A circle is the set of all points that are the same distance from a center point. Let's explore its properties and measurements.
Center: Fixed point in middle
Radius (r): Distance from center to edge
Diameter (d): Distance across center (d = 2r)
Circumference: Distance around circle
Arc: Part of circumference
Chord: Straight line between two points
Circumference: C = πd or C = 2πr
Area: A = πr²
π (pi) ≈ 3.14159...
Use π button on calculator or 3.14 for approximations
Circle theorems are powerful rules about angles and lengths in circles.
Angle in semicircle: Always 90°
Angles in same segment: Equal
Angle at center: Twice angle at circumference
Cyclic quadrilateral: Opposite angles sum to 180°
Tangent-radius: Meet at 90°
Two tangents from point: Equal length
Alternate segment: Tangent angle equals angle in alternate segment
We can find the length of arcs and areas of sectors using angles.
Arc length = (θ/360°) × 2πr
Where θ = angle at center in degrees
For angle in radians: Arc = θr
Sector area = (θ/360°) × πr²
Where θ = angle at center in degrees
For angle in radians: Area = ½θr²
1. Find the circumference of a circle with radius 7 cm (use π = 3.14)
cm2. Find the area of a circle with radius 5 cm (use π = 3.14)
cm²