The formula to calculate the circumradius of a rectangle given its length and the obtuse angle between its diagonals is:
\[ r_c = \frac{l}{2 \sin\left(\frac{\angle_d(\text{Obtuse})}{2}\right)} \]
Where:
The circumradius of a rectangle is the radius of the circle that contains the rectangle with all its vertices lying on the circle.
The length of the rectangle is any one of the pair of parallel sides which are longer than the remaining pair of parallel sides.
The obtuse angle between the diagonals of the rectangle is the angle made by the diagonals which is greater than 90 degrees.
Let's assume the following values:
Using the formula:
\[ r_c = \frac{8}{2 \sin\left(\frac{1.9198621771934}{2}\right)} \approx 4.8831 \, \text{meters} \]
The circumradius is approximately 4.8831 meters.
| Length (meters) | Obtuse Angle (radians) | Circumradius (meters) |
|---|---|---|
| 6 | 1.5 | 4.4012 |
| 6 | 1.6 | 4.1820 |
| 6 | 1.7 | 3.9932 |
| 6 | 1.8 | 3.8298 |
| 6 | 1.9 | 3.6882 |
| 7 | 1.5 | 5.1347 |
| 7 | 1.6 | 4.8790 |
| 7 | 1.7 | 4.6587 |
| 7 | 1.8 | 4.4681 |
| 7 | 1.9 | 4.3028 |
| 8 | 1.5 | 5.8682 |
| 8 | 1.6 | 5.5760 |
| 8 | 1.7 | 5.3242 |
| 8 | 1.8 | 5.1064 |
| 8 | 1.9 | 4.9175 |