

  4. 90o
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two equal circles touching externally at C with common tangent AB, the radii to the point of tangency are perpendicular to the tangent. This creates two radii OA and OB perpendicular to AB. Since O, C, and the point of tangency are collinear, triangle AOB is a right triangle with OA and OB as perpendicular legs, making angle ACB = 90°. Option D is correct.

Multiple choice

Two parallel chords are drawn in a circle of radius 15 cm. The length of one chord is 24 cm and the distance between two chords is 21 cm. The length of the other chord is-

  1. 18 cm

  2. 9 cm

  3. 12 cm

  4. 16 cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For first chord (24 cm) in circle of radius 15 cm: half chord = 12 cm. Distance from center d₁ = √(15² - 12²) = √(225-144) = √81 = 9 cm. Distance between chords is 21 cm, so d₂ = |21 ± 9| = 12 cm or 30 cm. Since 30 > 15, d₂ = 12 cm. Half of other chord = √(15² - 12²) = √(225-144) = 9 cm. Full chord = 2 × 9 = 18 cm. Option A is correct.

Multiple choice

Two circles touches each other externally at P. AB is direct comon tangent. Whose touch points are A and B if ∠PAB = 350, then ∠ABP is- दो वृत्त एक दूसरे को वाह्यतः P पर स्पर्श करते हैं। उन दोनों वृत्तों पर AB एक उभयनिष्ठ अनुस्पर्श रेखा है। जिसके स्पर्श बिन्दु A तथा B है। तद्नुसार यदि ∠PAB = 350 हो, तो ∠ABP का मान कितना होगा?

  1. 350

  2. 550

  3. 650

  4. 750
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given two circles touching externally at P, AB is direct common tangent. PA is tangent to circle A at A, so ∠PAB = 90° (tangent ⊥ radius). In triangle PAB: ∠PAB + ∠ABP + ∠BPA = 180°. If ∠PAB = 35° (likely 35°, not 350° which exceeds 180°), and since PB is also tangent, ∠PBA = 90°. However, with given ∠PAB = 35°, using angle sum: ∠ABP = 180° - 90° - 35° = 55°.

Multiple choice

A chord of length 30 cm is at a distance of 8 cm from the center of a circle. The area of the circle (in cm2) is: एक वृत्त के केंद्र से 8 सेमी की दूरी पर 30 सेमी लंबाई की एक जीवा स्थित है। वृत्त का क्षेत्रफल (सेमी2 में) हैः

  1. 289π

  2. 529π

  3. 441π

  4. 361π
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a chord at distance d from center, radius r = sqrt((L/2)² + d²). Here: r = sqrt(15² + 8²) = sqrt(225 + 64) = sqrt(289) = 17 cm. Area = πr² = π(289) = 289π. The perpendicular from center bisects the chord, forming a right triangle.