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The figure shows 6 circles, each of radius 7 cm. Each circle touches the circles next to it. (Take pi = 22/7). (a) Find the perimeter of the shaded part. (b) Find the area of the shaded part.

Mathematics

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2 parts
a

88 cm\text{88 cm}

b

308 cm2\text{308 cm}^2

Step-by-step solution

8 steps

Here's how to solve this problem:

Given:
Radius of each circle, r=7r = 7 cm.
Use π=227\pi = \frac{22}{7}.

Part a· 4 steps

Find the perimeter of the shaded part.

  1. Identify the components of the perimeter.

    The perimeter of the shaded part consists of arcs from the circles.
    There are 4 quarter-circle arcs (from the top-left, top-right, bottom-left, and bottom-right shaded regions) and 2 semi-circle arcs (from the middle-left and middle-right shaded regions).

  2. Calculate the length of one quarter-circle arc.

    The circumference of a full circle is 2πr2\pi r.
    Length of one quarter-circle arc = 14×2πr=12πr\frac{1}{4} \times 2\pi r = \frac{1}{2}\pi r.

  3. Calculate the length of one semi-circle arc.

    Length of one semi-circle arc = 12×2πr=πr\frac{1}{2} \times 2\pi r = \pi r.

  4. Calculate the total perimeter.

    Total perimeter = 4×(length of a quarter-circle arc)+2×(length of a semi-circle arc)4 \times (\text{length of a quarter-circle arc}) + 2 \times (\text{length of a semi-circle arc})

    Perimeter=4×(12πr)+2×(πr)\text{Perimeter} = 4 \times \left(\frac{1}{2}\pi r\right) + 2 \times (\pi r) Perimeter=2πr+2πr=4πr\text{Perimeter} = 2\pi r + 2\pi r = 4\pi r

    Substitute the given values for rr and π\pi:

    Perimeter=4×227×7\text{Perimeter} = 4 \times \frac{22}{7} \times 7 Perimeter=4×22=88 cm\text{Perimeter} = 4 \times 22 = 88 \text{ cm}

    The perimeter of the shaded part is 88 cm\boxed{\text{88 cm}}.

Part b

308 cm2\text{308 cm}^2

Find the area of the shaded part.

  1. Identify the components of the area.

    The shaded part consists of 4 quarter-circles (from the top-left, top-right, bottom-left, and bottom-right shaded regions) and 2 semi-circles (from the middle-left and middle-right shaded regions).

  2. Calculate the area of one quarter-circle.

    The area of a full circle is πr2\pi r^2.
    Area of one quarter-circle = 14πr2\frac{1}{4}\pi r^2.

  3. Calculate the area of one semi-circle.

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

  4. Calculate the total area.

    Total area = 4×(area of a quarter-circle)+2×(area of a semi-circle)4 \times (\text{area of a quarter-circle}) + 2 \times (\text{area of a semi-circle})

    Area=4×(14πr2)+2×(12πr2)\text{Area} = 4 \times \left(\frac{1}{4}\pi r^2\right) + 2 \times \left(\frac{1}{2}\pi r^2\right) Area=πr2+πr2=2πr2\text{Area} = \pi r^2 + \pi r^2 = 2\pi r^2

    Substitute the given values for rr and π\pi:

    Area=2×227×(7)2\text{Area} = 2 \times \frac{22}{7} \times (7)^2 Area=2×227×49\text{Area} = 2 \times \frac{22}{7} \times 49 Area=2×22×497\text{Area} = 2 \times 22 \times \frac{49}{7} Area=2×22×7\text{Area} = 2 \times 22 \times 7 Area=44×7=308 cm2\text{Area} = 44 \times 7 = 308 \text{ cm}^2

    The area of the shaded part is 308 cm2\boxed{\text{308 cm}^2}.

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