What is the difference between the sum of interior angles of a plane triangle and a spherical triangle for an area of 195 square kilometers on the Earth's surface?
🌍 Understanding Spherical Excess
This question highlights the fundamental difference between Plane Surveying and Geodetic Surveying.
- A plane triangle is drawn on a flat surface. Its interior angles always sum to exactly 180°.
- A spherical triangle is drawn on the surface of a sphere (like the Earth). Its sides are arcs of great circles, and its interior angles always sum to more than 180°.
🔬 Detailed Analysis of the Options
C. one second
This is the correct answer. For geodetic calculations, it is an established rule of thumb that for every 195 sq. km (or about 75.5 sq. miles) of area of a spherical triangle on the Earth's surface, the spherical excess is approximately one second (1"). This difference, while tiny, is critical for high-precision, large-scale surveys.
The relationship can be expressed as:
Sum of Spherical Angles - Sum of Plane Angles = Spherical Excess
$$ (\alpha' + \beta' + \gamma') - (\alpha + \beta + \gamma) = 1^{\prime\prime} $$
A. one degree & B. one minute
These are incorrect. A difference of one degree or even one minute represents a massive spherical excess, which would correspond to an enormous area, far larger than 195 sq. km. For context, the spherical excess for a triangle covering 1/8th of the Earth's surface is 90°.
D. one radian
This is incorrect. One radian is approximately 57.3 degrees, which is an extremely large angle in this context, making it a completely implausible answer.
🔗 Related Surveying Facts
The effect of Earth's curvature becomes noticeable in other measurements as well:
- Arc vs. Chord Length: For a 12 km distance measured along the Earth's curved surface (arc), the length is about 1 cm greater than the straight-line distance (chord) between the two endpoints.
- This demonstrates that for most typical engineering projects, the errors introduced by assuming a flat earth (plane surveying) are negligible. However, for geodetic control and large-scale mapping, these differences must be accounted for.
💡 Study Tips
- Memorize the Key Value: The relationship "195 sq. km ≈ 1 second of spherical excess" is a standard value often cited in surveying exams. Commit it to memory.
- Think Scale: The difference between plane and geodetic surveying is all about scale. For small areas, the Earth is "flat enough." For large areas, the curvature matters. The 1" difference is the very first level where this curvature becomes numerically significant in angular measurement.