Great Circle

Great Circle

A circle on the surface of the sphere that shares its center with the center of the sphere. An arc of such a circle is often called a great circle, even though it is not a full circle.

Note: The shortest path connecting two points on the surface of a sphere is a great circle.

Sphere with a great circle drawn as a full equatorial ellipse and a dot marking the shared center point.

Key Formula

d = r \cdot \theta

Where:

$d$ = The great circle arc distance between two points on the sphere's surface
$r$ = The radius of the sphere
$\theta$ = The central angle (in radians) between the two points, as measured from the center of the sphere

Worked Example

Problem: Two cities lie on the surface of the Earth (radius approximately 6,371 km). The central angle between them, measured from the Earth's center, is 30°. Find the great circle distance between the two cities.

Step 1: Identify the known values: the Earth's radius and the central angle.

r = 6{,}371 \text{ km}, \quad \theta = 30°

Step 2: Convert the central angle from degrees to radians.

\theta = 30° \times \frac{\pi}{180°} = \frac{\pi}{6} \approx 0.5236 \text{ radians}

Step 3: Apply the great circle arc distance formula.

d = r \cdot \theta = 6{,}371 \times \frac{\pi}{6}

Step 4: Calculate the result.

d = 6{,}371 \times 0.5236 \approx 3{,}336 \text{ km}

Answer: The great circle distance between the two cities is approximately 3,336 km.

Another Example

This example contrasts a great circle with a small circle, showing how to determine whether a cross-section of a sphere qualifies as a great circle.

Problem: A sphere has a radius of 10 cm. A plane cuts through the sphere 6 cm from the center. Is the resulting circle a great circle? Find the radius of the circle formed by the cut.

Step 1: Recall that a great circle must pass through the center of the sphere, meaning the cutting plane must be exactly 0 cm from the center.

Step 2: Since the plane is 6 cm from the center (not 0 cm), this is NOT a great circle. It is a small circle.

Step 3: Use the relationship between the sphere's radius, the distance from center to the cutting plane, and the radius of the resulting circle.

r_{\text{circle}} = \sqrt{R^2 - d^2}

Step 4: Substitute the known values.

r_{\text{circle}} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ cm}

Step 5: Compare this to the great circle radius. A great circle on this sphere would have a radius of 10 cm (equal to the sphere's radius). The small circle's radius of 8 cm is smaller, confirming it is not a great circle.

Answer: The circle is NOT a great circle. It is a small circle with radius 8 cm, compared to the great circle radius of 10 cm.

Frequently Asked Questions

What is the difference between a great circle and a small circle?

A great circle has the same center and the same radius as the sphere itself—it is the largest circle you can draw on the sphere's surface. A small circle is any other circle on the sphere whose center does not coincide with the sphere's center. Every plane that passes through the center of a sphere creates a great circle; a plane that misses the center creates a small circle.

Why is a great circle the shortest path between two points on a sphere?

A great circle is a geodesic on a sphere, meaning it represents the straightest possible path along the curved surface. Any deviation from a great circle arc would force the path to curve away from the most direct route, increasing the total distance. This is why airplane flight routes often follow great circle arcs rather than straight lines on a flat map.

Is the equator a great circle?

Yes, the equator is a great circle because its center coincides with the center of the Earth. However, lines of latitude other than the equator are small circles. Interestingly, all lines of longitude (meridians) are halves of great circles.

Great Circle vs. Small Circle

	Great Circle	Small Circle
Definition	A circle on a sphere whose center is the center of the sphere	A circle on a sphere whose center does NOT coincide with the sphere's center
Radius	Equal to the sphere's radius R	Less than the sphere's radius: √(R² − d²), where d is the distance from the sphere's center to the cutting plane
Cutting plane	Passes through the center of the sphere	Does not pass through the center of the sphere
Shortest path	Yes — an arc of a great circle is the shortest surface path between two points	No — following a small circle between two points is always a longer path
Example on Earth	The equator; any line of longitude	Any line of latitude other than the equator

Why It Matters

Great circles are fundamental in navigation and aviation. Pilots and ship captains plot great circle routes to minimize travel distance between distant cities, which is why flight paths on a flat map often appear curved. In geometry and trigonometry courses, great circles introduce the concept of geodesics—the generalization of straight lines to curved surfaces—and they form the basis of spherical trigonometry.

Common Mistakes

Mistake: Assuming that a straight line on a flat (Mercator) map represents the shortest path on the globe.

Correction: On a sphere, the shortest path is a great circle arc, which typically appears as a curve on most flat map projections. A straight line on a Mercator map usually corresponds to a path of constant compass bearing (a rhumb line), which is longer than the great circle route.

Mistake: Thinking that any circle drawn on a sphere is a great circle.

Correction: Only circles whose plane passes through the center of the sphere qualify as great circles. All other circles on the sphere are small circles with a smaller radius.

Related Terms

Circle — A great circle is a special circle on a sphere
Sphere — The surface on which great circles are defined
Arc of a Circle — Great circle paths are measured as arcs
Point — Great circles connect pairs of points on a sphere
Circumference — A great circle's circumference equals 2πr of the sphere
Radian — Central angles in radians are used to compute arc distance