Angle Measure

How to measure angles manually with a tool?

Angles can be measured manually by using a protractor. This is done by placing the protractor on one of the rays, with the 0 value is at the intersection of the two rays (common vertex) and while looking at which value the second ray reaches the protractor.

Representation of the correct way to use a protractor, mathbites.com

As you can see above, the angle formed between the two blue rays is 40°. With a protractor, angles are measured in degrees.

How to measure angles mathematically?

Angles can also be measured mathematically in many different ways. For example, using the fact that all angles along a straight line must add up to 180°, we can work out the values of missing angles.

What is the formula to measure angles?

To find missing angles in polygons, we can work out the sum of the interior angles by using the formula

$sumofinteriorangles=(n-2)\times 180°$,

where n is the number of sides of the polygon. From this, we can find the missing angle.

The sum of all the exterior angles of any polygon is always 360°. This is independent of the number of sides that the polygon has. Therefore, you can also use this fact to find missing exterior angles.

Angles in a triangle can be measured mathematically by using trigonometry. Trigonometry is the field of maths that relates angles and sides in triangles. In a right-angled triangle, for example, if we know the length of two sides of the triangle, we can work out any angle, $\theta$, by using SOH CAH TOA.

How to measure angles in a triangle?

If we have a right-angle triangle as below, and we label one angle θ, we must label the three sides of the triangle Opposite (for the only side that is opposite the angle θ and is not in contact with that angle), Hypotenuse (for the longest side, which is always the one opposite the 90 ° angle) and Adjacent (for the last side).

Labelling the sides of a right-angled triangle, StudySmarter Originals

The sine, cosine and tangent rations each relate the ratio of two sides in a right-angle triangle to one of the angles. To remember which functions involve which sides of the triangle, we use the acronym SOH CAH TOA. The S, C and T stand for Sine, Cosine and Tangent respectively, and the O, A and H for Opposite, Adjacent and Hypotenuse. So the Sine ratio involves the Opposite and the Hypotenuse, and so on.

SOH CAH TOA triangles for remembering trigonometric functions, StudySmarter Originals

All of the ratios sine, cosine and tangent are equal to the sides they involve divided by each other.

$\sin \theta =\frac{opposite}{hypotenuse},\cos \theta =\frac{adjacent}{hypotenuse},\tan \theta =\frac{opposite}{adjacent}$

What are the units for angle measure?

Angles can be measured in degrees and radians. Degrees range between 0 and 360° and radians between 0 and 2π. This unit might be more common, but you can easily convert between the two using the formula

$Radians=degrees\times \frac{\mathrm{\pi }}{180}$

Radians are often expressed in terms of π where possible.

How to measure acute angles?

Let's revisit its definition.

This type of angle can be measured in any of the ways mentioned above, just like obtuse angles or right angles.

An acute angle can be measured with a protractor, using trigonometry (SOH CAH TOA) in a triangle, or using the formula

$\frac{(n-2)\times 180°}{n}$

for regular polygons.