Mathematics
Circle and Disk
Length and Area
R is the radius of the circle. Circumference C = 2πR, area of the disk S = πR².
| Part | Formula |
|---|---|
| Arc length (α in degrees) | l = πRα / 180 |
| Sector area | S = πR²α / 360 |
| Segment area | sector area minus triangle area |
Angles
- A central angle is equal to the arc it subtends.
- Inscribed angle = half of the arc.
- Inscribed angles subtending the same arc are equal; an inscribed angle subtending a diameter is 90°.
- The angle between a tangent and a chord is equal to half of the arc.
Chords, Tangent, Secant
- Intersecting chords: AE · EB = CE · ED.
- A tangent is perpendicular to the radius at the point of tangency.
- Two tangent segments drawn from one point are equal.
- Tangent and secant: PT² = PA · PB (PT is the tangent segment, PA is the external part of the secant, PB is the whole secant).
- For intersecting chords, the product is the product of the parts, not of the whole chord.
- An inscribed angle is half the arc, a central angle equals the arc itself: do not equate the inscribed angle to the arc.
Relation to Triangles and Quadrilaterals
- Radius of the inscribed circle: r = S / p (p is the semiperimeter).
- Circumscribed circle: R = abc / 4S. In a right triangle R = c/2, r = (a + b − c)/2 (c is the hypotenuse).
- In an inscribed quadrilateral, the sum of opposite angles is 180°.
- In a circumscribed quadrilateral, the sums of opposite sides are equal: a + c = b + d.
- Mark the central and inscribed angles and the arcs on the diagram
- Choose the suitable theorem: chords, tangent, or the perpendicularity of radius and tangent
- Substitute into the formula and find the unknown
- Check the units (degrees or radians)
Examples
- R = 6, central angle 60°. Sector area 36π · 60 / 360 = 6π.
- Chords intersect, AE = 4, EB = 6, CE = 3: ED = 24 / 3 = 8.
- In a right triangle with legs 6 and 8, the hypotenuse is 10, r = (6 + 8 − 10)/2 = 2.