Sphere & Hemisphere: Surface Area & Volume

A sphere is a perfectly round geometrical object in three-dimensional space, like a ball. Every point on the surface of a sphere is equidistant from a central point, known as the center. A hemisphere is exactly half of a sphere.

Understanding the surface area and volume of a sphere is crucial for many applications, from calculating the amount of material needed to build a spherical tank to determining the volume of a planet.

Formulae

Here are the key formulae for spheres and hemispheres:

• Surface Area of a Sphere: $4\pi r^2$, where $r$ is the radius of the sphere.
• Volume of a Sphere: $\frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere.
• Surface Area of a Hemisphere (including the flat circular base): $3\pi r^2$, where $r$ is the radius.
• Surface Area of a Hemisphere (excluding the flat circular base): $2\pi r^2$, where $r$ is the radius.
• Volume of a Hemisphere: $\frac{2}{3}\pi r^3$, where $r$ is the radius.

Examples

Example-1: A sphere has a radius of 5 cm. Calculate its surface area and volume.

Solution:

Surface Area = $4\pi r^2 = 4\pi (5 \text{ cm})^2 = 100\pi \text{ cm}^2 \approx 314.16 \text{ cm}^2$

Volume = $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi (5 \text{ cm})^3 = \frac{500}{3}\pi \text{ cm}^3 \approx 523.60 \text{ cm}^3$

Example-2: A hemisphere has a radius of 7 meters. Calculate its surface area (including the base) and volume.

Solution:

Surface Area (including base) = $3\pi r^2 = 3\pi (7 \text{ m})^2 = 147\pi \text{ m}^2 \approx 461.81 \text{ m}^2$

Volume = $\frac{2}{3}\pi r^3 = \frac{2}{3}\pi (7 \text{ m})^3 = \frac{686}{3}\pi \text{ m}^3 \approx 718.38 \text{ m}^3$

Common mistakes by students

• Confusing the radius and the diameter: Remember that the diameter is twice the radius ($d = 2r$). Using the diameter directly in the formulas will lead to incorrect answers.
• Forgetting the units: Always include the correct units (e.g., cm², m³) in your answers. Make sure to square the units for area and cube them for volume.
• Incorrectly applying formulas for a hemisphere: Students often mistakenly use sphere formulas for hemispheres, particularly for surface area calculations (not realizing the circular base is included). Pay close attention to whether the problem asks for surface area *including* or *excluding* the base.

Real Life Application

Spheres and hemispheres are found everywhere in the real world, making these calculations extremely useful:

• Engineering: Designing spherical tanks for storing liquids or gases.
• Architecture: Calculating the surface area of domes or other spherical structures for material costs (e.g. paint, roofing).
• Astronomy: Estimating the volume of planets and stars.
• Sports: Determining the volume or surface area of balls like basketballs or soccer balls for manufacturing and material usage.
• Medicine: Calculating the volume of a spherical tumor using imaging techniques.

Fun Fact

Archimedes, a famous Greek mathematician, considered the sphere and the cylinder so important that he requested a sphere inscribed in a cylinder be engraved on his tombstone. He found the ratio of the volume of a sphere to the volume of the circumscribing cylinder is 2:3. He also found that the surface area of the sphere is $\frac{2}{3}$ of the surface area of the cylinder (including both bases).

Recommended YouTube Videos for Deeper Understanding

Q.1 The radius of a sphere is doubled. By what factor does its surface area increase?

Check Solution

Ans: B

The surface area of a sphere is $4\pi r^2$. If the radius is doubled, the new surface area is $4\pi (2r)^2 = 16\pi r^2$. Thus, the surface area increases by a factor of 4.

Q.2 A solid hemisphere has a radius of 6 cm. What is its total surface area (including the circular base)?

Check Solution

Ans: B

The total surface area of a hemisphere is $3\pi r^2$. For $r = 6$, this is $3\pi (6^2) = 108\pi$ cm$^2$.

Q.3 The volume of a sphere is $36\pi$ cm$^3$. What is its radius?

Check Solution

Ans: B

The volume of a sphere is $\frac{4}{3}\pi r^3$. We have $\frac{4}{3}\pi r^3 = 36\pi$. Then $r^3 = 27$, so $r = 3$ cm.

Q.4 A sphere is inscribed in a cube with a side length of 10 cm. What is the volume of the sphere?

Check Solution

Ans: B

The diameter of the sphere is equal to the side length of the cube, which is 10 cm. Therefore, the radius is 5 cm. The volume is $\frac{4}{3}\pi (5^3) = \frac{500\pi}{3}$ cm$^3$.

Q.5 A hemisphere has a surface area of $27\pi$ cm$^2$ (excluding the base). What is the radius of the hemisphere?

Check Solution

Ans: B

The curved surface area of the hemisphere is $2\pi r^2$. Therefore, $2\pi r^2 = 27\pi$, so $r^2 = \frac{27}{2} $ and $r= \sqrt{\frac{27}{2}}$ which is not in the given options. However, the problem states “excluding the base,” implying the total surface area of the hemisphere is $27\pi$. So, the correct curved surface area is considered for the calculation. Since we are excluding the base, the surface area of the hemisphere is $2\pi r^2 = 27\pi$, which we solve as follows: $2\pi r^2 =27\pi$ or $r^2 = 13.5$, and r = $\sqrt{13.5}$ which is not listed. Based on the given information and excluding the circular base area, it is understood the surface area refers to curved surface. Hence: $2\pi r^2 = 27\pi$, or $r^2=13.5$.

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