Measuring angles

A full turn around a vertex can be written $360^{\circ }$ or $2\pi$. Later pages of this math handbook write $\pi /2$, $\pi$, $3\pi /5$. If you only know $360$, those look like new angles. They are the same turn, read in the unit the circle itself supplies. This page writes angles in radians unless a degree sign is shown.

Counterclockwise is extra

An angle is two rays that share a vertex. Measuring it still needs a zero and a sign. From the positive $x$-axis, counterclockwise is positive and clockwise is negative. That convention is not in the two rays; it is added so a directed turn has one number.

The first board marks a right angle $\gamma$ (a quarter turn), a positive angle $\alpha$, and the smaller wedge toward $N$. Drag $P$ or $N$.

Three hundred and sixty pieces

A degree cuts a full turn into $360$ equal parts. A right angle is a quarter turn:

$1\text{ turn}=360^{\circ },90^{\circ }=\frac{\pi }{2}.$

The split into $360$ pieces, like the $60$ minutes in an hour, comes from base-$60$ counting. It is a useful ruler. It is not a length you can read off the circle.

The arc that matches the radius

A radian is geometric. On a circle of radius $r$, a central angle of $\alpha$ radians cuts an arc of length $\alpha r$. When that arc equals the radius, the angle is one radian.

The board below freezes an extra condition: both radii stay length $2$, so this is not the unit circle. Drag $A$; the thick arc is still as long as one radius.

On the unit circle the arc length is the angle. A full turn is the circumference $2\pi$. That is why later formulas write $\pi$ for a straight angle and $2\pi$ for a turn — they are counting in this unit, not translating $180$ after the fact.

The same turn, two readings

A degree measure ${\alpha }^{\circ }$ is the fraction ${\alpha }^{\circ }/360$ of a turn, so

$\alpha =2\pi \cdot \frac{{\alpha }^{\circ }}{360}={\alpha }^{\circ }\cdot \frac{\pi }{180},{\alpha }^{\circ }=\alpha \cdot \frac{180}{\pi }.$

The table is that one conversion, not a second topic.

Finer work — surveying, astronomy — splits a degree the same way an hour is split. Minutes and seconds of arc are a finer degree ruler, not a third geometric unit:

$1^{\circ }=60^{\prime },1^{\prime }=60^{\prime \prime },1^{\circ }=3600^{\prime \prime }.$

In radians,

$1^{\prime }=\frac{\pi }{10800}\approx 0.000291,1^{\prime \prime }=\frac{\pi }{648000}\approx 4.848\times 10^{-6}.$

A parsec is the distance at which one astronomical unit subtends one arcsecond. It is about $3.26$ light years. (The Sun’s disk is about $30^{\prime }$ across on the sky.) Drop that extra $60$-split and the radian formulas above do not change.

A triangle already fills a straight angle

In the Euclidean plane the interior angles of a triangle sum to a straight angle. A straight angle is $\pi$ in the unit just defined — that is the debt the conversion table paid. The formula is not a new $180^{\circ }$ fact:

$\alpha +\beta +\gamma =\pi .$

Drag a vertex. The three marks change; their sum does not. The formula is for triangles in the Euclidean plane of the next figures.

A quadrilateral splits into two triangles, so its interior angles sum to $2\pi$. An $n$-gon splits into $n-2$ triangles:

$\text{interior sum}=(n-2)\pi .$

A pentagon sums to $3\pi$; a hexagon to $4\pi$. That sum does not need equal sides.

The last board starts with an extra assumption: a regular pentagon, so each interior angle is $3\pi /5$. Drag a vertex. Regularity is gone; five interior angles still add to $3\pi$. The per-corner $3\pi /5$ was the extra freeze, not the sum.

The unit is now the one later pages use. Areas and perimeters of plane figures start with triangles, then quadrilaterals and regular n-gons.