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Mensuration - Area and Circumference of a Circle

Grade 9ICSE

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

🔑Concepts

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A circle is defined as the locus of a point moving in a plane such that its distance from a fixed point (the center) remains constant. Visually, imagine a perfectly round loop where every point on the edge is exactly the same distance, called the radius rr, from the center point OO.

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The diameter dd of a circle is a straight line segment passing through the center and connecting two points on the circumference. Visually, it is the longest possible straight line you can draw inside a circle, effectively cutting the circle into two equal halves. It is always twice the length of the radius (d=2rd = 2r).

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The circumference is the linear distance around the edge of the circle, also known as its perimeter. If you were to 'unroll' the circle into a straight line, the length of that line would be the circumference. It is directly proportional to the diameter by the constant factor π\pi.

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The constant π\pi (Pi) represents the ratio of any circle's circumference to its diameter. For calculations, it is approximately taken as 227\frac{22}{7} or 3.141593.14159. Visually, if you take the diameter and wrap it along the curve of the circle, it would fit slightly more than three times.

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The area of a circle measures the total surface region enclosed within its boundary. Visually, this is the amount of 'space' inside the circle. It can be visualized by dividing the circle into numerous thin wedges (sectors) and arranging them to form a shape that approximates a rectangle with length πr\pi r and height rr.

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A semi-circle is exactly half of a circle. Visually, it looks like a protractor or the letter 'D'. Its perimeter is unique because it includes both the curved boundary (half the circumference) and the straight boundary (the diameter).

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Concentric circles are two or more circles that share the same center point but have different radii. The region lying between two concentric circles is called a ring or an annulus. Visually, this looks like a flat donut or a washer, where the area is the difference between the outer circle's area and the inner circle's area.

📐Formulae

d=2rd = 2r

Circumference(C)=2πr=πdCircumference (C) = 2 \pi r = \pi d

Area(A)=πr2Area (A) = \pi r^2

Area of a Semi−circle=12πr2Area \ of \ a \ Semi-circle = \frac{1}{2} \pi r^2

Perimeter of a Semi−circle=πr+2rPerimeter \ of \ a \ Semi-circle = \pi r + 2r

Area of a Ring(Annulus)=πR2−πr2=π(R2−r2)Area \ of \ a \ Ring (Annulus) = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)

Radius(r)=AπRadius (r) = \sqrt{\frac{A}{\pi}}

💡Examples

Problem 1:

Find the circumference and the area of a circle whose radius is 77 cm. (Take π=227\pi = \frac{22}{7})

Solution:

Step 1: Identify the given values. Given, radius r=7r = 7 cm.

Step 2: Calculate the circumference using the formula C=2πrC = 2 \pi r. C=2×227×7C = 2 \times \frac{22}{7} \times 7 C=2×22=44C = 2 \times 22 = 44 cm.

Step 3: Calculate the area using the formula A=πr2A = \pi r^2. A=227×7×7A = \frac{22}{7} \times 7 \times 7 A=22×7=154A = 22 \times 7 = 154 cm2^2.

Explanation:

This problem demonstrates the direct application of basic circle formulas. We substitute the known radius into the standard equations for circumference and area to find the results.

Problem 2:

The circumference of a circle is 8888 cm. Find its area.

Solution:

Step 1: Use the circumference to find the radius rr. 2πr=882 \pi r = 88 2×227×r=882 \times \frac{22}{7} \times r = 88 447×r=88\frac{44}{7} \times r = 88 r=88×744=2×7=14r = 88 \times \frac{7}{44} = 2 \times 7 = 14 cm.

Step 2: Use the radius to find the area. Area=πr2Area = \pi r^2 Area=227×14×14Area = \frac{22}{7} \times 14 \times 14 Area=22×2×14Area = 22 \times 2 \times 14 Area=44×14=616Area = 44 \times 14 = 616 cm2^2.

Explanation:

In this problem, the radius is not given directly. We must first use the given circumference to solve for the unknown radius. Once the radius is found, we can then proceed to calculate the area.