How are the inscribed and the central angle related?

The inscribed angle is half the central angle standing on the same arc: α = 2β. It follows that all inscribed angles on the same arc are equal, wherever on the circle their vertex sits. The relation works both ways, so from the inscribed angle you double, and from the central one you halve.

Why is an angle over a diameter a right angle?

Because the central angle over a diameter is a straight angle of 180°, and the inscribed angle is half of it — hence 90°. This is Thales’s theorem on the circle, and its practical use is this: any triangle inscribed in a circle with a diameter as one of its sides is a right triangle.

What are the properties of a tangent to a circle?

A tangent meets the circle in exactly one point and is perpendicular to the radius drawn to the point of tangency. From a point outside the circle two tangents can be drawn, and the segments from that point to the two points of tangency are equal, with length t = √(d² − r²).

How do you compute arc length and sector area?

Both are the fraction of the whole circle set by the central angle: l = 2πr · α / 360° and P = πr² · α / 360°. For 90° that gives a quarter of the circumference and a quarter of the area. The only thing to keep straight is that an arc is measured in units of length and a sector in units of area.

What is the difference between an incircle and a circumcircle?

The incircle lies inside the polygon and touches all of its sides — its radius is the apothem, the distance from the centre to a side. The circumcircle passes through every vertex, so its radius is the distance from the centre to a vertex. For a regular polygon the two share a centre, and the circumradius is always the larger of the two.

Intermediate level

The circle: angles, tangent and regular polygons

An inscribed angle is always half the central angle on the same arc — and from that one sentence follow the right angle over a diameter, the construction of a tangent and half of every exam question about circles. Plus chords, tangent segments, regular polygons, and the length of an arc and the area of a sector.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Where this is used

Real situations where you count exactly the way this lesson teaches:

A clock and the angle between the hands
A clock face is a circle cut into 12 equal arcs of 30°. At 4 o’clock the hands are separated by a central angle of 4 · 30° = 120°, and on a face of radius 12 cm the arc between them measures 2π · 12 · 120 / 360 ≈ 25.1 cm.
CNC milling an arc
An operator cuts an arc of radius 250 mm through 72°. The tool path is 2π · 250 · 72 / 360 ≈ 314 mm, and at a feed rate of 600 mm/min that move takes about 31 seconds — which is exactly how cycle time is estimated before the machine starts.
Transmitter range and a tangent segment
A transmitter stands 15 km from the centre of a protected zone of radius 9 km. The tangent segment — the farthest point of that zone seen "along the tangent" — measures √(15² − 9²) = √144 = 12 km. The same 9, 12, 15 is an ordinary Pythagorean triple.
A nut and a spanner
An M10 nut is a regular hexagon with a 17 mm across-flats size — that is the diameter of the incircle, so r = 8.5 mm. The circumradius, the distance from the centre to a corner, is R = 2r / √3 ≈ 9.81 mm, and it is that number that decides whether the nut fits into a recess.
Pie charts and sectors
A sector representing 15% of the data takes an angle of 0.15 · 360° = 54°. On a chart of radius 5 cm its area is π · 25 · 54 / 360 ≈ 11.8 cm², exactly 15% of the whole disc — which is the reason a pie chart works as a picture of proportions at all.

All formulas

Central and inscribed angle
$\alpha = 2\beta$
both standing on the same arc
Angle over a diameter
$\beta = 90^\circ \iff AB \ \text{is a diameter}$
Thales's theorem on the circle
Tangent segment
$t^2 = d^2 - r^2$
d is the distance from the point to the centre
Chord and its distance from the centre
$\left(\frac{c}{2}\right)^2 + h^2 = r^2$
the perpendicular from the centre bisects the chord
Interior angle of a regular polygon
$\alpha_n = \frac{(n - 2) \cdot 180^\circ}{n}$
the central angle is 360°/n
Arc length
$l = 2\pi r \cdot \frac{\alpha}{360^\circ}$
the fraction of the circumference set by the angle
Area of a sector
$P = \pi r^2 \cdot \frac{\alpha}{360^\circ}$
the same fraction, of the area this time

This lesson is about the circle, the line itself, rather than about the disc, the region inside it. The circumference and the area are already known — what matters here is what happens on the circle: angles, chords, tangents, and the polygons that can be inscribed in it.

The whole topic follows from one theorem, so that is where to start.

An inscribed angle is half the central one

A central angle has its vertex at the centre of the circle; an inscribed angle has it on the circle itself. If both stand on the same arc, the relation that is the heart of this lesson holds:

\alpha = 2\beta

The central angle ASB and the inscribed angle ACB stand on the same arc AB. The angle at C is exactly half the angle at S.

A second property follows at once — less obvious and more useful: all inscribed angles standing on the same arc are equal. It makes no difference where on the circle the vertex is placed; as long as the arms pass through the same two points, the measure is the same.

A central angle standing on arc AB is 130°. What is the inscribed angle on the same arc?

An angle over a diameter is 90°

Take the special case of an arc that is half the circle. The central angle over a diameter is a straight angle, $180^\circ$ , so the inscribed angle is half of it:

\beta = \frac{180^\circ}{2} = 90^\circ

Any triangle inscribed in a circle with a diameter as one side has a right angle at the third vertex. The right-angle mark is not declared here — it follows from where the points sit.

This is Thales's theorem on the circle, and in exercises it acts as a switch: the moment a diameter and a point on the circle appear in the statement, a right triangle appears in the drawing, and with it the whole Pythagorean theorem.

A chord and its distance from the centre

A chord is a segment joining two points of the circle. The perpendicular drawn from the centre bisects the chord, so a right triangle appears with legs $\tfrac{c}{2}$ and $h$ and hypotenuse $r$ :

\left(\frac{c}{2}\right)^2 + h^2 = r^2

A chord of a circle of radius 13 cm lies 5 cm from the centre. How long is the chord?

The tangent

A tangent meets the circle in exactly one point and is perpendicular to the radius drawn to the point of tangency. That is all there is to know about it — the rest is consequence.

The tangent k at T and the radius ST meet at a right angle. A line closer to the centre than r cuts the circle twice, and one further than r not at all.

From a point outside the circle two tangents can be drawn, and the segments from that point to the points of tangency are equal. Since the radius and such a segment meet at a right angle, Pythagoras gives the length of the tangent segment:

t^2 = d^2 - r^2

where $d$ is the distance from the point to the centre.

The relative position of a line and a circle therefore comes down to comparing one number with the radius:

Distance of the line from the centre	Common points	Name
$h < r$	two	secant
$h = r$	one	tangent
$h > r$	none	line missing the circle

Regular polygons, incircle and circumcircle

A regular polygon has all sides and all angles equal. A circle can be circumscribed about it (through the vertices, of radius $R$ ) and one inscribed in it (touching the sides, of radius $r$ , called the apothem) — both with the same centre.

A regular hexagon. The radii to two neighbouring vertices form a central angle of 360° / 6 = 60°, and the apothem meets the middle of a side at a right angle.

Two angles are enough to describe such a figure:

\text{central angle} = \frac{360^\circ}{n}, \qquad \alpha_n = \frac{(n - 2) \cdot 180^\circ}{n}

For a triangle ( $n = 3$ ) that gives $60^\circ$ , for a square $90^\circ$ , for a hexagon $120^\circ$ . The more sides, the closer the angle to $180^\circ$ — and the more the polygon looks like a circle.

Find the circumradius of a regular hexagon with side 6 cm.

Arc length and sector area

A central angle $\alpha$ cuts an arc out of the circle and a sector out of the disc. Both are the same fraction of the whole, given by the ratio $\tfrac{\alpha}{360^\circ}$ :

l = 2\pi r \cdot \frac{\alpha}{360^\circ}, \qquad P = \pi r^2 \cdot \frac{\alpha}{360^\circ}

There is nothing to memorise here beyond the circumference and the area of a circle: $90^\circ$ is a quarter, $180^\circ$ a half, $60^\circ$ a sixth.

Find the arc length and the sector area for an angle of 120° in a circle of radius 9 cm.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

the inscribed angle, when the central one is 80°

Common mistakes

Doubling the inscribed angle instead of halving the central one — it is the inscribed angle that is half, not the other way round.
Stopping at half the chord — the right triangle inside the circle gives $\tfrac{c}{2}$ , so the result still has to be doubled.
Confusing the inradius with the circumradius — the apothem meets the middle of a side, the circumradius reaches a vertex.
Computing an arc with the area formula — an arc comes from the circumference $2\pi r$ , a sector from the area $\pi r^2$ .
Assuming a tangent is perpendicular to a chord — a tangent is perpendicular to the radius at the point of tangency.
Looking for a right angle where the chord is not a diameter — Thales's theorem on the circle holds for a diameter only.

Formula card

Topic: The circle: angles, tangent, regular polygons

Central and inscribed angle
$\alpha = 2\beta$
both standing on the same arc
Angle over a diameter
$\beta = 90^\circ \iff AB \ \text{is a diameter}$
Thales's theorem on the circle
Tangent segment
$t^2 = d^2 - r^2$
d is the distance from the point to the centre
Chord and its distance from the centre
$\left(\frac{c}{2}\right)^2 + h^2 = r^2$
the perpendicular from the centre bisects the chord
Interior angle of a regular polygon
$\alpha_n = \frac{(n - 2) \cdot 180^\circ}{n}$
the central angle is 360°/n
Arc length
$l = 2\pi r \cdot \frac{\alpha}{360^\circ}$
the fraction of the circumference set by the angle
Area of a sector
$P = \pi r^2 \cdot \frac{\alpha}{360^\circ}$
the same fraction, of the area this time

The central angle ASB and the inscribed angle ACB stand on the same arc AB. The inscribed one is always half of the central one.

A regular hexagon with its circumcircle of radius R and its incircle of radius r. The central angle is 360° / 6 = 60°.

Frequently asked questions

A luminous right triangle with squares built on its sides drawn as a constellation, beside a circle with its radius, against a cosmic nebula in violet and magenta

Geometry

Why we measure area in squares

Perimeter is counted in metres, area in square metres — and that single exponent is behind most of the mistakes people make in plane geometry. It is why 1 m² is not 100 cm², why two plots with identical area can need wildly different amounts of fencing, and why the Pythagorean theorem is really a statement about the areas of squares rather than the lengths of sides.

11 min