Circles

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Parts
of a
Circle

Perimeters $C = π d$

Areas $A = π r^{2}$

Circle Theorems

Equation of a Circle $x^{2} + y^{2} = r^{2}$

The Circle (Simple Wiki).: Measurement of a Circle: a treatise of three propositions by Archimedes.

Circumference and other Perimeters

Circumference

C = π d = 2 π r

where
$C$ is the circumference of the circle;
$d$ is the diameter of the circle;
and $r$ is the radius of the circle.

Practice Questions
Textbook Exercise: (care of corbettmaths.com)

A Definition for π.

The above gives us a simple definition for π

π = \frac{C}{d}

That is, if we take the measurements of the curcumference and diameter, then find their ratio. This ratio is an empirical value for π

Other definitions can be found at en.wikipedia.org

Arc Length

The length of the arc $AB$ formed by a sector is is given by

l = \frac{π r θ}{180}

Practice Questions
Textbook Exercise: (care of corbettmaths.com)

Parts of a Circle
(care of www.cuemath.com)

parts of a circle: radius, diameter, circumference, chord, sector, segment, tangent, arc

Circles: parts: Practice Questions; Textbook Exercise; (care of corbettmaths.com)

Circles: perimeter - semi-circle: Practice Questions; Textbook Exercise; (care of corbettmaths.com)

Circles: perimeter - quarter-circle: Practice Questions; (care of corbettmaths.com)

Equation of a Circle

Prerequiste materials:

Parts of a Circle

Pythagorus's Theorem

Formula that Scribes a Circle

For a circle centered on the origin with a radius, $r$ :

$x^{2} + y^{2} = r^{2}$

Videos and Questions:

Equation of a Circle Video: Practice Questions; Textbook Exercise

Equation of a circle centred at $(a, b)$ in the cartesian plan

{(x - a)}^{2} + {(y - b)}^{2} = r^{2}

If you multiply out the brackets this equation takes the form of

x^{2} + y^{2} + j x + k y + l = 0

To go from this form to the original just complete the square twice once with respect to $x$ and then with respect to $y$ .

Questions
(Care of www.revisely.co.uk)

The parametric equation of a circle is given by the following two formulae:

x = a + r \cos θ

y = b + r \sin θ

where
$θ$ is the angle around the circle between the x-axis and the radius, $r$ to the point $(x, y)$ ;
and
$(a, b)$ are the co-ordinates of the centre of the circle;

Questions:

Question 1

Given

x = 4 + 10 \cos θ

y = 7 + 10 \sin θ

Write down the co-ordinates of the centre of the circle and its radius.

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