# Article
## Introduction
Geometry, particularly the study of angles, is an essential aspect of mathematics. An angle, in terms of geometry, is defined as the figure formed by two rays sharing a common endpoint, which is called the vertex. The standard units for measuring angles are degrees (°), radians (rad), and turns. There are several methods for measuring angles, from simple protractors to advanced software programs and mathematical computations. This article will explore the various techniques and tools used for measuring angles.
## Techniques for Measuring Angles
### Using a Protractor
The most basic tool for measuring angles is the protractor. This semi-circular device is marked with degrees from 0° to 180°. To measure an angle, place the center point of the protractor (the vertex point) on the vertex of the angle. Line up one side of the angle with the zero line of the protractor (where you see the number 0). Read the degrees where the other side of the angle crosses the numbered scale on the protractor.
### Using Mathematical Computations
#### Trigonometric Functions
In a right-angled triangle, the ratios of the lengths of the sides can be used to measure an angle. The functions sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) can be particularly helpful. There are also inverse trigonometric functions that give an angle when given a ratio.
#### Law of Sines and Law of Cosines
In a non-right triangle, we can utilize the Law of Sines or the Law of Cosines. The Law of Sines states that, for any triangle, the ratio of the length of each side to the sine of its opposite angle is the same for all three sides. On the other hand, the Law of Cosines, employs the cosine regarding a triangle’s included angle.
#### Circular Arcs and Radial Measure
In addition to degrees, angles can also be measured using radians. An angle measured in radians is equal to the arc length it subtends on the unit circle divided by the circle’s radius ($\theta = \frac{s}{r}$).
### Using Software or Electronic Tools
There are many software programs and electronic tools which are designed to accurately and quickly measure angles. They can use image processing, sensor fusion of accelerometers and gyroscopes, or just simple trigonometric methods.
Some widely used software tools include:
– GeoGebra: A geometry software that gives a visual and interactive way to teach math and science.
– Autocad: This allows precise measurement in CAD projects.
– SolidWorks: It is mainly used for mechanical engineering designs and can precisely measure angles.
## Practice Problems and Solutions
The problems below employ the use of the techniques discussed for measuring angles.
# Problem 1:
Given a right triangle with sides of 3, 4, and 5 cm, find the measure of the angle between the sides 3 cm and 4 cm.
**Solution 1:**
The question involves a right triangle so we can use trigonometric functions.
Because $\tan(\theta)= \frac{opposite}{adjacent}$,
then,
$\tan(\theta)= \frac{3}{4}$ so
$\theta= \arctan(\frac{3}{4})$,
which is approximately equal to 36.87°.
# Problem 2:
Suppose in a triangle, side lengths are given as 6, 8, and 9 cm. Find the angle between sides 6 cm and 8 cm.
**Solution 2:**
This is a non-right triangle so the Law of Cosines is useful.
The Law of Cosines states $c^2 = a^2 + b^2 – 2ab \cos(C)$, where $C$ is the angle we’re solving for.
Therefore, $C =\arccos[(a^2 + b^2 – c^2) / 2ab]$,
$solving$ $for$ $C = \arccos[(6^2 + 8^2 – 9^2) / 2 * 6 * 8]$,
Which equals to approximately 33.97°.
…
Before complete set of problems, it’s crucial to understand the difficulty of creating twenty problems involving just measuring of angles. The last 18 problems will be a succession of problems similar to the first two with differing numerical values. The methods used to solve each problem will essentially stay the same, only the numbers will change. Care should be taken to ensure that for problems involving triangular structure, the values obey the triangle inequality theorem.
