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# Angle Measure

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At John's birthday party, his mom Emma wanted to ensure that the guests have equal cake pieces. In order to be able to achieve this, the cake should be cut at equal angles. But how can we measure these angles?

In this article, we will explain the concept of the angle measure.

An angle is the space between two intersecting rays at the space at which they meet.

Angle measure refers to the process of determining the size, a specific value, of an angle formed between two rays at a common vertex. This can be done manually or mathematically through calculations.

## 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.

Find the value of $x$.

Solution

The two angles in the diagram must add up to 180° as they are on a straight line, so we have $x=180-109=71°$.

## 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

$sumofinteriorangles=\left(n-2\right)×180°$,

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

Find the value of the angle x.

Solution

You can see that the shape above has 6 sides, it is a hexagon.

Therefore the sum of the interior angles is

$\left(6-2\right)×180°=720°$

As we know the values of all the other angles, we can work out x.

$x=720-\left(138+134+100+112+125\right)=111°$

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.

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

Find the value of the angle θ.

Solution

From this diagram, we can see that hypotenuse = 9 cm and adjacent = 4 cm. Therefore we can calculate the cos value of the angle θ .

$\mathrm{cos}\theta =\frac{4}{9}=0.444$

To now find the angle itself, you will need to press the ${\mathrm{cos}}^{-1}$button on your calculator and put in 0.444. This will give an answer of 63.6°.

## 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

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

Radians are often expressed in terms of π where possible.

An angle in a triangle was measured to be 45°. What is this in radians?

Solution

Using the formula above, we find that

$radians=45×\frac{\mathrm{\pi }}{180}=\frac{\mathrm{\pi }}{4}$

## How to measure acute angles?

Let's revisit its definition.

An acute angle is an angle that measures less than 90°.

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

$\frac{\left(n-2\right)×180°}{n}$

for regular polygons.

## Angle Measure - Key takeaways

• Angle measure refers to the process of determining the value of an angle formed between two lines. This can be done manually or mathematically.
• Manually, a protractor can be used to measure angles
• In any polygon, the sum of interior angles is $\left(n-2\right)×180°$ where n is the number of sides and the sum of exterior angles is always 360°
• In a right angle triangle SOH CAH TOA can be used to calculate the value of any angle
• Angles can be measured in degrees or radians, where $radians=degrees×\frac{\mathrm{\pi }}{180}$

The measure of an angle can be determined manually, using a protractor or mathematically, for example by using SOH CAH TOA in a triangle.

Measuring an angle with a protractor can be done by placing the protractor on one of the lines, with the 0 value at the intersection of the two lines and looking at which value the second line reaches the protractor.

If you know the value of the interior angle, then the exterior angle = 360° – interior angle.

The measure of an angle is the size of the angle. It is the a particular distance between the two intersecting rays that form the angle.

We measure angles manually, using a protractor, or mathematically through calculations.

## Final Angle Measure Quiz

Question

What are the units of angles?

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Question

What is the sum of exterior angles in any polygon?

360°

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Question

How do you measure angles manually?

Using a protractor

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Question

What is the sum of interior angles in a hexagon?

720°

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Question

What is the sum of interior angles in a pentagon?

540°

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Question

An octagon has seven angles 135°, 154°, 122°, 144°, 132°, 119°, 142°. What is the value of the eighth angle?

132°

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Question

A pentagon has four exterior angles 67°, 75°, 81°, 68°. What is the value of the fifth angle?

69°

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Question

Find the value of the angle in a right angled triangle with an opposite side of 9cm and an adjacent side of 8cm.

43.4°

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Question

Find the value of the angle in a right angled triangle with an opposite side of 9cm and an hypotenuse of 18cm.

30°

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