Volume of a Sphere
The volume of a sphere is calculated using the formula V = (4/3) x pi x r^3, where r is the radius. This formula requires cubing the radius, multiplying by pi, and then by 4/3. For example, a gumball with a 1 cm radius has a volume of about 4.19 cubic centimeters. This Grade 8 math skill from Yoshiwara Core Math Chapter 3 extends students' knowledge of volume to curved three-dimensional shapes, building on understanding of rectangular volume. The sphere volume formula appears in geometry, physics, and engineering contexts including calculating the capacity of spherical tanks and the size of planets.
Key Concepts
Property The volume of a sphere is given by $$ \text{Volume} = \dfrac{4}{3} \times \pi r^3 $$ where $r$ is the radius of the sphere. Recall that $r^3$, which we read as '$r$ cubed,' means $r \times r \times r$.
Examples A gumball has a radius of 1 centimeter. Its volume is $V = \frac{4}{3} \pi (1)^3 = \frac{4}{3}\pi \approx 4.19$ cubic centimeters. A soccer ball has a diameter of 22 cm, so its radius is 11 cm. Its volume is $V = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) \approx 5575.28$ cubic centimeters. A spherical ornament has a volume of $36\pi$ cubic inches. To find its radius, solve $36\pi = \frac{4}{3}\pi r^3$, which simplifies to $27 = r^3$, so the radius is $r = \sqrt[3]{27} = 3$ inches.
Explanation Volume measures the space inside a 3D shape, like a ball or a planet. For a sphere, you cube the radius (multiply it by itself three times), then multiply by pi ($\pi$), and finally multiply by the fraction $\frac{4}{3}$.
Common Questions
What is the formula for the volume of a sphere?
The volume of a sphere is V = (4/3) x pi x r^3, where r is the radius. Cube the radius first, then multiply by pi, then multiply by 4/3. For example, a sphere with radius 3 has volume (4/3) x pi x 27 = 36pi, or about 113.1 cubic units.
How do you calculate the volume of a sphere step by step?
Step 1: Find the radius. Step 2: Cube the radius (multiply it by itself three times). Step 3: Multiply the result by pi. Step 4: Multiply by 4/3. For a sphere with radius 6: 6^3 = 216, then 216 x pi x 4/3 = 904.8 cubic units.
What does r cubed mean in the sphere volume formula?
r cubed means r raised to the third power, or r x r x r. In the sphere volume formula V = (4/3) x pi x r^3, you multiply the radius by itself three times before multiplying by pi and 4/3.
When do 8th graders learn about the volume of a sphere?
Students study the volume of a sphere in Grade 8 math as part of Chapter 3 of Yoshiwara Core Math, which covers measurement including areas and volumes of two- and three-dimensional shapes.
If you know the volume of a sphere, how do you find the radius?
Rearrange the formula: r = cube root of (3V / 4pi). For example, if volume = 36pi, then 3 x 36pi / 4pi = 27, and the cube root of 27 is 3, so the radius is 3.
What is the difference between volume and surface area of a sphere?
Volume measures the space inside the sphere in cubic units, using V = (4/3) x pi x r^3. Surface area measures the outside skin in square units, using SA = 4 x pi x r^2. These are different calculations answering different questions.