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Measurement - Circumference and Area of Circles

Grade 6IB

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

🔑Concepts

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The Radius (rr) is the distance from the center of a circle to any point on its boundary, while the Diameter (dd) is a straight line passing through the center connecting two points on the boundary.

A circle showing the relationship between radius and diameter.
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The Circumference (CC) is the perimeter or distance around the circle. It is calculated using the formula C=2πrC = 2 \pi r or C=πdC = \pi d.

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The Area (AA) measures the region enclosed within the circle's boundary. It is calculated using the formula A=πr2A = \pi r^2.

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The constant π\pi (Pi) represents the ratio of the circumference to the diameter. For calculations, we commonly use π≈3.14\pi \approx 3.14 or π≈227\pi \approx \frac{22}{7}.

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A Semicircle is exactly half of a circle. Its area is 12πr2\frac{1}{2} \pi r^2 and its curved boundary is πr\pi r.

A semicircle diagram.

📐Formulae

d=2rd = 2r

r=d2r = \frac{d}{2}

C=πdC = \pi d

C=2πrC = 2 \pi r

A=πr2A = \pi r^2

π≈3.14\pi \approx 3.14 or 227\frac{22}{7}

💡Examples

Problem 1:

A circular clock has a radius of 7cm7 cm. Calculate its circumference. (Use π=227\pi = \frac{22}{7})

Solution:

  1. Identify the given value: r=7cmr = 7 cm.
  2. Choose the circumference formula involving radius: C=2πrC = 2 \pi r.
  3. Substitute the values: C=2×227×7C = 2 \times \frac{22}{7} \times 7.
  4. Simplify: The 77 in the numerator and denominator cancel out, leaving C=2×22C = 2 \times 22.
  5. Calculate the final value: C=44cmC = 44 cm.

Explanation:

Since the radius is a multiple of 77, using the fraction 227\frac{22}{7} for π\pi makes the calculation easier through cancellation.

Problem 2:

A circular garden has a diameter of 10m10 m. Find the total area of the garden. (Use π=3.14\pi = 3.14)

Solution:

  1. Identify the given value: d=10md = 10 m.
  2. Find the radius (rr): r=d2=102=5mr = \frac{d}{2} = \frac{10}{2} = 5 m.
  3. Use the area formula: A=πr2A = \pi r^2.
  4. Substitute the values: A=3.14×(5)2A = 3.14 \times (5)^2.
  5. Calculate the square: 52=255^2 = 25.
  6. Multiply: A=3.14×25=78.5m2A = 3.14 \times 25 = 78.5 m^2.

Explanation:

Always convert diameter to radius first before calculating area, as the area formula requires r2r^2 rather than d2d^2.

Problem 3:

A circular table top has a diameter of 1.4m1.4 m. Find its circumference using π=227\pi = \frac{22}{7}.

Circle with diameter labeled 1.4m

Solution:

Given d=1.4md = 1.4 m and π=227\pi = \frac{22}{7}. C=πdC = \pi d C=227×1.4C = \frac{22}{7} \times 1.4 C=22×0.2C = 22 \times 0.2 C=4.4mC = 4.4 m

Explanation:

To find the circumference when the diameter is known, we multiply the diameter by π\pi. Since 1.41.4 is a multiple of 77 (0.7×20.7 \times 2), using the fraction 227\frac{22}{7} makes the calculation simpler.

Problem 4:

A circular ring has an inner radius of 66 cm and an outer radius of 1010 cm. Calculate the area of the shaded region (the ring) between the two circles. (Use π=3.14\pi = 3.14)

Concentric circles representing a ring with inner radius 6 cm and outer radius 10 cm.

Solution:

Outer radius (R)=10 cm\text{Outer radius } (R) = 10 \text{ cm} Inner radius (r)=6 cm\text{Inner radius } (r) = 6 \text{ cm} Area of Outer Circle =πR2=3.14×102=3.14×100=314 cm2\text{Area of Outer Circle } = \pi R^2 = 3.14 \times 10^2 = 3.14 \times 100 = 314 \text{ cm}^2 Area of Inner Circle =πr2=3.14×62=3.14×36=113.04 cm2\text{Area of Inner Circle } = \pi r^2 = 3.14 \times 6^2 = 3.14 \times 36 = 113.04 \text{ cm}^2 Area of Ring=Outer Area−Inner Area\text{Area of Ring} = \text{Outer Area} - \text{Inner Area} Area of Ring=314−113.04=200.96 cm2\text{Area of Ring} = 314 - 113.04 = 200.96 \text{ cm}^2

Explanation:

To find the area of the ring (annulus), we calculate the area of the larger circle and subtract the area of the smaller inner circle. Both calculations use the formula A=πr2A = \pi r^2.