7th Grade - Spheres

Introduction

• A sphere is a rounded three-dimensional solid.
• It is perfectly symmetrical.
• All the points on its surface are at an equal distance from its center. This distance is called the radius of the sphere.

• Half of a sphere is called a hemisphere.

Volume of a Sphere

• If the radius $\left(r\right)$ of a sphere is known, its volume can be found by using the following formula:

$Volum{e}_{sphere} = \frac{4}{3}{\mathrm{\pi r}}^3$

• If the diameter $\left(d\right)$ of a sphere is known, its volume can be found by using the following formula:

$Volum{e}_{sphere} = \frac{4}{3}\pi \times {\left(\frac{d}{2}\right)}^3$$= \frac{4}{3}\pi \times {\frac{d}{8}}^3$$= \frac{1}{6}\pi d^3$

• The volume of a sphere is measured in cubic units.

Surface Area of a Sphere

• The surface area of a sphere is the total area covered by its outer surface.
• If the radius $\left(r\right)$ of a sphere is known, its surface area can be calculated using the formula mentioned below.

$Are{a}_{sphere}=4{\mathrm{\pi r}}^2$

Solved Examples

Question 1: Find the volume of a sphere with a radius of 3 cm.

Solution: $Volum{e}_{sphere} = \frac{4}{3}{\mathrm{\pi r}}^3$$=\frac{4}{3}\mathrm{\pi } \times {\left(3\right)}^3$$=4\mathrm{\pi } \times 9$$=36\mathrm{\pi } c{m}^3$

 

Question 2: If the radius of a sphere is 20 cm, find its surface area.

Solution: $Are{a}_{sphere}=4{\mathrm{\pi r}}^2$$=4\mathrm{\pi } \times {\left(20\right)}^2$$=4\mathrm{\pi } \times 400$$=1600\mathrm{\pi } {\mathrm{cm}}^2$

Cheat Sheet

• $Volum{e}_{sphere} = \frac{4}{3}{\mathrm{\pi r}}^3$$= \frac{1}{6}\pi d^3$
• $Are{a}_{sphere}=4{\mathrm{\pi r}}^2$

 

Blunder Areas

• The curved surface area and total surface area of a sphere are the same. But the curved surface area and total surface of a hemisphere are not the same.