Circumference and area of a circle

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MODULE: Planimetry (2D Geometry)

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✖️ 1. The definition of Pi (π) as a ratio (C/d)

🥧 The Definition of Pi as a Ratio

  • Pi (π\pi) is the ratio of any circle's circumference to its diameter.
  • Formula: π=Cd\pi = \frac{C}{d} where CC is circumference and dd is diameter.
  • This ratio is always the same for every circle (approximately 3.14159).
  • Pi is irrational so it never ends or repeats.
  • We use π≈3.14\pi \approx 3.14 or π≈227\pi \approx \frac{22}{7} for quick estimates.

If a circle has circumference 31.4 cm and diameter 10 cm, then π≈31.410=3.14\pi \approx \frac{31.4}{10} = 3.14

💡 Pi is the "stretch factor" from diameter to circumference—always about 3 times around!

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1. The definition of Pi (π) as a ratio (C/d)

The Definition of Pi (π) as a Ratio (C/d)

Pi (π\pi) is defined as the ratio of a circle's circumference (CC) to its diameter (dd): π=C/d\pi = C/d. This ratio is constant for all circles, regardless of size.

Intuition: No matter how large or small a circle is, if you measure the distance around it and divide by the distance across its center, you always get the same number: approximately 3.14159.

Core Rules:

  • π\pi is irrational: it cannot be expressed as a fraction of integers and its decimal expansion never repeats
  • π≈3.14159\pi \approx 3.14159 (commonly approximated as 3.14 or 22/7 for calculations)
  • The relationship C=πdC = \pi d holds for every circle
  • Since diameter d=2rd = 2r, we can also write π=C/(2r)\pi = C/(2r)

Consequence: Because π\pi is the same for all circles, it serves as a universal constant linking circular measurements. This allows us to derive formulas for circumference and area.

Example: If a circle has circumference 31.4 cm and diameter 10 cm, then π=31.4/10=3.14\pi = 31.4/10 = 3.14.

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A circular pool has a diameter of 8 meters. Using the approximation π≈3.14\pi \approx 3.14, calculate the circumference of the pool in meters.

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✖️ 2. Circumference formula: C = 2πr

📏 Circumference Formula

  • Circumference is the distance around a circle.
  • Formula: C=2πrC = 2\pi r where rr is the radius.
  • Alternative form: C=πdC = \pi d where dd is the diameter (since d=2rd = 2r).
  • Double the radius means double the circumference.
  • Units of circumference match the units of radius (cm to cm, m to m).

A circle with radius 5 m has circumference C=2π(5)=10π≈31.4C = 2\pi(5) = 10\pi \approx 31.4 m

💡 Think: wrap a string around the circle—that length is 2πr2\pi r!

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2. Circumference formula: C = 2πr

Circumference Formula: C = 2πr

The circumference of a circle is the total distance around its boundary, calculated as C=2πrC = 2\pi r, where rr is the radius. This formula derives directly from C=πdC = \pi d by substituting d=2rd = 2r.

Intuition: The circumference is proportional to the radius—doubling the radius doubles the distance around the circle. The factor 2π2\pi (approximately 6.28) tells us the circumference is always about 6.28 times the radius.

Core Rules:

  • Input: radius rr must be positive (r>0r > 0)
  • Units: CC has the same linear units as rr (e.g., if rr is in meters, CC is in meters)
  • Equivalent form: C=πdC = \pi d where d=2rd = 2r
  • For practical calculations, use π≈3.14\pi \approx 3.14 or the π button on a calculator

Consequence: Knowing either the radius or diameter allows immediate calculation of how far you would travel walking once around the circle.

Example: A circle with radius 5 cm has circumference C=2π(5)=10π≈31.4C = 2\pi(5) = 10\pi \approx 31.4 cm.

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A circle has a radius of 10 meters. Calculate its circumference in meters.

Use π=3.14\pi = 3.14 for your calculation.

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✖️ 3. Area formula: A = πr²

🟢 Area Formula

  • Area is the space inside a circle.
  • Formula: A=πr2A = \pi r^2 where rr is the radius.
  • The radius is squared so doubling radius makes area four times larger.
  • Units are always square units (cm² if radius is in cm).
  • Area grows much faster than circumference as radius increases.

A circle with radius 3 cm has area A=π(3)2=9π≈28.3A = \pi(3)^2 = 9\pi \approx 28.3 cm²

💡 Remember: radius gets squared—area explodes compared to circumference!

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3. Area formula: A = πr²

Area Formula: A = πr²

The area of a circle is the amount of two-dimensional space enclosed within its boundary, given by A=πr2A = \pi r^2, where rr is the radius. The radius is squared, making area grow much faster than circumference as radius increases.

Intuition: Area measures how many square units fit inside the circle. Since area is two-dimensional, it depends on r2r^2 rather than just rr—doubling the radius quadruples the area.

Core Rules:

  • Input: radius rr must be positive (r>0r > 0)
  • Units: AA has square units (e.g., if rr is in meters, AA is in square meters)
  • Growth rate: Area increases with the square of radius, so AA is proportional to r2r^2
  • No diameter form commonly used: convert diameter to radius first (r=d/2r = d/2)

Consequence: Small changes in radius produce large changes in area, which is critical in applications like material usage or land coverage.

Example: A circle with radius 3 m has area A=π(3)2=9π≈28.3A = \pi(3)^2 = 9\pi \approx 28.3 square meters.

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A circular garden has a radius of 5 meters. Using π=3.14\pi = 3.14, calculate the area of the garden in square meters.

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✖️ 4. Calculating arc length and sector area (fractional parts of a circle)

🍕 Arc Length and Sector Area

  • An arc is a piece of the circumference (like a pizza crust slice).
  • Arc length formula: L=θ360∘×2πrL = \frac{\theta}{360^\circ} \times 2\pi r where θ\theta is the central angle in degrees.
  • A sector is a pie-slice shaped region of the circle.
  • Sector area formula: Asector=θ360∘×πr2A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2.
  • Both formulas use the fraction θ360∘\frac{\theta}{360^\circ} of the full circle.

A 90-degree sector of a circle with radius 4 cm has area 90360×π(4)2=14×16π=4π≈12.6\frac{90}{360} \times \pi(4)^2 = \frac{1}{4} \times 16\pi = 4\pi \approx 12.6 cm²

💡 Think pizza slices: angle fraction × full circle measurement!

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4. Calculating arc length and sector area (fractional parts of a circle)

Calculating Arc Length and Sector Area

An arc is a portion of a circle's circumference, and a sector is a pie-slice region bounded by two radii and an arc. Both are calculated as fractions of the full circle based on the central angle θ\theta.

Intuition: If a central angle is θ\theta degrees out of 360 degrees total, then the arc is θ/360\theta/360 of the full circumference, and the sector is θ/360\theta/360 of the full area.

Core Rules:

  • Arc length: L=θ360⋅2πrL = \frac{\theta}{360} \cdot 2\pi r (when θ\theta is in degrees)
  • Sector area: Asector=θ360⋅πr2A_{\text{sector}} = \frac{\theta}{360} \cdot \pi r^2 (when θ\theta is in degrees)
  • Radian form: If θ\theta is in radians, use L=rθL = r\theta and Asector=12r2θA_{\text{sector}} = \frac{1}{2}r^2\theta
  • The fraction θ/360\theta/360 represents the proportion of the circle

Consequence: These formulas allow calculation of partial circular measurements, essential in engineering and design.

Example: A sector with radius 6 cm and central angle 60 degrees has area A=60360⋅π(6)2=6π≈18.8A = \frac{60}{360} \cdot \pi(6)^2 = 6\pi \approx 18.8 square cm.

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Find the arc length of a circle with radius 10 and central angle 90 degrees. Use π≈3.14\pi \approx 3.14.

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✖️ 5. Applications: Designing circular irrigation systems and calculating torque in mechanics

🚜 Applications in Irrigation and Mechanics

  • Circular irrigation systems spray water in a circle from a central pivot.
  • Area watered equals πr2\pi r^2 where rr is the spray radius.
  • Torque (rotational force) depends on distance from the rotation center.
  • Torque formula: τ=r×F\tau = r \times F where rr is radius and FF is force applied.
  • Longer radius means greater torque for the same force (like a longer wrench).

An irrigation system with 50 m radius waters an area of π(50)2=2500π≈7850\pi(50)^2 = 2500\pi \approx 7850 m²

💡 Bigger circles = more coverage in farming, more leverage in mechanics!

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5. Applications: Designing circular irrigation systems and calculating torque in mechanics

Applications: Circular Irrigation Systems and Torque in Mechanics

Circular irrigation systems use rotating sprinklers that water a circular field. The area formula A=πr2A = \pi r^2 determines coverage, while arc length calculations help design partial-circle systems.

Intuition: A center-pivot irrigation system with arm length rr waters a circular area. Knowing AA helps farmers estimate water needs and crop yield. In mechanics, torque τ=rF\tau = rF involves circular motion, where arc length relates to work done.

Core Rules:

  • Irrigation coverage: Area A=πr2A = \pi r^2 gives total watered land; rr is the sprinkler arm length
  • Partial coverage: For a sector angle θ\theta, watered area is θ360⋅πr2\frac{\theta}{360} \cdot \pi r^2
  • Torque and rotation: Work done rotating through angle θ\theta involves arc length s=rθs = r\theta
  • Material costs: Circumference C=2πrC = 2\pi r determines piping or fencing needed

Consequence: These formulas enable precise resource planning in agriculture and accurate force calculations in mechanical systems.

Example: An irrigation system with 200 m radius covers A=π(200)2≈125664A = \pi(200)^2 \approx 125664 square meters of farmland.

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A center-pivot irrigation system has an arm length of 100 meters. Calculate the total circumference of the watered area to determine the length of fencing needed. Use π≈3.14\pi \approx 3.14.

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