On a flat surface, the angles of a triangle always add up to 180^∘ but on a spherical surface th

step 1 Question number 20 is asking us to draw two triangles on a sphere, one whose internal angles add

On a flat surface, the angles of a triangle always add up to...

On a flat surface, the angles of a triangle always add up to $180^{\circ}$ but on a spherical surface they may add up to a larger number. a. Find a triangle on a spherical surface that contains three $90^{\circ}$ angles. Illustrate your example. b. What's the largest sum of the angles you can find for a triangle on a spherical surface?

On a flat surface, the angles of a triangle always add up to $180^{\circ}$ but on a spherical surface they may add up to a larger number.
a. Find a triangle on a spherical surface that contains three $90^{\circ}$ angles. Illustrate your example.
b. What's the largest sum of the angles you can find for a triangle on a spherical surface?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Spherical Geometry

Spherical geometry is the study of figures on the surface of a sphere, where the usual rules of Euclidean geometry do not apply. On a sphere, the familiar properties of lines and angles are replaced by great circles and spherical angles, leading to unique results such as triangles with angle sums greater than 180°.

Spherical Triangle

A spherical triangle is formed by the intersection of three great circle arcs on a sphere. Unlike Euclidean triangles, the sum of its angles always exceeds 180° and this excess is directly related to the area of the triangle on the curved surface.

Great Circle

A great circle is the largest possible circle that can be drawn on a sphere, whose center coincides with the center of the sphere. Great circles represent the shortest path between two points on the sphere and serve as the analogues to straight lines in Euclidean geometry.

Spherical Excess

Spherical excess is the amount by which the sum of the angles of a spherical triangle exceeds 180°. It is directly proportional to the area of the triangle on the sphere, and in the extreme, the maximum spherical excess for a triangle is 360°, leading to a maximum angle sum of 540°.

Transcript

Please give Ace some feedback