Mathematics · Physics

Sphere Geometry

183 Questions

Sphere geometry deals with calculating the volume and surface area of round objects. It includes problems on spherical shells, recasting spheres, and understanding radius variations. These mathematical formulas are essential for competitive exam preparation.

Volume of a sphereSurface area calculationsSpherical shellsRadius ratio variationsRecasting spheres

Sphere Geometry Questions

Multiple choice

Find the surface area of a sphere with radius 4 cm.

  1. 251.2 cm²

  2. 125.6 cm²

  3. 62.8 cm²

  4. 31.4 cm²
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a sphere is given by SA = 4πr², where r is the radius. In this case, r = 4 cm, so SA = 4π(4²) = 251.2 cm².

Multiple choice

A sphere has a radius of 4 cm. What is the volume of the sphere?

  1. 268.08 cm³

  2. 134.04 cm³

  3. 67.02 cm³

  4. 33.51 cm³
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a sphere is given by V = ½πr³, where r is the radius. In this case, r = 4 cm, so V = ½π(4³) = 268.08 cm³.

Multiple choice

Which mathematical concept is used to calculate the volume of a sphere?

  1. Volume = 4/3πr³

  2. Volume = πr²h

  3. Volume = πd³

  4. Volume = πd²h
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a sphere is calculated using the formula Volume = 4/3πr³, where r is the radius of the sphere.

Multiple choice

Which mathematical concept is used to calculate the surface area of a sphere?

  1. Surface Area = 4πr²

  2. Surface Area = πr²h

  3. Surface Area = πd²

  4. Surface Area = πd³
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a sphere is calculated using the formula Surface Area = 4πr², where r is the radius of the sphere.

Multiple choice

What formula did Dandin provide for the volume of a sphere?

  1. $\frac{4}{3}\pi r^3$
  2. $\frac{1}{3}\pi r^3$
  3. $\frac{2}{3}\pi r^3$
  4. $\pi r^3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Dandin gave the formula for the volume of a sphere as (\frac{4}{3}\pi r^3).

Multiple choice

A spherical balloon is being inflated with air. If the radius of the balloon is increasing at a rate of 2 cm/s, how fast is the volume of the balloon increasing when the radius is 10 cm?

  1. 320π cubic centimeters per second

  2. 640π cubic centimeters per second

  3. 800π cubic centimeters per second

  4. 1280π cubic centimeters per second
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We can use calculus to find the relationship between the radius and volume of the balloon, and then differentiate to find the rate of change of volume with respect to time.

Multiple choice

What is Bhaskara II's formula for finding the volume of a sphere?

  1. $\frac{4}{3} * \pi * radius^3$
  2. $\frac{4}{3} * \pi * diameter^3$
  3. $\frac{4}{3} * \pi * radius * diameter^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Bhaskara II's formula for finding the volume of a sphere is (\frac{4}{3} * \pi * radius^3).

Multiple choice

Calculate the volume of a sphere with radius 10 cm.

  1. 4/3π(1000) cm^3

  2. 4/3π(100) cm^3

  3. 4/3π(10) cm^3

  4. 4/3π cm^3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the volume of a sphere, use the formula: Volume = (4/3)πr^3. Substitute the radius r = 10 cm: Volume = (4/3)π(10 cm)^3 = 4/3π(1000) cm^3. Therefore, the answer is 4/3π(1000) cm^3.

Multiple choice

What is the Euler characteristic of a sphere?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Euler characteristic of a space is a topological invariant that is defined as the alternating sum of the number of vertices, edges, and faces in a triangulation of the space. The Euler characteristic of a sphere is 2 because it can be triangulated into two triangles.

Multiple choice

What is the formula for the volume of a sphere?

  1. $V = 4\pi r^3$
  2. $V = \pi r^2$
  3. $V = \frac{4}{3}\pi r^3$
  4. $V = \frac{1}{3}\pi r^2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$, where r is the radius of the sphere.

Multiple choice

What is the formula for the surface area of a sphere?

  1. $S = 4\pi r^2$
  2. $S = \pi r^2$
  3. $S = \frac{4}{3}\pi r^2$
  4. $S = \frac{1}{3}\pi r^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the surface area of a sphere is $S = 4\pi r^2$, where r is the radius of the sphere.

Multiple choice

What is the formula for the volume of a sphere with radius (r)?

  1. \(\frac{4}{3}\pi r^3\)
  2. \(\frac{1}{3}\pi r^3\)
  3. \(\pi r^3\)
  4. \(2\pi r^3\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a sphere is given by the formula (\frac{4}{3}\pi r^3), where (r) is the radius of the sphere.

Multiple choice

What is the formula for the surface area of a sphere with radius (r)?

  1. \(4\pi r^2\)
  2. \(2\pi r^2\)
  3. \(\pi r^2\)
  4. \(\frac{1}{2}\pi r^2\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a sphere is given by the formula (4\pi r^2), where (r) is the radius of the sphere.

Multiple choice

What is the formula for the volume of a hemisphere with radius (r)?

  1. \(\frac{1}{2}\cdot\frac{4}{3}\pi r^3\)
  2. \(\frac{1}{2}\cdot\pi r^3\)
  3. \(\frac{2}{3}\pi r^3\)
  4. \(\pi r^3\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a hemisphere is given by the formula (\frac{1}{2}\cdot\frac{4}{3}\pi r^3), where (r) is the radius of the hemisphere.

Multiple choice

What is the general formula for the Madhava series for the volume of a sphere?

  1. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  2. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  3. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  4. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
Explanation

The general formula for the Madhava series for the volume of a sphere is $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$.