What is a spherical polygon?

A polygon on the surface of the sphere defined by a number of great-circle arcs.

What are great-circle arcs?

The intersection of the surface with planes through the center of the sphere.

The intersection on the sides of the sphere.

The intersection of the angles with planes through the center of the sphere.

The intersection of the median with planes through the center of the sphere.

What is the formula for the spherical law of sines?

Sin A/ Sin a = Sin B/ Sin b= Sin C/ Sin c

How Much Do You Know About Spherical Trigonometry?

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Quizzes Created: 202 | Total Attempts: 609,163

Questions: 10 | Attempts: 429 | Updated: Mar 22, 2023

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How Much Do You Know About Spherical Trigonometry? - Quiz

Spherical trigonometry is the branch of Spherical geometry which deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons defined by a number of intersecting great circles on the sphere. If you like geometry, then complete our quiz now!

Questions and Answers

1.

Where did spherical trigonometry originate from?
- A.
  
  Greece
- B.
  
  Ancient Egypt
- C.
  
  Mesopotamia
- D.
  
  Turkey
Correct Answer
A. Greece

Explanation
Spherical trigonometry originated from Greece. This branch of mathematics deals with the relationships between the angles and sides of triangles on a sphere's surface. The ancient Greeks, particularly mathematicians like Hipparchus and Ptolemy, made significant contributions to the development of spherical trigonometry. They applied this knowledge to various fields such as astronomy and navigation, where the curvature of the Earth's surface needed to be taken into account. The Greeks' advancements in this field laid the foundation for further exploration and understanding of spherical trigonometry.

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2.

What is a spherical polygon?
- A.
  
  A sphere.
- B.
  
  A cube
- C.
  
  A triangle.
- D.
  
  A polygon on the surface of the sphere defined by a number of great-circle arcs.
Correct Answer
D. A polygon on the surface of the sphere defined by a number of great-circle arcs.

Explanation
A spherical polygon is a polygon on the surface of a sphere that is defined by a number of great-circle arcs. Unlike a traditional polygon in Euclidean geometry, the sides of a spherical polygon are not straight lines but rather arcs along the surface of the sphere. The vertices of the polygon are connected by these arcs, which are the shortest paths between the points on the sphere. This concept is important in various fields such as geography, astronomy, and computer graphics, where the Earth or other celestial bodies are represented as spheres.

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3.

What are great-circle arcs?
- A.
  
  The intersection of the surface with planes through the center of the sphere.
- B.
  
  The intersection on the sides of the sphere.
- C.
  
  The intersection of the angles with planes through the center of the sphere.
- D.
  
  The intersection of the median with planes through the center of the sphere.
Correct Answer
A. The intersection of the surface with planes through the center of the sphere.

Explanation
Great-circle arcs are the intersection of the surface of a sphere with planes that pass through its center. These arcs are the shortest paths between two points on the surface of the sphere, and they divide the sphere into two equal hemispheres. They are commonly used in navigation and geodesy to calculate distances and directions on the Earth's surface.

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4.

What's the other name for a lune?
- A.
  
  Diagon
- B.
  
  Diagonal
- C.
  
  Median
- D.
  
  Perimeter
Correct Answer
A. Diagon

Explanation
A lune is a geometric shape formed by two circular arcs, and it is also known as a "diagon." Therefore, the other name for a lune is "diagon."

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5.

What defines a spherical triangle?
- A.
  
  Not much
- B.
  
  3 planes
- C.
  
  4 planes
- D.
  
  5 planes
Correct Answer
B. 3 planes

Explanation
A spherical triangle is defined by the intersection of three planes on the surface of a sphere. These planes form the three sides of the triangle. Therefore, the correct answer is "3 planes".

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6.

What defines a spherical quadrilateral?
- A.
  
  4 planes
- B.
  
  6 planes
- C.
  
  7 planes
- D.
  
  Nothing
Correct Answer
A. 4 planes

Explanation
A spherical quadrilateral is defined by the intersection of four planes on a sphere. These four planes create a closed figure with four sides and four vertices on the surface of the sphere. The intersection of these planes forms the edges of the quadrilateral, and the points where the edges meet are the vertices. Therefore, the correct answer is 4 planes.

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7.

What is the formula for the spherical law of sines?
- A.
  
  Sin A/ Sin a = Sin B/ Sin b= Sin C/ Sin c
- B.
  
  Sin A/ Sin B
- C.
  
  Sin A/ Sin B= Sin b/Sin a
- D.
  
  Sin B/ Sin b= Sin C/ Sin c
Correct Answer
A. Sin A/ Sin a = Sin B/ Sin b= Sin C/ Sin c

Explanation
The formula for the spherical law of sines states that the ratio of the sine of an angle to the sine of its corresponding side length is equal for all angles and side lengths in a spherical triangle. This means that Sin A/ Sin a is equal to Sin B/ Sin b, which is also equal to Sin C/ Sin c. This formula allows for the calculation of unknown angles or side lengths in a spherical triangle when given the values of the other angles and side lengths.

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8.

What's the formula for the scalar triple product?
- A.
  
  OA.OB/OC
- B.
  
  OA. (OB x OC)
- C.
  
  OB.OA.OC
- D.
  
  (0A.OB.OC)^2
Correct Answer
B. OA. (OB x OC)

Explanation
The scalar triple product is calculated by taking the dot product of one vector with the cross product of the other two vectors. In this case, the correct answer is OA. (OB x OC), which follows the formula for the scalar triple product.

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9.

How can you represent the six parts of a triangle in cyclic order?
- A.
  
  (aCbAcB)
- B.
  
  Acbabc
- C.
  
  ACBACB
- D.
  
  CAaBbC
Correct Answer
A. (aCbAcB)

Explanation
The correct answer is (aCbAcB) because it represents the six parts of a triangle in cyclic order. The lowercase letters represent the vertices of the triangle in counterclockwise order, while the uppercase letters represent the sides of the triangle in counterclockwise order. The cyclic order ensures that the parts of the triangle are represented in a consistent and logical sequence.

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10.

Which mathematician has his name assigned to a Pentagon?
- A.
  
  Napier
- B.
  
  Newton
- C.
  
  Pythagoras
- D.
  
  Pascal
Correct Answer
A. Napier

Explanation
The mathematician who has his name assigned to a Pentagon is Napier. This is referring to the Napier's pentagon, which is a geometric construction named after John Napier. It is a method used to calculate trigonometric functions using a regular pentagon.

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