Measuring Space: Perimeter and Area | Exercise 6.3

Question 9

A square is inscribed in a circle of radius rr. Show that the ratio of the area of the square to the area of the circle is equal to 2π≈0.637\dfrac{2}{\pi} \approx 0.637.

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Solution
Understand the Question
  • A square inscribed in a circle has its four vertices touching the circle.
  • The diagonal of the inscribed square passes through the center of the circle and is equal to the diameter of the circle (2r2r).
  • To find the ratio of their areas, compute the area of the square using its diagonal and the area of the circle using its radius rr, then divide the area of the square by the area of the circle.

Step 1 · Find the Area of the Inscribed Square

For a circle of radius rr, the diameter is 2r2r. The diagonal of the inscribed square is equal to the diameter of the circle.Diagram 1

Diameter of circle=2r\text{Diameter of circle} = 2r Diagonal of square=2r\text{Diagonal of square} = 2r

Let the side of the square be aa. By Pythagoras theorem in the right triangle formed by two adjacent sides and the diagonal

a2+a2=(diagonal)22a2=(2r)22a2=4r2a2=4r22=2r2\begin{aligned} a^2 + a^2 &= (\text{diagonal})^2 \\ 2a^2 &= (2r)^2 \\ 2a^2 &= 4r^2 \\[0.6em] a^2 &= \dfrac{4r^2}{2} = 2r^2 \end{aligned}

Area of square=a2=2r2\text{Area of square} = a^2 = 2r^2

Step 2 · Calculate the Area of the Circle and the Ratio

The area of the circle is given by Area of circle=πr2\text{Area of circle} = \pi r^2

Now, calculate the ratio of the area of the square to the area of the circle

Ratio=Area of squareArea of circle=2r2πr2=2π\begin{aligned} \text{Ratio} &= \dfrac{\text{Area of square}}{\text{Area of circle}} \\[0.6em] &= \dfrac{2r^2}{\pi r^2} \\[0.6em] &= \dfrac{2}{\pi} \end{aligned}

Using π≈3.14159\pi \approx 3.14159

Ratio≈23.14159≈0.636619≈0.637\begin{aligned} \text{Ratio} &\approx \dfrac{2}{3.14159} \\[0.6em] &\approx 0.636619 \approx 0.637 \end{aligned}
Answer

2π≈0.637\dfrac{2}{\pi} \approx 0.637

Common Mistakes
  • Diagonal vs. Radius: Mistaking the diagonal of the inscribed square for the radius rr instead of the diameter 2r2r.
  • Inverted Ratio: Computing the ratio of the circle's area to the square's area (π2≈1.571\dfrac{\pi}{2} \approx 1.571) instead of square to circle.

More questions in Exercise 6.3

Q1

Unless stated otherwise, use the approximation 227\dfrac{22}{7} for π\pi.

Find the area of a sector of a circle with radius 7 cm7\text{ cm} if the angle of the sector is 60∘60^\circ.

Q2

Find the area of a quadrant of a circle whose circumference is 44 cm44\text{ cm}.

Q3

The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes.

Q4

A chord of a circle of radius 10 cm10\text{ cm} subtends 90∘90^\circ at the centre. Find the area of the corresponding:

(i) minor sector (that subtends 90∘90^\circ at the centre), and (ii) major sector (that subtends 270∘270^\circ at the centre). (Use π≈3.14\pi \approx 3.14.)

Q5

A chord of a circle of radius 15 cm15\text{ cm} subtends an angle of 60∘60^\circ at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use π≈3.14\pi \approx 3.14 and 3≈1.73\sqrt{3} \approx 1.73.)

Q6

A car has two wipers which do not overlap. Each wiper has a blade of length 28 cm28\text{ cm} and sweeps through an angle of 120∘120^\circ. Find the total area cleaned at each sweep of the blades.

Q7

A chord of a circle of radius rr subtends an angle of 60∘60^\circ at the centre of the circle. Show that the area of the corresponding minor segment of the circle is equal to πr2(16−34)\pi r^2 \left( \dfrac{1}{6} - \dfrac{\sqrt{3}}{4} \right).

Q8

An equilateral triangle is inscribed in a circle of radius rr. Show that the ratio of the area of the triangle to the area of the circle is equal to 334π≈0.413\dfrac{3\sqrt{3}}{4\pi} \approx 0.413.

Q9

A square is inscribed in a circle of radius rr. Show that the ratio of the area of the square to the area of the circle is equal to 2π≈0.637\dfrac{2}{\pi} \approx 0.637.

Q10

A hexagon is inscribed in a circle of radius rr. Show that the ratio of the area of the hexagon to the area of the circle is equal to 332π≈0.827\dfrac{3\sqrt{3}}{2\pi} \approx 0.827. Can you see why the answer is exactly twice the answer to Question 8?

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