Sphere Worksheet

**Page 1**

1.

If the radius of any great circle on a sphere is 5 cm, then what is the surface area of that sphere?

a. | 50$\pi $ cm ^{2} | ||

b. | $\frac{500\pi}{3}$ cm ^{2} | ||

c. | 100$\pi $ cm ^{2} | ||

d. | $\frac{100\pi}{3}$ cm ^{2} |

Since the radius of any great circle on the sphere is 5 cm, the radius of sphere,

Surface area of sphere = 4

= 4 ×

= 100

The surface area of the sphere is 100

Correct answer : (3)

2.

Which of the following statements is correct?

I. There is a unique straight line passing through any two points on a plane, in plane Euclidean Geometry.

II. There are finite number of straight lines passing through any two points on a plane, in plane Euclidean Geometry.

III. There is a unique circle passing through any two non-polar points on sphere, in Spherical Geometry.

IV. There are finite number of great circles passing through any two non-polar points on a sphere, in Spherical Geometry.

I. There is a unique straight line passing through any two points on a plane, in plane Euclidean Geometry.

II. There are finite number of straight lines passing through any two points on a plane, in plane Euclidean Geometry.

III. There is a unique circle passing through any two non-polar points on sphere, in Spherical Geometry.

IV. There are finite number of great circles passing through any two non-polar points on a sphere, in Spherical Geometry.

a. | IV | ||

b. | I | ||

c. | II | ||

d. | III |

There is a unique great circle passing through any two points on sphere, in Spherical Geometry.

Correct answer : (2)

3.

Which of the following statements is correct?

I. There are finite number of staright lines passing through any two points on a plane, in plane Euclidean Geometry.

II.There are finite number of great circles passing through any two non-polar points on a sphere, in spherical seometry.

III. There are infinite number of straight lines passing through any two points on a sphere, in spherical geometry.

IV. There are infinite number of great circles passing through any non-polar point on a sphere, in spherical geometry.

I. There are finite number of staright lines passing through any two points on a plane, in plane Euclidean Geometry.

II.There are finite number of great circles passing through any two non-polar points on a sphere, in spherical seometry.

III. There are infinite number of straight lines passing through any two points on a sphere, in spherical geometry.

IV. There are infinite number of great circles passing through any non-polar point on a sphere, in spherical geometry.

a. | IV | ||

b. | II | ||

c. | III | ||

d. | I |

There is a unique great circle passing through any two non-polar points on a sphere, in Spherical Geometry.

There are infinite number of great circles passing through any non-polar point on a sphere, in spherical geometry.

Correct answer : (1)

4.

Which of the following statements is correct?

I. Every great circle of a sphere intersects all the other great circles of a sphere in exactly one point.

II. Every great circle of a sphere intersects all the other great circles of a sphere in exactly two points.

III. Every great circle of a sphere intersects all the other great circles of a sphere in finite number of points.

IV. Every great circle of a sphere intersects all the other great circles of a sphere in infinite number of points.

I. Every great circle of a sphere intersects all the other great circles of a sphere in exactly one point.

II. Every great circle of a sphere intersects all the other great circles of a sphere in exactly two points.

III. Every great circle of a sphere intersects all the other great circles of a sphere in finite number of points.

IV. Every great circle of a sphere intersects all the other great circles of a sphere in infinite number of points.

a. | II | ||

b. | IV | ||

c. | III | ||

d. | I |

Correct answer : (1)

5.

The parallel postulate for Spherical Geometry is:

a. | through a point not on a line (great circle), there are finite number of lines (great circles) parallel to the given line(great circle) | ||

b. | through a point not on a line (great circle), there is no line (great circle) parallel to the given line(great circle) | ||

c. | through a point not on a line (great circle), there is two lines (great circles) parallel to the given line(great circle) | ||

d. | through a point not on a line (great circle), there is unique line (great circle) parallel to the given line(great circle) |

So, there are no parallel lines in Spherical Geometry.

Through a point not on a line (great circle), there in no line (great circle) parallel to the given line (great circle).

Correct answer : (2)

6.

Which of the following statements is correct?

I. Two distinct lines in Euclidean Geometry intersect in at most one point.

II. Two distinct lines in Euclidean Geometry intersect in finite number of points.

III. Two distinct lines (great circles) in Spherical Geometry intersect in at most one point.

IV. Two distinct lines (great circles) in Spherical Geometry intersect in finite number of points.

I. Two distinct lines in Euclidean Geometry intersect in at most one point.

II. Two distinct lines in Euclidean Geometry intersect in finite number of points.

III. Two distinct lines (great circles) in Spherical Geometry intersect in at most one point.

IV. Two distinct lines (great circles) in Spherical Geometry intersect in finite number of points.

a. | IV | ||

b. | II | ||

c. | I | ||

d. | III |

Two distinct lines (great circles) in Spherical Geometry intersect in exactly two points.

Correct answer : (3)

7.

Find the surface area of the spherical triangle with angles A = 90°, B = 60° and C = 45°, which sits on a sphere of radius R = 6 cm.

a. | $\frac{25\pi}{2}$ cm ^{2} | ||

b. | 3$\pi $ cm ^{2} | ||

c. | 75$\pi $ cm ^{2} | ||

d. | $\frac{\pi}{2}$ cm ^{2} |

Δ = R

A = 90° =

Required area, Δ = 6 × 6 (

=

=

= 3

So, the area of spherical triangle is 3

Correct answer : (2)

8.

The sum of the angles of a spherical triangle is between:

a. | $\pi $ and 2$\pi $ | ||

b. | 2$\pi $ and 3$\pi $ | ||

c. | 2$\pi $ and 4$\pi $ | ||

d. | $\pi $ and 3$\pi $ |

Correct answer : (4)

9.

Which of the following is not a possible sum of angles of a spherical triangle?

a. | 420° | ||

b. | 360° | ||

c. | 560° | ||

d. | 500° |

560° is not between 180° and 540°.

Correct answer : (3)

10.

The amount by which the sum of the angles of a triangle exceeds 180° is called :

a. | Spherical defect | ||

b. | Spherical excess | ||

c. | Defect sum | ||

d. | Excess sum |

Correct answer : (2)