Circles: Circumference and Area
Master the key circle formulas β circumference, area, arc length, and sector area β and apply them to real-world problems involving wheels, pipes, and pizza.
Circles are everywhere in the built world: wheels, pipes, tanks, plates, domes, and satellite dishes. Engineers calculate the capacity of cylindrical tanks using circle area. Mechanics work with gear circumferences and wheel diameters. Chefs calculate pizza area to compare sizes. Understanding circles β and their one irrational constant Ο β is essential for any technical or practical field.
- Calculate circumference and area of a circle from radius or diameter
- Find arc length and sector area from a central angle
- Apply circle formulas to real-world problems
ΞΈ = central angle in degrees; r = radius
A circle has a radius of 7 cm. Find its circumference and area.
\[ C = 2\pi(7) = 14\pi \approx 43.98 \text{ cm} \] \[ A = \pi(7)^2 = 49\pi \approx 153.94 \text{ cm}^2 \]Leave answers in exact form (\(14\pi\), \(49\pi\)) when asked for exact values, or use Ο β 3.14159 for decimal approximations.
A circle has radius 10 m. Find the arc length and sector area for a 72Β° central angle.
\[ L = \frac{72}{360} \times 2\pi(10) = \frac{1}{5} \times 20\pi = 4\pi \approx 12.57 \text{ m} \] \[ A_s = \frac{72}{360} \times \pi(10)^2 = \frac{1}{5} \times 100\pi = 20\pi \approx 62.83 \text{ m}^2 \]A 10-inch pizza (diameter) costs $10. A 14-inch pizza costs $16. Which is the better value?
- 10-inch: radius = 5 in, area = \(\pi(5)^2 = 25\pi \approx 78.5\) inΒ² β cost per inΒ² = \(\$10/78.5 \approx \$0.127\)
- 14-inch: radius = 7 in, area = \(\pi(7)^2 = 49\pi \approx 153.9\) inΒ² β cost per inΒ² = \(\$16/153.9 \approx \$0.104\)
The 14-inch pizza is the better value at about $0.10/inΒ² vs. $0.13/inΒ². Area scales with radius squared β the larger pizza gives nearly twice the area for only 1.6Γ the price.
- A circle has diameter 18 ft. Find its circumference and area.
- Find the arc length of a 120Β° arc in a circle with radius 6 cm.
- A circular pool has radius 12 ft. What is the area of the pool cover needed?
βΆ Show Answers
- r = 9 ft; \(C = 18\pi \approx 56.55\) ft; \(A = 81\pi \approx 254.47\) ftΒ².
- \(L = \frac{120}{360} \times 2\pi(6) = \frac{1}{3} \times 12\pi = 4\pi \approx\) 12.57 cm.
- \(A = \pi(12)^2 = 144\pi \approx\) 452.4 ftΒ².
- Using diameter instead of radius in the area formula: \(A = \pi r^2\) uses the radius. If you're given the diameter, divide by 2 first.
- Squaring Ο instead of r: \(\pi r^2\) means \(\pi \times r^2\), NOT \((\pi r)^2\). Square only the radius.
- Confusing circumference (linear) and area (square) units: Circumference is in cm, ft, etc. Area is in cmΒ², ftΒ², etc.
- Circumference = \(2\pi r\) (or \(\pi d\)) β the "perimeter" of a circle.
- Area = \(\pi r^2\) β always use the radius, not the diameter.
- Arc and sector are fractions of the full circle: multiply by \(\frac{\theta}{360Β°}\).
- Ο β 3.14159 β for exact answers, leave in terms of Ο; for decimals, use Ο on your calculator.
Mechanical engineers calculate the displacement of engine pistons (circular cross-sections), gear ratios (based on circumference), and pipe flow rates (cross-sectional area). Civil engineers size culverts and water mains using pipe area. A pipe with twice the radius carries four times the volume flow β because area scales as rΒ². This non-linear relationship is why understanding circle area matters in engineering design.
Calculator Connection
The Circle Calculator computes circumference, area, diameter, and radius from any single input. For arc length, use the Arc Length Calculator; for sector area, use the Sector Area Calculator.
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Apply what you've learned with these tools.
Circles: Circumference and Area β Quiz
5 questions per attempt Β· Beginner Β· Pass at 70%
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