In geodetic surveying, the sum of all the internal angles (degree) of a spherical triangle should be ........
🌍 Understanding Geodetic Surveying
The key difference between Plane Surveying and Geodetic Surveying is how they treat the shape of the Earth.
- Plane Surveying: Assumes the Earth is a flat surface. This is accurate for small areas where curvature is negligible. Lines are considered straight, and angles are calculated using plane geometry.
- Geodetic Surveying: Accounts for the true curved (spheroidal) shape of the Earth. This is essential for large-scale projects. Lines are treated as arcs, and calculations involve spherical trigonometry.
🔬 Detailed Analysis of the Options
C. greater than 180
This is the correct answer. Because geodetic surveying considers the Earth's curvature, any large triangle drawn on its surface is a spherical triangle, not a plane triangle. The sides of a spherical triangle are arcs of great circles, and its angles are measured on the curved surface.
A fundamental property of spherical geometry is that the sum of the internal angles of a spherical triangle is always greater than 180° and less than 540°. The amount by which the sum exceeds 180° is known as the "spherical excess."
A. equal to 180
This is incorrect. The sum of interior angles is exactly 180° only for a triangle on a flat plane (a plane triangle), which is the assumption made in plane surveying, not geodetic surveying.
B. equal to 360 & D. less than 180
These are incorrect. The sum of angles in any triangle, plane or spherical, cannot be 360° or less than 180°.
📊 Summary: Plane vs. Spherical Triangles
| Property | Plane Triangle (Plane Surveying) | Spherical Triangle (Geodetic Surveying) |
|---|---|---|
| Surface | Flat (plane) | Curved (sphere) |
| Sides | Straight lines | Arcs of great circles |
| Sum of Interior Angles | Exactly 180° | Always > 180° |
| Geometry Used | Plane Geometry & Trigonometry | Spherical Geometry & Trigonometry |
💡 Study Tips
- Geodetic = Globe: Associate Geodetic surveying with the globe. If you try to draw a large triangle on a ball, you'll see the sides bulge outwards, making the corner angles larger.
- Plane = Paper: Associate Plane surveying with a flat sheet of paper, where the rules of standard school geometry apply.
- Spherical Excess: Remember this key term. It is the reason the sum is greater than 180° and is directly related to the area of the triangle. Larger area = larger spherical excess.