🧮 Sphere Volume Practice Problems
Let's dive into some practice problems to help you master calculating the volume of a sphere. Remember, the formula for the volume of a sphere is:
$V = \frac{4}{3} \pi r^3$
Where:
* $V$ = Volume
* $\pi$ ≈ 3.14159
* $r$ = Radius
Problem 1: Basic Calculation 📏
A sphere has a radius of 5 cm. What is its volume?
Solution:
1. Apply the formula:
$V = \frac{4}{3} \pi (5)^3$
2. Calculate $5^3 = 125$
$V = \frac{4}{3} \pi (125)$
3. Multiply by $\frac{4}{3}$:
$V = \frac{500}{3} \pi$
4. Approximate $\pi$ as 3.14159:
$V ≈ \frac{500}{3} * 3.14159 ≈ 523.6$ cm³
Therefore, the volume of the sphere is approximately 523.6 cm³.
Problem 2: Diameter Given 🔍
A sphere has a diameter of 12 inches. Find its volume.
Solution:
Remember that the radius is half the diameter. So, $r = \frac{12}{2} = 6$ inches.
1. Apply the formula:
$V = \frac{4}{3} \pi (6)^3$
2. Calculate $6^3 = 216$
$V = \frac{4}{3} \pi (216)$
3. Multiply by $\frac{4}{3}$:
$V = \frac{864}{3} \pi = 288\pi$
4. Approximate $\pi$ as 3.14159:
$V ≈ 288 * 3.14159 ≈ 904.78$ in³
Therefore, the volume of the sphere is approximately 904.78 in³.
Problem 3: Volume to Radius 🔄
A sphere has a volume of 904.78 cm³. What is its radius?
Solution:
1. Start with the volume formula:
$V = \frac{4}{3} \pi r^3$
2. Plug in the given volume:
$904.78 = \frac{4}{3} \pi r^3$
3. Multiply both sides by $\frac{3}{4}$:
$904.78 * \frac{3}{4} = \pi r^3$
$678.585 = \pi r^3$
4. Divide both sides by $\pi$ (approximately 3.14159):
$\frac{678.585}{3.14159} = r^3$
$216 ≈ r^3$
5. Take the cube root of both sides:
$\sqrt[3]{216} = r$
$r = 6$ cm
Therefore, the radius of the sphere is 6 cm.
Problem 4: Real-World Application 🌍
A spherical water tank has a diameter of 10 meters. How many cubic meters of water can it hold?
Solution:
1. Find the radius: $r = \frac{10}{2} = 5$ meters
2. Apply the volume formula:
$V = \frac{4}{3} \pi (5)^3$
$V = \frac{4}{3} \pi (125)$
$V ≈ 523.6$ m³
Therefore, the tank can hold approximately 523.6 cubic meters of water.
Problem 5: Comparing Volumes ⚖️
Sphere A has a radius of 3 cm. Sphere B has a radius of 6 cm. How many times greater is the volume of Sphere B compared to Sphere A?
Solution:
1. Calculate the volume of Sphere A:
$V_A = \frac{4}{3} \pi (3)^3 = 36\pi$
2. Calculate the volume of Sphere B:
$V_B = \frac{4}{3} \pi (6)^3 = 288\pi$
3. Find the ratio of the volumes:
$\frac{V_B}{V_A} = \frac{288\pi}{36\pi} = 8$
Therefore, the volume of Sphere B is 8 times greater than the volume of Sphere A.
💡 Key Takeaways
- Remember the formula: $V = \frac{4}{3} \pi r^3$
- Radius is half the diameter.
- Practice converting between volume and radius.
- Be mindful of units (cm³, m³, etc.).
Keep practicing, and you'll ace those geometry problems! 🚀
redswan742
• Jan 11, 2026