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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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A circle is a closed plane figure where every point on the boundary is at a constant distance from a fixed internal point called the center. Visually, it looks like a perfectly round loop with no corners.

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The Radius (rr) is the straight-line segment connecting the center to any point on the boundary. The Diameter (dd) is a straight line passing through the center with both endpoints on the boundary. Visually, the diameter acts as a line of symmetry that cuts the circle into two equal halves, and its length is exactly twice the radius (d=2rd = 2r).

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Circumference (CC) is the total length of the boundary of the circle, also known as its perimeter. If you were to cut the circle at one point and straighten it out into a line, the length of that line would be the circumference.

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The mathematical constant π\pi (Pi) represents the ratio of any circle's circumference to its diameter. Regardless of the circle's size, this ratio remains constant, approximately equal to 227\frac{22}{7} or 3.143.14.

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The Area (AA) of a circle is the total region or space enclosed within its boundary. Visually, if you were to color the entire inside of the circle, the amount of surface covered is the area.

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A Semi-circle is exactly half of a circle, formed by cutting a circle along its diameter. Visually, it looks like a protractor or a half-moon. Its perimeter is unique because it includes the curved boundary (half the circumference) and the straight flat edge (the diameter).

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Concentric Circles are two or more circles that share the same center but have different radii. Visually, this looks like a target or a ring (annulus), where the space between the outer and inner circle forms a circular path.

📐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.