Circle or Semi Circle MCQ Quiz - Objective Question with Answer for Circle or Semi Circle - Download Free PDF

Given:

Area of circular field = 6 × area of triangular field

Formula used:

Area of circular field = πr², where r is a radius of circle.

Area of triangle = √s (s - a) (s - b) (s - c),

where s = (a + b + c)/2

a, b and c are sides of triangle respectively.

Calculation:

a = 35 m, b = 53 m and c = 66 m

s = (35 + 53 + 66)/2

⇒ s = 77

Area of triangle = √s(s - a) (s - b) (s - c)

⇒ √77 (77 - 35) (77 - 53) (77 - 66)

⇒ √77 × 42 × 24 × 11

⇒ √7 × 11 × 2 × 3 × 7 × 2 × 3 × 2 × 2 × 11

⇒ 11 × 7 × 2 × 2 × 3

⇒ 924 m²

Now, 6 × Area of triangle = Area of circle

⇒ 924 × 6 = πr²

⇒ (924 × 6 × 7)/22 = r²

⇒ 1764 = r²

⇒ 42 = r

∴ Radius of circle is 42 m.

Given:

Number of revolutions = 5000

Distance covered = 11 km = 1100000 cm

Formula used:

Circumference of circle = 2πr

Calculation:

Circumference of circle = 1100000/5000

⇒ 220 cm

According to the question,

⇒ 2πr = 220 cm

⇒ 2 × (22/7) × r = 220 cm

⇒ r/7 = 5 cm

⇒ r = 35 cm

Diameter = 2r

⇒ 2 × 35

⇒ 70 cm

∴ The diameter of circle is 70 cm

Given:

Diameter of the wheel = 154 cm

Radius of the wheel = 154/2 = 77 cm

Speed = 33km/h

Formula used:

Circumference of a circle = 2πr

where r → radius of a circle

Distance covered = speed × time

Calculation:

Distance covered in one revolution = Circumference of wheel

⇒ 2πr = 2 × 22/7 × 77 = 484 cm

Speed = 33 km/hr = 33 × 100000/60 = 55000 cm/min

∴ Total Revolutions covered in one min

⇒ 55000/484 = 113.63 ≈ 114

Given:

Area of a square shape wire = 81 cm2

Formulas used:

Perimeter of a square with side 'a' = 4a 

Area of square with side 'a' = a²

Perimeter of a semicircle = 2πr/2 + 2r  = πr + 2r 

Calculation:

Area of square = 81 cm2

⇒ a2 = 9²

⇒ a = 9 cm 

Perimeter of square = 4 × 9 = 36 cm 

⇒ Perimeter of square = Perimeter of semicirlce 

⇒ 36 cm = 22/7 × r + 2r 

⇒ 36 = (22r + 14r)/7 

⇒ 36 = 36r/7 

⇒ 1 = r/7 

⇒ r = 7 cm 

∴ The radius of the semicircle is 7 cm.

Given:

Area of circle = 154 cm²

Formula used:

Area of circle = πr²

Circumference of circle = 2πr

Calculation:

154 = πr²

⇒ r² = 154 × (7/22)

⇒ r² = 7 × 7

⇒ r² = 49

⇒ r = √49

⇒ r = 7 cm

Circumference = 2πr

⇒ Circumference = 2 × (22/7) × 7

⇒ Circumference = 44 cm

∴ The correct answer is option 2.

Given:

Diameter of semicircle = 14√2 cm

Radius = 14√2/2 = 7√2 cm

Total no. of chords = 6

Concept:

Since the chords are equal in length, they will subtend equal angles at the centre. Calculate the area of one sector and subtract the area of the isosceles triangle formed by a chord and radius, then multiply the result by 6 to get the desired result.

Formula used:

Area of sector = (θ/360°) × πr²

Area of triangle = 1/2 × a × b × Sin θ

Calculation:

The angle subtended by each chord = 180°/no. of chord

⇒ 180°/6

⇒ 30°

Area of sector AOB = (30°/360°) × (22/7) × 7√2 × 7√2

⇒ (1/12) × 22 × 7 × 2

⇒ (77/3) cm²

Area of triangle AOB = 1/2 × a × b × Sin θ

⇒ 1/2 × 7√2 × 7√2 × Sin 30°

⇒ 1/2 × 7√2 × 7√2 × 1/2

⇒ 49/2 cm²

∴ Area of shaded region = 6 × (Area of sector AOB - Area of triangle AOB)

⇒ 6 × [(77/3) – (49/2)]

⇒ 6 × [(154 – 147)/6]

⇒ 7 cm²

∴ Area of shaded region is 7 cm²

Given : 

Length of an arc of a circle is 4.5π.

Area of ​​the sector circumscribed by it is 27π cm2.

Formula Used : 

Area of sector = θ/360 × πr²

Length of arc = θ/360 × 2πr

Calculation : 

According to question,

⇒ 4.5π = θ/360 × 2πr 

⇒ 4.5 = θ/360 × 2r   -----------------(1)

⇒ 27π = θ/360 × πr2 

⇒ 27 = θ/360 × r2       ---------------(2)

Doing equation (1) ÷ (2)

⇒ 4.5/27 = 2r/πr²

⇒ 4.5/27 = 2/r

⇒ r = (27 × 2)/4.5

⇒ Diameter = 2r = 24

∴ The correct answer is 24.

Given:

Radius of the wheel of car = 14 cm

Speed of car = 132 km/hr

Formula Used:

Circumference of the wheel = \(2\pi r\) 

1 km = 1000 m

1m = 100 cm

1hr = 60 mins.

Calculation:

Distance covered by the wheel in one minute = \(\frac{132 \times 1000 \times 100}{60}\) = 220000 cm.

Circumference of the wheel = \(2\pi r\) = \(2\times \frac{22}{7} \times 14\) = 88 cm

∴ Distance covered by wheel in one revolution = 88 cm

∴ The number of revolutions in one minute = \(\frac{220000}{88}\) = 2500.

∴ Therefore the correct answer is 2500.

Given:

∠ROS = 42º

Concept used:

The sum of the angles of a triangle = 180°

Exterior angle = Sum of opposite interior angles

Angle made by an arc at the centre = 2 × Angle made by the same arc at any point on the circumference of the circle

Calculation:

Join RQ and RS

According to the concept,

∠RQS = ∠ROS/2

⇒ ∠RQS = 42°/2 = 21°   .....(1)

Here, PQ is a diameter.

So, ∠PRQ = 90°  [∵ Angle in the semicircle = 90°]

In ΔRQT, ∠PRQ is an exterior angle

So, ∠PRQ = ∠RTQ + ∠TQR

⇒ 90° = ∠RTQ + 21°  [∵ ∠TQR = ∠RQS = 21°]

⇒ ∠RTQ = 90° - 21° = 69°

⇒ ∠PTQ = 69°

∴ The measure of  ∠PTQ is 69°

Given:

AB is a diameter of a circle with a center O

∠APC = 62º

Concept used:

The radius/diameter of a circle is always perpendicular to the tangent line.

Sum of all three angles of a triangle = 180°

Calculation:

Minor arc AC will create angle CBA

∠APC = 62º = ∠APB

∠BAP = 90° (diameter perpendicular to tangent)

In Δ APB,

∠APB + ∠BAP + ∠PBA = 180° 

⇒ ∠PBA = 180° - (90° + 62°)

⇒ ∠PBA = 28° 

∴ The measure of minor arc AC is 28° 

Mistake PointsMeasure of the minor arc AC is asked,

∠ABC marks arc AC, 

∴ ∠ABC is the correct angle to show a measure of arc AC

This is a previous year's question, and according to the commission, this is the correct answer.

The two sides holding the right angle in a right-angled triangle are 3 cm and 4 cm long,

⇒ Length of hypotenuse = (3² + 4²)^1/2 = 5 cm

⇒ Radius of circum-circle = 5/2 = 2.5 cm

Given:

Length of an arc = 23.1 cm

Angle subtended on center by arc = 18°

Formula used:

Length of an arc = (2 × π × R × θ)/360

Area of circle = π × R²

Where, R = radius

Calculation:

Length of an arc = (2 × π × R × θ)/360

⇒ 23.1 = (2 × 22 × R × 18)/(360 × 7)

⇒ 23.1 = (22 × R)/(10 × 7)

⇒ R = (2.1 × 70)/2 = 73.5 cm

Area of circle = π × R2

⇒ (22/7) × 73.5 × 73.5

⇒ 22 × 10.5 × 73.5

⇒ 16978.50 cm2

∴ The correct answer is 16978.50 cm2.

Given:

Diameter of pizza = 28cm

Formula:

Circumference of circle = πd

Calculation:

Radius of pizza = 28/2 = 14cm

Total circumference of pizza = 22/7 × 28 = 88cm

Circumference of 3/4 of pizza = 88 × 3/4 = 66cm

∴ Perimeter of remaining pizza = 66 + 14 + 14 = 94cm

Given:

A play ground has a circular path with a certain width around it.

Difference between the circumference of the outer and inner circle is 144 cm

Formula used:

Circumference of a circle = 2πr unit

where r → radius of the circle.

Calculation:

Let the inner radius and outer radius be r cm and R cm respectively.

The width of the path will be (R - r) cm

Difference between the circumference of the outer and inner circle = 144 cm

⇒ 2πR - 2πr = 144

⇒ 2π(R - r) = 144

⇒ R - r = (144 × 7)/44

⇒ R - r = 22.9 ≈ 23

∴ The width of the path is 23 cm.

Given:

The circumference of the two circles is 198 cm and 352 cm respectively.

Concept used:

Circumference of the two circles = 2πr

Where, r = radius

Calculation:

Let the radius of two circle is r₁ & r₂ 

According to the question,

2πr2 - 2πr1 = 352 - 198

⇒ 2π(r2 - r1) = 154

⇒ π(r2 - r1) = 77

⇒ r2 - r1 = 77 × 7/22

⇒ r2 - r1 = 49/2

⇒ r2 - r1 = 24.5

∴ The required answer is 24.5 cm