Spherical Geometry
Geometry on a sphere where lines are great circles.
Key idea
Geometry on a sphere where lines are great circles.
In depth
On a sphere, the straight lines are great circles, and angles of a triangle sum to more than 180°. The excess is proportional to the area: A = R²(α + β + γ − π). Used in navigation and astronomy.
Worked example
An octant triangle on a unit sphere has three 90° angles and area π/2.
Formula
Real-world uses
- Navigation on Earth and flight planning
- Calculating great-circle routes between cities
Common mistake
Applying plane triangle rules to a sphere.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Spherical Geometry?
Geometry on a sphere where lines are great circles. It is a topic at the expert level in Geometry.
What is the formula for Spherical Geometry?
The key formula is angle sum > 180°.
Where is Spherical Geometry used in real life?
Real-world examples: Navigation on Earth and flight planning; Calculating great-circle routes between cities.
What should I learn before Spherical Geometry?
Make sure you are comfortable with Non-Euclidean Geometry, the previous topic in Geometry.
What is a common mistake with Spherical Geometry?
Applying plane triangle rules to a sphere.
How can I practise Spherical Geometry?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Spherical Geometry?
Move on to Projective Geometry next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.