Solved Problems: Volume of Cylinders, Cones, Spheres for Grade 8

📚 Understanding Volume: Cylinders, Cones, and Spheres

Volume is the amount of space a three-dimensional object occupies. Calculating the volume of cylinders, cones, and spheres is a fundamental concept in geometry. Let's explore each shape!

📜 Historical Context

The study of volumes dates back to ancient civilizations. Archimedes, a Greek mathematician, made significant contributions to calculating the volumes of spheres and cylinders. His work laid the foundation for modern geometry and calculus.

📏 Key Principles and Formulas

• Cylinder: A cylinder has two parallel circular bases connected by a curved surface. The volume of a cylinder is given by the formula: $V = \pi r^2 h$, where $r$ is the radius of the base and $h$ is the height.
• Cone: A cone has a circular base and tapers to a single point called the apex. The volume of a cone is given by the formula: $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius of the base and $h$ is the height.
• Sphere: A sphere is a perfectly round three-dimensional object. The volume of a sphere is given by the formula: $V = \frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere.

➕ Solved Problems

Cylinder Example

Problem: A cylinder has a radius of 5 cm and a height of 10 cm. Find its volume.

Solution:

• 🔑 Identify the variables: $r = 5$ cm, $h = 10$ cm
• ✍️ Apply the formula: $V = \pi r^2 h$
• 🔢 Substitute the values: $V = \pi (5)^2 (10) = 250\pi$
• ✅ Calculate the volume: $V ≈ 785.4 \text{ cm}^3$

Cone Example

Problem: A cone has a radius of 3 cm and a height of 8 cm. Find its volume.

Solution:

• 🔑 Identify the variables: $r = 3$ cm, $h = 8$ cm
• ✍️ Apply the formula: $V = \frac{1}{3} \pi r^2 h$
• 🔢 Substitute the values: $V = \frac{1}{3} \pi (3)^2 (8) = 24\pi$
• ✅ Calculate the volume: $V ≈ 75.4 \text{ cm}^3$

Sphere Example

Problem: A sphere has a radius of 6 cm. Find its volume.

Solution:

• 🔑 Identify the variables: $r = 6$ cm
• ✍️ Apply the formula: $V = \frac{4}{3} \pi r^3$
• 🔢 Substitute the values: $V = \frac{4}{3} \pi (6)^3 = 288\pi$
• ✅ Calculate the volume: $V ≈ 904.8 \text{ cm}^3$

💡 Real-World Applications

• Cylinders: 🧪 Storage tanks, beverage cans, and pipes are common examples of cylinders.
• Cones: 🚧 Traffic cones, ice cream cones, and funnels are real-world applications of cones.
• Spheres: ⚽ Basketballs, marbles, and planets are examples of spheres.

📝 Practice Quiz

Calculate the volume for each of the following problems:

1. A cylinder with a radius of 4 cm and a height of 7 cm.
2. A cone with a radius of 6 cm and a height of 9 cm.
3. A sphere with a radius of 2 cm.

Solutions:

1. $V = \pi (4^2)(7) = 112\pi ≈ 351.9 \text{ cm}^3$
2. $V = \frac{1}{3} \pi (6^2)(9) = 108\pi ≈ 339.3 \text{ cm}^3$
3. $V = \frac{4}{3} \pi (2^3) = \frac{32}{3}\pi ≈ 33.5 \text{ cm}^3$

заключение

Understanding how to calculate the volume of cylinders, cones, and spheres is crucial for various applications in mathematics, science, and engineering. By mastering these formulas and practicing with examples, you can confidently solve volume-related problems.