1. Arcs and Sectors
are
Fractions of a Circle

2. C = πD or C = 2πr For π use the π button on your calculator or 3·14
Remember
The CIRCUMFERENCE of a circle is the distance around the perimeter •
The CIRCUMFERENCE FORMULA is
C = πD or C = 2πr D r
For π use the π button on your calculator or 3·14

3. A = πr2 For π use the π button on your calculator or 3·14
Remember
The AREA of a circle is the space inside the perimeter •
The AREA FORMULA is
A = πr2 r
For π use the π button on your calculator or 3·14

4. A sector is a fraction of the area
Arcs and Sectors
An arc is a fraction of the circumference
arc
Sector
A sector is a fraction of the area

5. The fraction of the circle is a fraction of 360°
Finding the Fraction
The fraction of the circle is a fraction of 360° A B O x

6. To Find the Arc Length AB
Remember diameter
= 2 x radius, hence 34 cm

7. To Find the Area of the Sector
All calculations should have three lines of working

8. Angle in a Semi-circle
When a diameter is drawn and the ends joined to any point on the circumference, the angle formed is always a right angle.

9. Tangent to a Circle tangent tangent
A tangent is a line which just touches the circle
The radius drawn from the centre to the point of contact of the tangent is always perpendicular to the tangent
tangent

10. Tangent Radius Problems
Tangent radius problems involve right angled triangles:
Sum of the angles in a triangle = 180°
Pythagoras: a2 = b2 + c2
SOH CAH TOA

11. Chords and Isosceles Triangles
A line which joins any two points on the circumference of a circle is called a chord •
Chord
Joining the ends of a chord to the centre always creates an isosceles triangle:
2 equal sides, 2 equal angles
Chord

12. Chord, Centre, Perpendicular Bisector Properties
The perpendicular bisector of the chord passes through the centre of the circle
Drawing any radius to the ends of the chord creates a right angled triangle

13. Typical Perpendicular Bisector Problem
The diagram shows a water pipe of radius 50 cm.
The distance across the surface is 84 cm
Calculate the depth of water in the pipe.
Draw bisector and use Pythagoras in ∆OSQ to calculate OS O P Q R S
50 cm
14 cm
42 cm
84 cm
Since OR is another radius, SR, the depth of water, is = 36 cm

14. What I Should Know find the length of an arc of a circle
How to:
find the length of an arc of a circle
find the area of a sector of a circle
How to use the properties of circles:
relationship between tangent and radius
angle in a semi-circle
the interdependence of the centre, bisector of a chord and a perpendicular to a chord