﻿ Sphere Worksheet | Problems & Solutions Sphere Worksheet

Sphere Worksheet
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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$\pi$ cm2 b. $\frac{500\pi }{3}$ cm2 c. 100$\pi$ cm2 d. $\frac{100\pi }{3}$ cm2

Solution:

If a plane passes through the center of the sphere, the section is called a great circle. Its center and radius are the same as those of the sphere itself.

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

Surface area of sphere = 4πr2

= 4 × π × 5 × 5

= 100π cm2

The surface area of the sphere is 100π cm2.

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. a. IV b. I c. II d. III

Solution:

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

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

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. a. IV b. II c. III d. I

Solution:

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

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.

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. a. II b. IV c. III d. I

Solution:

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

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)

Solution:

In Spherical Geometry, two lines (great circles) always intersect.

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).

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. a. IV b. II c. I d. III

Solution:

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

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

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. $\frac{25\pi }{2}$ cm2 b. 3$\pi$ cm2 c. 75$\pi$ cm2 d. $\frac{\pi }{2}$ cm2

Solution:

The surface area of a spherical triangle with angles A, B and C, which rests on a sphere of radius R is
Δ = R2[(A + B + C) - π]

A = 90° = π2 radians, B = 60° = π3 radians and C = 45° = π4 radians

Required area, Δ = 6 × 6 (π2+π3+π4-π)

= π × 6 × 6 (1 / 2 + 1 / 3 + 1 / 4 - 1)

= π × 6 × 6 × 1 / 12

= 3π cm2

So, the area of spherical triangle is 3π cm2.

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$\pi$

Solution:

The sum of the angles of a spherical triangle is between π and 3π radians (180° and 540°).

9.
Which of the following is not a possible sum of angles of a spherical triangle? a. 420° b. 360° c. 560° d. 500°

Solution:

The sum of the angles of a spherical triangle is between π (180°) and 3π (540°).

560° is not between 180° and 540°. a. Spherical defect b. Spherical excess c. Defect sum d. Excess sum