Circle basics

A circle is measured from its centre. The radius $r$ is the distance from the centre to the edge, and the diameter is twice that, $2r$. The distance all the way around the edge is the circumference, $C = 2\pi r$, and the space inside is the area, $A = \pi r^2$.

For example, a circle of radius $5$ has circumference $2\pi \times 5 = 10\pi$ and area $\pi \times 5^2 = 25\pi$.

A slice of a circle is a sector, and part of the edge is an arc. The central angle $\theta$ of a sector is a fraction of the full turn of $360^\circ$, and the sector takes exactly that fraction of the whole circle.
For a central angle of $90^\circ$, the fraction is $\frac{90}{360} = \frac{1}{4}$, so the sector is a quarter of the area and the arc is a quarter of the circumference.

In general, the sector area is $\frac{\theta}{360} \times \pi r^2$ and the arc length is $\frac{\theta}{360} \times 2\pi r$.

A useful fact about chords: if you draw a perpendicular from the centre of a circle to a chord, it cuts the chord exactly in half.
In the figure below, the perpendicular from $O$ meets the chord $AB$ at $D$, so $AD = DB$.

A quadrilateral whose four corners all lie on a circle is called a cyclic quadrilateral, and its opposite angles always add up to $180^\circ$.
In the figure below, $a + c = 180^\circ$ and $b + d = 180^\circ$.

A circle has radius $7$. What is its area?
Use $A = \pi r^2$:
$\pi \times 7^2 = 49\pi$.

A circle has radius $4$. What is its circumference?
Use $C = 2\pi r$:
$2\pi \times 4 = 8\pi$.

A sector has central angle $120^\circ$ in a circle of radius $6$. What is its area?
The fraction of the circle is $\frac{120}{360} = \frac{1}{3}$.
Sector area $= \frac{1}{3} \times \pi \times 6^2 = \frac{1}{3} \times 36\pi = 12\pi$.

In a circle of radius $5$, an arc has central angle $90^\circ$. What is its length?
The fraction is $\frac{90}{360} = \frac{1}{4}$.
Arc length $= \frac{1}{4} \times 2\pi \times 5 = 2.5\pi$.

A chord $AB$ is $16$ long, and a perpendicular is drawn to it from the centre $O$, meeting it at $D$. What is $AD$?
The perpendicular from the centre bisects the chord, so $AD = \frac{16}{2} = 8$.

A cyclic quadrilateral has one angle of $70^\circ$. What is the angle opposite it?
Opposite angles of a cyclic quadrilateral add to $180^\circ$.
So the opposite angle is $180 - 70 = 110^\circ$.