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Mensuration - Circles: Circumference and Area

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The diameter of a circle is twice its radius, and the radius is the distance from the center to any point on the boundary.

A circle showing radius from center to edge and diameter across the center.
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The circumference is the perimeter or boundary length of the circle, calculated using C=2πrC = 2\pi r.

A circle illustrating the boundary as the circumference.
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The area is the region enclosed within the circle, calculated as A=πr2A = \pi r^2.

A filled circle representing the enclosed area.
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A semi-circle is half a circle. Its perimeter includes the curved arc length πr\pi r and the straight diameter 2r2r.

A semi-circle showing the curved part and the diameter.
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A ring or annulus is the region between two concentric circles. Area =π(R2−r2)= \pi(R^2 - r^2).

Two concentric circles showing inner radius r and outer radius R.

📐Formulae

d=2rd = 2r

r=d2r = \frac{d}{2}

Circumference (C)=2πr=πd\text{Circumference } (C) = 2 \pi r = \pi d

Area (A)=πr2\text{Area } (A) = \pi r^2

π≈227≈3.14\pi \approx \frac{22}{7} \approx 3.14

Perimeter of a Semi-circle=πr+2r=r(π+2)\text{Perimeter of a Semi-circle} = \pi r + 2r = r(\pi + 2)

Area of a Semi-circle=12πr2\text{Area of a Semi-circle} = \frac{1}{2} \pi r^2

Area of a Ring (Annulus)=π(R2−r2)\text{Area of a Ring (Annulus)} = \pi(R^2 - r^2), where RR is the outer radius and rr is the inner radius

💡Examples

Problem 1:

Find the circumference and the area of a circle whose radius is 10.510.5 cm. (Take π=227\pi = \frac{22}{7})

Solution:

Given: r=10.5r = 10.5 cm = 212\frac{21}{2} cm.

  1. To find Circumference: C=2πrC = 2 \pi r C=2×227×212C = 2 \times \frac{22}{7} \times \frac{21}{2} C=22×3=66 cmC = 22 \times 3 = 66 \text{ cm}

  2. To find Area: A=πr2A = \pi r^2 A=227×(10.5)2A = \frac{22}{7} \times (10.5)^2 A=227×110.25A = \frac{22}{7} \times 110.25 A=227×4414A = \frac{22}{7} \times \frac{441}{4} A=11×632=6932=346.5 cm2A = \frac{11 \times 63}{2} = \frac{693}{2} = 346.5 \text{ cm}^2

Explanation:

We use the standard formulas for circumference and area by substituting the given radius. Converting decimal 10.510.5 to fraction 212\frac{21}{2} makes the calculation with 227\frac{22}{7} easier through cancellation.

Problem 2:

The circumference of a circle is 176176 cm. Find its radius and its area.

Solution:

Given: C=176C = 176 cm.

  1. Find Radius (rr): C=2πrC = 2 \pi r 176=2×227×r176 = 2 \times \frac{22}{7} \times r 176=447×r176 = \frac{44}{7} \times r r=176×744r = \frac{176 \times 7}{44} r=4×7=28 cmr = 4 \times 7 = 28 \text{ cm}

  2. Find Area (AA): A=πr2A = \pi r^2 A=227×28×28A = \frac{22}{7} \times 28 \times 28 A=22×4×28A = 22 \times 4 \times 28 A=88×28=2464 cm2A = 88 \times 28 = 2464 \text{ cm}^2

Explanation:

First, we rearrange the circumference formula to solve for the unknown radius rr. Once the radius is found, we substitute it into the area formula to calculate the total space enclosed.

Problem 3:

A circular park of radius 2121 m has a 77 m wide path running around it on the outside. Find the area of the path.

A circular park with an outer path showing radii.

Solution:

Inner radius (rr) = 2121 m Width of path = 77 m Outer radius (RR) = 21+7=2821 + 7 = 28 m Area of path = Outer Area−Inner Area\text{Outer Area} - \text{Inner Area} Area =πR2−πr2=π(R2−r2)= \pi R^2 - \pi r^2 = \pi(R^2 - r^2) Area =227×(282−212)= \frac{22}{7} \times (28^2 - 21^2) Area =227×(784−441)= \frac{22}{7} \times (784 - 441) Area =227×343= \frac{22}{7} \times 343 Area =22×49=1078= 22 \times 49 = 1078 m2^2

Explanation:

To find the area of the path, we subtract the area of the smaller inner circle from the area of the larger outer circle formed by adding the path's width to the radius.

Problem 4:

The area of a semi-circular plate is 7777 cm2^2. Find its perimeter. (Take π=227\pi = \frac{22}{7})

A semi-circular plate with its area labeled.

Solution:

Area of semi-circle = 12πr2=77\frac{1}{2} \pi r^2 = 77 12×227×r2=77\frac{1}{2} \times \frac{22}{7} \times r^2 = 77 117×r2=77\frac{11}{7} \times r^2 = 77 r2=77×711=7×7=49r^2 = \frac{77 \times 7}{11} = 7 \times 7 = 49 r=49=7r = \sqrt{49} = 7 cm Perimeter of semi-circle =πr+2r= \pi r + 2r Perimeter =(227×7)+(2×7)= (\frac{22}{7} \times 7) + (2 \times 7) Perimeter =22+14=36= 22 + 14 = 36 cm

Explanation:

First, use the area formula for a semi-circle to find the radius. Then, use the perimeter formula, which accounts for both the curved boundary and the straight diameter.