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Measuring Space: Perimeter and Area - π Is Irrational

Grade 9CBSE

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

🔑Concepts

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The number π\pi (pi) is defined as the ratio of the circumference CC of a circle to its diameter dd, such that π=Cd\pi = \frac{C}{d}.

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π\pi is an irrational number. This means it cannot be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0.

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The decimal expansion of π\pi is non-terminating and non-recurring (3.14159265...3.14159265...).

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For practical calculations, we use rational approximations like 227\frac{22}{7} or 3.143.14, but these are not the exact values of π\pi.

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The perimeter of a circle is called its circumference. For a circle of radius rr, the circumference is C=2πrC = 2\pi r.

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The area of a circle with radius rr is given by A=πr2A = \pi r^2.

📐Formulae

π=CircumferenceDiameter\pi = \frac{\text{Circumference}}{\text{Diameter}}

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

Area(A)=πr2\text{Area} (A) = \pi r^2

Perimeter of a semi-circle=πr+2r\text{Perimeter of a semi-circle} = \pi r + 2r

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

💡Examples

Problem 1:

Calculate the circumference and area of a circular garden whose radius is 14 m14 \text{ m}. (Take π=227\pi = \frac{22}{7})

Solution:

Given radius r=14 mr = 14 \text{ m}. Using the formula for circumference: C=2πrC = 2\pi r C=2×227×14C = 2 \times \frac{22}{7} \times 14 C=2×22×2=88 mC = 2 \times 22 \times 2 = 88 \text{ m}

Using the formula for area: A=πr2A = \pi r^2 A=227×14×14A = \frac{22}{7} \times 14 \times 14 A=22×2×14=616 m2A = 22 \times 2 \times 14 = 616 \text{ m}^2

Explanation:

We apply the standard circular measurement formulae by substituting the given radius and the rational approximation of π\pi.

Problem 2:

If the circumference of a circle is 31.4 cm31.4 \text{ cm}, find its radius. (Take π=3.14\pi = 3.14)

Solution:

Given C=31.4 cmC = 31.4 \text{ cm} and π=3.14\pi = 3.14. We know C=2πrC = 2\pi r 31.4=2×3.14×r31.4 = 2 \times 3.14 \times r 31.4=6.28×r31.4 = 6.28 \times r r=31.46.28r = \frac{31.4}{6.28} r=5 cmr = 5 \text{ cm}

The radius of the circle is 5 cm5 \text{ cm}.

Explanation:

To find the radius, we rearrange the circumference formula r=C2πr = \frac{C}{2\pi} and substitute the known values.

Problem 3:

Explain why π\pi is irrational even though we often use 227\frac{22}{7} in calculations.

Solution:

An irrational number has a non-terminating and non-recurring decimal expansion. The actual value of π\pi is 3.14159265...3.14159265... which continues forever without a repeating pattern. The fraction 227\frac{22}{7} is a rational number (its decimal is 3.142857‾3.\overline{142857}), which is used only as a convenient approximation. Since π≠227\pi \neq \frac{22}{7}, the use of the fraction does not make π\pi rational.

Explanation:

This distinguishes between the exact mathematical definition of an irrational constant and the rational approximations used for engineering and school mathematics.