**Figure (1): Angle given in the question statement**

The aim of this question is to develop the **ability to estimate angles to the nearest half radian** just by visualizing them.

To estimate such angles, we need to **imagine a circular scale** of our choice in accordance with our required **precision**.

If we **choose a circular grading** of $ \dfrac{ 1 }{ 2 } \pi $ radians, then the **scale** looks something like the following **figure (2)**:

**Figure (2): Angles with a circular grading of $ \dfrac{ 1 }{ 2 } \pi $ radians**

Where 1, 2, 3 and 4 represent the angles $ \dfrac{ 1 }{ 2 } \pi, \ \pi, \ \dfrac{ 3 }{ 2 } \pi, \text{ and } 2 \pi $ radians, respectively.

Similarly, if we **choose a circular grading** of $ \dfrac{ 1 }{ 2 } \pi $ radians, then the **scale looks** something like the following **figure (3)**:

F**igure (3): Angles with a circular grading of $ \dfrac{ 1 }{ 4 } \pi $ radians**

Where 1, 2, 3, 4, 5, 6, 7, and 8 represent the angles $ \dfrac{ 1 }{ 4 } \pi, \ \dfrac{ 1 }{ 2 } \pi, \ \dfrac{ 3 }{ 4 } \pi, \pi, \dfrac{ 5 }{ 4 } \pi, \ \dfrac{ 3 }{ 2 } \pi, \ \dfrac{ 7 }{ 4 } \pi, \ \text{ and } 2 \pi $ radians, respectively.

In practice, we use the **protractor scale** to **estimate the angles** to the **nearest degree** in the lab or in the field. Since **modern drawing applications** use state-of-the-art **computer software**, such scales have very little use in the industry.

## Expert Answer

Drawing the **gird angles with a circular grading** of $ \dfrac{ 1 }{ 4 } \pi $ radians on top of the given angle is drawn below in **figure (4)**:

**Figure (4): Given angle with a circular grading of $ \dfrac{ 1 }{ 4 } \pi $ radians**

Now here we can easily **visualize** that the **nearest half angle** when the circular grading is $ \dfrac{ 1 }{ 4 } \pi $ radians can be **approximated to** the $ 2^{ nd } $ grading which is in-turn **equal to** the $ \dfrac{ 1 }{ 4 } \pi $ radians.

## Numerical Result

\[ \text{ Estimated angle } \ = \ \dfrac{ 1 }{ 4 } \pi \ radians\]

## Example

Estimate the **nearest half angle** of the following angle:

**Figure (5): Angle given in the example statement**

Drawing the **gird angles with a circular grading** of $ \dfrac{ 1 }{ 4 } \pi $ radians on top of the given angle is drawn below in **figure (6)**:

**Figure (6): Given angle with a circular grading of $ \dfrac{ 1 }{ 4 } \pi $ radians**

Now here we can easily **visualize** that the **nearest half angle** when the circular grading is $ \dfrac{ 1 }{ 4 } \pi $ radians can be **approximated to** the $ 4^{ th } $ grading which equals the $ \dfrac{ 3 }{ 4 } \pi $ radians.

*Images/Mathematical drawings are created with Geogebra.*