Circles: arcs, sectors, inscribed angles and tangents

Everything starts from the radius

A circle is every point at one fixed distance, the radius $r$, from its centre. The diameter is $2r$. Two formulas carry most questions:

$$C = 2\pi r = \pi d \qquad A = \pi r^2$$

Whatever the question gives you (a diameter, a circumference, an area), turn it into the radius first; from there every other measure is one step away. A circumference of $10\pi$ means $r = 5$ and an area of $25\pi$. An area of $49\pi$ means $r = 7$ and a circumference of $14\pi$.

Answers to circle questions are usually left in terms of $\pi$. When you need a size, $\pi \approx 3.14$, a little more than 3.

Because circumference grows with $r$ and area with $r^2$, doubling the radius doubles the circumference but multiplies the area by four. All circles are similar to one another, so this is the scaling rule from the similarity lesson at work.

Arcs and sectors are fractions of the whole

A central angle has its vertex at the centre. Its two radii cut off an arc of the circle and a sector (a pie slice) of its area. Both are the same fraction of the whole as the angle is of $360^\circ$:

$$\text{arc length} = \frac{\theta}{360} \times 2\pi r \qquad \text{sector area} = \frac{\theta}{360} \times \pi r^2$$

An arc has two different measures, and a question can ask for either. Its degree measure equals the central angle, whatever the size of the circle. Its length is a distance, and depends on the radius. A $60^\circ$ arc is one sixth of any circle, but it is $2\pi$ long on a circle of radius 6 and $\pi$ long on a circle of radius 3.

The perimeter of a sector is the arc plus the two radii. Forgetting the radii is a common way to lose an easy mark.

Inscribed angles

An inscribed angle has its vertex on the circle and its sides along two chords. It measures half the central angle that cuts off the same arc. If central angle $AOB$ is $100^\circ$, then every inscribed angle $ACB$ with $C$ on the far side of the circle (the longer arc) is $50^\circ$, wherever $C$ sits on that arc.

Two consequences come up often:

• An angle inscribed in a semicircle is a right angle. If $AB$ is a diameter, the central angle is $180^\circ$, so any angle $ACB$ with $C$ on the circle is $90^\circ$. A triangle drawn inside a circle with one side along a diameter is therefore a right triangle, and the Pythagorean theorem applies.
• Inscribed angles on the same arc are equal. Points $C$ and $D$ on the same side of chord $AB$ see it at the same angle.

If $C$ is on the short arc instead, angle $ACB$ cuts off the other arc, $360 - 100 = 260^\circ$, and measures $130^\circ$.

Chords, radii and tangents

Any two radii are equal, so a triangle with two vertices on a circle and the third at the centre is isosceles. Look for those triangles whenever a chord and the centre appear together. The line from the centre perpendicular to a chord cuts the chord in half; with the radius as hypotenuse, that is a right triangle.

A tangent touches the circle at exactly one point, and it is perpendicular to the radius drawn to that point. The radius, the tangent segment and the line from the centre to the outside point form a right triangle whose hypotenuse runs from the centre to the outside point.

Describing a circle figure in words

A circle figure is fully described by its centre, its radius or diameter, the points that lie on the circle, and the chords, radii and tangents drawn between them. When you meet one, write down each fact in that form ("$O$ is the centre, $OA = OB = 6$, $PT$ is tangent at $T$") before you start; most of the work is spotting the equal radii and the right angles those facts imply.