During the celebration of John's birthday, his mother Emma was keen to ensure every guest received an identical portion of the cake. To achieve this, the cake needed to be sliced at consistent angles. However, the challenge arises in accurately measuring these angles.

This guide will delve into the principle of angle measurement.

An **angle** is defined as the space created by the intersection of two rays at the point where they meet.

The term **angle measure** signifies the technique used to ascertain the magnitude or specific value of an angle where two rays converge at a shared point. This determination can be carried out by hand or through mathematical computation.

## Manually measuring angles using an instrument?

To manually measure angles, one can employ a **protractor**. This involves positioning the protractor along one ray, ensuring the zero mark aligns with the junction of the two rays (shared point), and observing the value on the protractor where the second ray crosses.

Demonstration of proper protractor usage, mathbites.com

As demonstrated, the angle created by the two blue rays measures 40°. Measurement of angles with a protractor is expressed in **degrees**.

## Mathematically measuring angles?

Angles can be mathematically measured using various methods. For instance, knowing that the sum of angles along a straight line equals 180° allows for the calculation of unknown angles.

Determine the magnitude of $x$.

**Solution**

Given the two angles are connected by a straight line, they sum up to 180°, hence $x=180-109=71\xb0$.

## What formula is used to measure angles?

The sum of interior angles in **polygons** can be calculated using the formula

$sumofinteriorangles=(n-2)\times 180\xb0$,

with **n** representing the polygon’s side count. This aids in identifying missing angles.

Calculate the angle x.

**Solution**

Observing the given shape has 6 edges, indicating a hexagon.

Thus, the total interior angle sum is

$(6-2)\times 180\xb0=720\xb0$

Knowing the magnitude of the remaining angles allows for calculation of x.

$x=720-(138+134+100+112+125)=111\xb0$

The **sum of all exterior angles** of any polygon consistently equals 360°, regardless of side count, which can be used to determine missing exterior angles.

Angles within a triangle can be measured mathematically through **trigonometry**, a branch of mathematics correlating angles and triangle sides. Utilizing known side lengths in a right-angled triangle, any angle, $\theta $, can be calculated using SOH CAH TOA.

**Methods for measuring triangle angles**

In a right-angled triangle as presented, labeling one angle as θ mandates the identification of the three sides as **Opposite** (the side facing θ without touching it), **Hypotenuse** (the longest side, opposite the 90° angle), and **Adjacent** (the side next to θ).

The ratios of **sine, cosine,** and** tangent** connect two triangle sides to a specific angle.

To recall which functions correspond to which sides, the mnemonic **SOH CAH TOA** is used. Here, S stands for Sine, C for Cosine, T for Tangent, and O, A, H represent Opposite, Adjacent, and Hypotenuse respectively. For instance, the Sine ratio includes the Opposite and Hypotenuse.

These ratios, sine, cosine, and tangent, are calculated by dividing the lengths of the involved sides.

Evaluate the angle θ.

**Solution**

In this diagram, the hypotenuse measures 9 cm and the adjacent 4 cm, allowing for the cosine value of θ to be determined.

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

To find the specified angle, use the ${\mathrm{cos}}^{-1}$ key on a calculator with 0.444 as input, yielding a result of 63.6°.

## What units are used in angle measurement?

Angles are quantified in **degrees** and **radians**. Degrees are measured from 0 to 360°, and radians from 0 to 2π. Though degrees are widely used, conversion between them and radians is straightforward using the equation

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

Where feasible, radians are ideally stated in terms of π.

A 45° angle in a triangle translates to what in radians?

**Solution**

Applying the aforementioned conversion formula yields

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

## Measuring acute angles?

Reviewing its definition,

An **acute angle** is one that is less than 90° in measure.

Such angles can be determined using any previously mentioned methods, analogous to obtuse or right angles.

An acute angle's measurement is possible with a protractor, employing trigonometry (SOH CAH TOA) in triangles, or through the formula

$\frac{(n-2)\times 180\xb0}{n}$

for regular polygons.

## Key points on Angle Measure

- Angle measure involves determining the magnitude of an angle formed by two lines, either manually or through mathematical methods.
- Angles are traditionally measured using a protractor when done by hand
- For any polygon, the sum of the interior angles equals $(n-2)\times 180\xb0$, where n is the count of sides, and exterior angle sums always equal 360°
- Within a right-angled triangle, SOH CAH TOA is applicable to deduce any angle's value
- Angles measured in degrees or radians, where $radians=degrees\times \frac{\mathrm{\pi}}{180}$