Math Core

"Acute" and "obtuse" tell you roughly how wide an angle is. To say exactly how wide, you need a unit of measure, just like you use inches for length. For angles, that unit is the degree.

Angles are turns

Picture one ray spinning around its endpoint like the hand of a clock. The angle measures how far it turns. One full trip all the way around a circle is split into 360 equal tiny turns. Each tiny turn is 1 degree, written $1^\circ$ .

Definition

Degree

A one-degree angle turns through $\dfrac{1}{360}$ of a full circle. An angle that turns through $n$ one-degree angles has a measure of $n$ degrees, written $n^\circ$ .

Some turns are worth remembering:

Turn	Degrees	Kind of angle
$\dfrac{1}{4}$ turn	$90^\circ$	right
$\dfrac{1}{2}$ turn	$180^\circ$	straight
full turn	$360^\circ$	a full circle

Sorting angles by degrees

Acute: less than $90^\circ$
Right: exactly $90^\circ$
Obtuse: more than $90^\circ$ but less than $180^\circ$
Straight: exactly $180^\circ$

Worked example: A fraction of a turn

A wheel makes $\dfrac{1}{6}$ of a full turn. How many degrees does it turn?

A full turn is $360^\circ$ . Split it into 6 equal parts: $360 \div 6 = 60$ .

The wheel turns $60^\circ$ .

Using a protractor

A protractor is a tool for measuring angles. It is shaped like half a circle, with marks from $0$ to $180$ . Most protractors have two rows of numbers: one row counts up from the right, and the other counts up from the left.

To measure an angle:

Put the center dot of the protractor on the vertex of the angle.
Line up the bottom edge so that one side of the angle passes through a $0$ .
Find where the other side crosses the marks. Read the number on the row that starts at $0$ on your first side.

Worked example: Reading a protractor

What is the measure of this angle?

One side lies along the 0 on the right. The other side crosses the outer row at 50.

One side points right, through the $0$ of the outer row. So read the outer row. The other side crosses it at $50$ .

The angle measures $50^\circ$ . That is less than $90^\circ$ , so it is acute.

Worked example: An angle that opens to the left

What is the measure of this angle?

One side lies along the 0 on the left. The other side crosses the outer row at 60 and the inner row at 120.

This time the first side points left. The row that has $0$ on the left is the inner row. The other side crosses the inner row at $120$ .

The angle measures $120^\circ$ . Check: the angle is wider than a square corner, so it should be more than $90^\circ$ . It is.

Common mistake

The most common mistake is reading the wrong row of numbers. In the last example, the outer row says $60$ , but the angle is clearly obtuse, so $60^\circ$ can't be right. Before you read, decide if the angle is acute or obtuse. Then pick the number that makes sense.

Worked example: Neither side on zero

The sides of an angle cross the outer row of a protractor at $30$ and at $110$ . What does the angle measure?

The angle covers the marks from $30$ to $110$ . Subtract: $110 - 30 = 80$ .

The angle measures $80^\circ$ .

Practice

Practice 1

How many degrees are in a right angle?

Type a number.

Practice 2

A ray makes a half turn around its endpoint. How many degrees does it turn?

Type a number.

Practice 3

An angle turns through 45 one-degree angles. What is its measure, in degrees?

Type a number.

Practice 4

What kind of angle measures $130^\circ$ ?

Practice 5

What is the measure of this angle, in degrees?

Type a number.

Practice 6

What is the measure of this angle, in degrees?

Type a number.

Practice 7

A spinner turns $\dfrac{1}{3}$ of a full circle. How many degrees does it turn?

Type a number.

Practice 8

What is the measure of this angle, in degrees?

Type a number.