Mathematics · Physics

Sphere Geometry

157 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

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 $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$.

Multiple choice

What is the formula for calculating the radius of a Hill Sphere?

  1. $R_H = (1 + m/3M)^{1/3} a$
  2. $R_H = (1 + m/M)^{1/3} a$
  3. $R_H = (1 + 3m/M)^{1/3} a$
  4. $R_H = (1 + m/M)^{2/3} a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for calculating the radius of a Hill Sphere is $R_H = (1 + m/3M)^{1/3} a$, where $R_H$ is the radius of the Hill Sphere, $m$ is the mass of the orbiting body, $M$ is the mass of the central body, and $a$ is the distance between the two bodies.

Multiple choice

What is the formula for the volume of a sphere, as given by Bhaskara II?

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

Bhaskara II's formula for the volume of a sphere is $\text{Volume} = \frac{4}{3} \times \pi \times \text{radius}^3$. This formula is still used today in geometry.

Multiple choice

What was the Babylonian method for calculating the volume of a sphere?

  1. The Babylonian Method

  2. The Volume of a Sphere Formula

  3. The Pythagorean Theorem

  4. The Surface Area of a Sphere Formula
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Babylonians had a method for calculating the volume of a sphere that involved using a series of approximations to find the solution.

Multiple choice

What was the Babylonian method for calculating the surface area of a sphere?

  1. The Babylonian Method

  2. The Surface Area of a Sphere Formula

  3. The Pythagorean Theorem

  4. The Volume of a Sphere Formula
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Babylonians had a method for calculating the surface area of a sphere that involved using a series of approximations to find the solution.

Multiple choice

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

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

Bhaskara II's formula for the volume of a sphere is given by $$V = \frac{4}{3}\pi r^3$$, where $$r$$ is the radius of the sphere.

Multiple choice

Find the volume of a sphere with radius (r = 5 cm).

  1. \(100\pi cm^3\)
  2. \(200\pi cm^3\)
  3. \(300\pi cm^3\)
  4. \(400\pi cm^3\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find the volume of a sphere, we can use the formula: (Volume = \frac{4}{3}\pi r^3). Plugging in the value of (r = 5 cm), we get: (Volume = \frac{4}{3}\pi (5 cm)^3 = \frac{4}{3}\pi (125 cm^3) = 300\pi cm^3).

Multiple choice

What is the formula for the volume of a sphere?

  1. V = 4/3πr^3

  2. V = πr^2h

  3. V = πr^3

  4. V = 2πr^3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the volume of a sphere is V = 4/3πr^3, where V is the volume, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

Multiple choice

What is Brahmagupta's rule for finding the volume of a sphere?

  1. The volume of a sphere is equal to four-thirds the product of its radius and its surface area.

  2. The volume of a sphere is equal to two-thirds the product of its radius and its surface area.

  3. The volume of a sphere is equal to one-third the product of its radius and its surface area.

  4. The volume of a sphere is equal to half the product of its radius and its surface area.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's rule for finding the volume of a sphere states that the volume of a sphere is equal to four-thirds the product of its radius and its surface area.

Multiple choice

What is Brahmagupta's rule for finding the volume of a sphere?

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

Brahmagupta's rule for finding the volume of a sphere states that $V = \frac{4}{3} \pi r^3$.

Multiple choice

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

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

The formula for finding the volume of a sphere with radius (r) is (\frac{4}{3}\pi r^3).

Multiple choice

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

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

The formula for finding the surface area of a sphere with radius (r) is (4\pi r^2).