Circles are all around us, from pizzas to Ferris wheels. Their unique properties make them essential in geometry. Understanding how to calculate their circumference and area unlocks a world of practical applications.

Measuring circles involves key formulas using radius or diameter. These allow us to find circumference, area, and even work backwards to determine a circle's size from its measurements. We'll also explore partial circles like semicircles and quarter-circles.

Circumference and Area of Circles

Circumference calculation using diameter or radius

• Circumference represents the distance around the outside edge of a circle (equator, racetrack)
• Formula using diameter $C = \pi d$
  • $C$ circumference
  • $d$ diameter across the widest part of the circle passing through the center (width of a circular table)
  • $\pi$ mathematical constant pi equal to ratio of a circle's circumference to its diameter approximately 3.14159
• Formula using radius $C = 2\pi r$
  • $r$ radius distance from center of circle to outer edge (spoke of a bicycle wheel)
  • Radius equals half the diameter so doubling the radius gives the diameter

Area of circles from radius or diameter

• Area represents the space inside a circle's boundary (pizza, dartboard)
• Formula using radius $A = \pi r^2$
  • $A$ area
  • $r$ radius
  • $r^2$ radius squared by multiplying it by itself (3 cm radius squared equals 9 cm²)
• Formula using diameter $A = \frac{\pi d^2}{4}$
  • $d$ diameter
  • $d^2$ diameter squared by multiplying it by itself (10 inch diameter squared equals 100 in²)
  • Dividing by 4 needed since diameter is twice as long as radius

Radius and diameter from circumference or area

• Find radius from circumference $r = \frac{C}{2\pi}$
  • Dividing circumference by $2\pi$ isolates radius (hula hoop with 6.28 ft circumference has 1 ft radius)
• Find diameter from circumference $d = \frac{C}{\pi}$
  • Dividing circumference by $\pi$ isolates diameter (belt with 31.4 inch circumference has 10 inch diameter)
• Find radius from area $r = \sqrt{\frac{A}{\pi}}$
  • Dividing area by $\pi$ and taking square root isolates radius (circular rug with 78.5 ft² area has 5 ft radius)
• Find diameter from area $d = 2\sqrt{\frac{A}{\pi}}$
  • Dividing area by $\pi$, taking square root, and doubling result gives diameter (circular mirror with 50.24 cm² area has 16 cm diameter)

Perimeter and area of partial circles

• Semi-circle equals half a circle divided along the diameter (folding fan, rainbow)
  • Semi-circle perimeter $P = \frac{1}{2}(\pi d) + d$
    • Adding half the circumference to the diameter (semi-circular window with 4 ft diameter has perimeter of 8.28 ft)
  • Semi-circle area $A = \frac{1}{2}(\pi r^2)$
    • Taking half the area of a full circle (semi-circular flower bed with 3 m radius has area of 14.13 m²)
• Quarter-circle equals one-fourth of a circle divided by two perpendicular radii (piece of pie, corner of a square)
  • Quarter-circle perimeter $P = \frac{1}{4}(\pi d) + 2r$
    • Adding one-fourth the circumference to twice the radius (quarter-circle tabletop with 5 ft diameter has perimeter of 9.71 ft)
  • Quarter-circle area $A = \frac{1}{4}(\pi r^2)$
    • Taking one-fourth the area of a full circle (quarter-circle pizza slice with 6 inch radius has area of 28.26 in²)