Measuring Space: Perimeter and Area | Exercise 6.1

Question 7

Find the total perimeter of all the petals in each of the given flowers.

Question diagram 1Question diagram 2
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Solution
Understand the Question
  • Each flower petal is bounded by circular arcs.
  • The perimeter of a single petal is the sum of the curved lengths of all the arcs forming that petal.
  • To find the total perimeter of all the petals in a figure, calculate the total length of all boundary arcs formed across the shape using the arc length formula: Arc length=(θ360∘)×2πr\text{Arc length} = \left(\dfrac{\theta}{360^\circ}\right) \times 2\pi r

(i) Find the total perimeter of all the petals in Figure (i) (flower in a square of side 14 cm14\text{ cm}).

Step 1 · Calculate the Total Perimeter of the Petals

Diagram 1

Given the side length of the square is 14 cm14\text{ cm}.

The radius of each circular arc is half of the side length: r=14 cm2=7 cmr = \dfrac{14\text{ cm}}{2} = 7\text{ cm}

Each of the 44 petals consists of two quarter-circle arcs, which equals the length of one semicircle: Perimeter of 1 petal=2×(90∘360∘×2πr)=πr\text{Perimeter of 1 petal} = 2 \times \left(\dfrac{90^\circ}{360^\circ} \times 2\pi r\right) = \pi r

Total perimeter of all 44 petals:

Total perimeter=4×(πr)=4×227×7=4×22=88 cm\begin{aligned} \text{Total perimeter} &= 4 \times (\pi r) \\[0.6em] &= 4 \times \dfrac{22}{7} \times 7 \\[0.6em] &= 4 \times 22 \\[0.6em] &= 88\text{ cm} \end{aligned}
Answer

(i) 88 cm88\text{ cm}

(ii) Find the total perimeter of all the petals in Figure (ii) (flower in a regular hexagon of side 42 cm42\text{ cm}).

Step 1 · Calculate the Total Perimeter of the Petals

Diagram 2

Given the side length of the regular hexagon is 42 cm42\text{ cm}, the radius of each arc is: r=42 cmr = 42\text{ cm}

Each interior arc subtends an angle θ=60∘\theta = 60^\circ at the center.

Length of one arc: Length of one arc=(60∘360∘)×2πr=πr3\text{Length of one arc} = \left(\dfrac{60^\circ}{360^\circ}\right) \times 2\pi r = \dfrac{\pi r}{3}

Each petal consists of 22 such arcs, giving: Perimeter of 1 petal=2×(πr3)=2πr3\text{Perimeter of 1 petal} = 2 \times \left(\dfrac{\pi r}{3}\right) = \dfrac{2\pi r}{3}

Total perimeter of all 66 petals:

Total perimeter=6×(2πr3)=4πr=4×227×42=4×22×6=88×6=528 cm\begin{aligned} \text{Total perimeter} &= 6 \times \left(\dfrac{2\pi r}{3}\right) \\[0.6em] &= 4\pi r \\[0.6em] &= 4 \times \dfrac{22}{7} \times 42 \\[0.6em] &= 4 \times 22 \times 6 \\[0.6em] &= 88 \times 6 \\[0.6em] &= 528\text{ cm} \end{aligned}
Answer

(ii) 528 cm528\text{ cm}

Common Mistakes
  • Radius Misidentification: In Figure (i), the radius is half the side (r=7 cmr = 7\text{ cm}), whereas in Figure (ii), the radius equals the full side length of the hexagon (r=42 cmr = 42\text{ cm}).
  • Counting Arcs: Forgetting that each petal is formed by two curved boundaries/arcs rather than just one.
  • Angle at Center: Using 120∘120^\circ (the interior angle of the hexagon) instead of the 60∘60^\circ angle subtended by the arc at the vertex.

More questions in Exercise 6.1

Q1

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

  1. The perimeter of a circle is 44 cm. What is its radius?
Q2

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

  1. Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius 7 cm

(ii) radius 10 cm

(iii) radius 12 cm.

Q3

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

  1. Calculate the length of the arc of a circle if:

(i) the radius is 3.5 cm3.5\text{ cm} and the angle at the centre is 60∘60^\circ, and

(ii) the radius is 6.3 m6.3\text{ m} and the angle at the centre is 120∘120^\circ.

Q4

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

  1. Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius 14 cm14\text{ cm} and sector angle 75∘75^\circ.
Q5

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

  1. Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14i to 6.14ix):
Q6

If the diameter of a car tyre is 56 cm56\text{ cm}, then:

(i) How far does the car need to travel for the tyre to complete one revolution?

(ii) How many revolutions does the tyre make if the car travels 10 km10\text{ km}?

Q7

Find the total perimeter of all the petals in each of the given flowers.

Q8

The ratio of the perimeters of two circles is 5:45 : 4. What is the ratio of their radii?

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