The area of a spherical triangle ABC whose parts are

A=95°, B=65°, and C=118° is 2931.6 m2. Find its radius.

A. 43.2 m C. 41.4 m

B. 42.3 m D. 44.1 m

To find the radius of the sphere given the area of the spherical triangle and its angles, we can use the following formula for the area \( A_{\text{sphere}} \) of a spherical triangle:

\[
A_{\text{sphere}} = E \cdot r^2
\]
where:
- \( A_{\text{sphere}} \) is the area of the spherical triangle.
- \( r \) is the radius of the sphere.
- \( E \) is the spherical excess, calculated as:

\[
E = A + B + C - 180^\circ
\]

### Step 1: Calculate the spherical excess \( E \)
Given:
- \( A = 95^\circ \)
- \( B = 65^\circ \)
- \( C = 118^\circ \)

\[
E = 95^\circ + 65^\circ + 118^\circ - 180^\circ = 98^\circ
\]

Convert the spherical excess from degrees to radians, since the formula requires angles in radians:
\[
E = 98^\circ \times \frac{\pi}{180^\circ} = \frac{98\pi}{180} \approx 1.71 \, \text{radians}
\]

### Step 2: Solve for the radius \( r \)

Using the formula for the area of a spherical triangle:
\[
A_{\text{sphere}} = E \cdot r^2
\]
Given \( A_{\text{sphere}} = 2931.6 \, \text{m}^2 \), substitute the values:
\[
2931.6 = 1.71 \cdot r^2
\]

Solve for \( r^2 \):
\[
r^2 = \frac{2931.6}{1.71} \approx 1714.96
\]

Now, take the square root of both sides:
\[
r \approx \sqrt{1714.96} \approx 41.4 \, \text{m}
\]

### Final Answer: 
The radius of the sphere is approximately **41.4 m (Option C)**.

Would you like more details or further clarification?

Here are five related questions you might find interesting:
1. How is the spherical excess related to the area of spherical polygons?
2. Can this method be applied to any spherical triangle, and why?
3. How does the radius affect the size of a spherical triangle's area?
4. What is the difference between plane and spherical triangles in terms of angle sums?
5. Can the formula for spherical triangles be applied to Earth-like calculations?

**Tip:** Always ensure that angles are converted to radians when using trigonometric functions or formulas in spherical geometry.

The area of a spherical triangle ABC whose parts are A = 95°, B = 65°, and C = 118° is 2931.6 m². Find its radius. A. 43.2 m, B. 42.3 m, C. 41.4 m, D. 44.1 m.

Learn how to calculate the radius of a sphere given the area of a spherical triangle with angles A = 95°, B = 65°, and C = 118°. This solution covers key spherical geometry concepts, such as the spherical excess theorem and formulas for the area of spherical triangles. Suitable for students in Grades 10-12.

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