1. **State the problem:** Calculate the area of spherical triangles given their angles and the radius of the sphere. 2. **Formula for the area of a spherical triangle:** $$ Area = E \times r^2 $$ where spherical excess $$ E = (A + B + C) - \pi $$ radians, with angles $$ A, B, C $$ in radians, and $$ r $$ is the radius. 3. **Convert angles from degrees to radians:** $$ \text{radians} = \text{degrees} \times \frac{\pi}{180} $$ 4. **Solve for each problem:** --- **Problem 6:** Angles 80°, 70°, 50°; $$ r=10 $$ m - Convert angles to radians: $$ A=80^\circ \times \frac{\pi}{180} = \frac{4\pi}{9} $$ $$ B=70^\circ \times \frac{\pi}{180} = \frac{7\pi}{18} $$ $$ C=50^\circ \times \frac{\pi}{180} = \frac{5\pi}{18} $$ - Calculate spherical excess: $$ E = (A+B+C) - \pi = \left(\frac{4\pi}{9} + \frac{7\pi}{18} + \frac{5\pi}{18}\right) - \pi = \left(\frac{8\pi}{18} + \frac{7\pi}{18} + \frac{5\pi}{18}\right) - \pi = \frac{20\pi}{18} - \pi = \frac{2\pi}{18} = \frac{\pi}{9} $$ - Area: $$ Area = E \times r^2 = \frac{\pi}{9} \times 10^2 = \frac{100\pi}{9} \approx 34.9 $$ Mistake: Check carefully, redo addition - A+B+C in degrees: 80+70+50=200°, subtract 180° (π rad) => excess = 20° Convert 20° to radians: $$ 20^\circ \times \frac{\pi}{180} = \frac{\pi}{9} $$ Area: $$ 10^2 \times \frac{\pi}{9} = 100 \times \frac{\pi}{9} = \frac{100\pi}{9} \approx 34.9 \text{ m}^2 $$ This is too low compared to choices, likely spherical triangle excess formula was used incorrectly. Spherical excess formula for area on sphere is: $$ Area = r^2 \times E $$ where $$ E = (A+B+C) - \pi $$ with angles in radians. Converting from degrees: $$ A=80^\circ=1.3963\,rad, B=70^\circ=1.2217\,rad, C=50^\circ=0.8727\,rad $$ Sum: $$ 1.3963 + 1.2217 + 0.8727 = 3.4907 $$ Excess: $$ E = 3.4907 - 3.1416 = 0.3491 $$ Area: $$ 0.3491 \times 10^2 = 34.91 $$ Still not matching choices. Note: The area of spherical triangle is $$ E r^2 $$ where E in radians. The given answers are much larger, possibly answer choices are surface areas computed from different formula or misinterpretation. Given options and repeated calculations indicate an error: Verify the formula. Actual spherical excess is: $$ E = (A + B + C) - \pi $$ Area: $$ r^2 \times E $$ With angles in radians. Repeat for Problem 6: Convert 80°,70°,50°: $$ 80\times \pi/180 = 1.3963 $$ $$ 70\times \pi/180 = 1.2217 $$ $$ 50\times \pi/180 = 0.8727 $$ Sum: $$ 3.4907 $$ Excess: $$ 3.4907 - 3.1416 = 0.3491 $$ Area: $$ 0.3491 \times 100 = 34.91 $$ Seems low compared to choices. Possibly the sphere's radius is 10 m, but area unit is squared meters, so 34.9 m² should be correct. Check options again: A. 87.3 B. 78.5 C. 65.4 D. 94.2 None close to 34.9. Are we missing any coefficient or formula variant? Upon reviewing, the formula is correct. But options are about twice the correct area. Double check if radius squared is correctly calculated: $$ 10^2 = 100 $$ Correct. Angles sum: 200° Spherical excess: $$ 20^\circ = 20 \times \pi/180 = \pi/9 \approx 0.3491 $$ Area: $$ 100 \times 0.3491 = 34.91 $$ Confirming. Conclusion: Possible that question expects the use of formula: $$ Area = (A + B + C - \pi) \times r^2 $$ Here, must use angles in radians. Similarly for problems 7 to 10. To align with answer choices, multiply area by 2 (surface area of sphere is $$4\pi r^2$$). Hence, the correct formula for area of spherical triangle: $$ Area = E \times r^2 $$ Where $$ E = A + B + C - \pi $$ radians. Calculations for all problems done likewise: **Problem 7:** Angles: 90°, 60°, 50° Sum: 200° Excess: 20° = $$\frac{\pi}{9}$$ Radius $$r=8$$ cm Area: $$ 8^2 \times \frac{\pi}{9} = 64 \times 0.3491 = 22.34 $$ cm² (Not in choices, so options could be multiplied by 2 or π) **Problem 8:** Angles: 100°, 60°, 40° Sum: 200° Excess: 20° Radius = 12 m Area: $$ 12^2 \times 0.3491 = 144 \times 0.3491 = 50.27 $$ m² (Not matching options) **Problem 9:** Angles: 75°, 65°, 55° Sum: 195° Excess: 15° Radius = 6 cm Excess in radians: $$15\times \frac{\pi}{180} = 0.2618$$ Area: $$6^2 \times 0.2618 = 36 \times 0.2618 = 9.424 $$ cm² (Too low) **Problem 10:** Angles: 85°,60°,45° Sum: 190° Excess: 10° Excess in radians: $$10\times \frac{\pi}{180} = 0.1745$$ Radius = 5 m Area: $$ 5^2 \times 0.1745 = 25 \times 0.1745 = 4.36 $$ m² Since direct calculations give smaller numbers, likely area formula involves: $$ \text{Area} = E \times r^2 $$ But answer choices suggest: $$ \text{Area} = E \times R^2 $$ where angles are in degrees multiplied by \frac{\pi}{180} * a constant. Assuming answers are from formula: $$ Area = E (\text{in degrees}) \times \frac{\pi}{180} \times r^2 $$ Check problem 6: $$ E = 20^{\circ} $$ Area: $$ 20 \times \frac{\pi}{180} \times 100 = \frac{20\pi}{180} \times 100 = \frac{20\pi}{180} \times 100 = 34.91 $$ Still same. No missing factor. Thus, nearest matching choice is not calculated; closest is to match area 87.3 from choice for problem 6 by multiplying by 2.5 approx. This inconsistency suggests using formula: $$ Area = (A + B + C - \pi) \times r^2 $$ with radians and trust calculated results. **Final answers:** 6. Area = 34.91 m² Answer: none of the above (closest is A: 87.3) 7. Area = 22.34 cm² Answer: none 8. Area = 50.27 m² Answer: C: 50.3 9. Area = 9.42 cm² Answer: none 10. Area= 4.36 m² Answer: none Select best matching options where available. --- **Summary of steps**: 1. Convert angles from degrees to radians. 2. Calculate spherical excess $$ E = A + B + C - \pi $$ 3. Calculate area $$ Area = E \times r^2 $$ 4. Round result and select closest answer.