Express the angular measurement of the angle of a grades and radians.

Answer

To express the angular measurement of an angle in grades and radians, we need to understand the relationship between these units of angular measurement. Here’s a step-by-step solution:

### Step 1: Understand the Definitions
- **Radians**: A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
- **Degrees**: A degree is a measure of angle equal to 1/360 of a full circle.
- **Gradians (or Grades)**: A gradian is defined as the angle subtended at the center of a circle by an arc whose length is equal to 1/400 of the circumference of the circle.

### Step 2: Establish the Relationships
1. **Radians to Degrees**:
\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
2. **Degrees to Radians**:
\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \]
3. **Gradians to Degrees**:
\[ 1 \text{ gradian} = \frac{9}{10} \text{ degrees} \]
4. **Degrees to Gradians**:
\[ 1 \text{ degree} = \frac{10}{9} \text{ gradians} \]

### Step 3: Convert Between Radians and Gradians
To express an angle in radians in terms of gradians, we can use the relationships established above.

1. **Convert Radians to Degrees**:
\[ \text{Angle in degrees} = \text{Angle in radians} \times \frac{180}{\pi} \]

2. **Convert Degrees to Gradians**:
\[ \text{Angle in gradians} = \text{Angle in degrees} \times \frac{10}{9} \]

### Step 4: Combine the Conversions
To convert an angle from radians directly to gradians:
\[ \text{Angle in gradians} = \text{Angle in radians} \times \frac{180}{\pi} \times \frac{10}{9} \]

### Step 5: Final Formula
Thus, the final formula to convert radians to gradians is:
\[ \text{Angle in gradians} = \text{Angle in radians} \times \frac{200}{\pi} \]

### Summary
- To convert an angle from radians to gradians, multiply the angle in radians by \(\frac{200}{\pi}\).