The Volume of a Sphere - Numberphile

TL;DR

Archimedes used a cylinder, sphere, and double napped cone to prove that the volume of a sphere is two-thirds the volume of the cylinder that encloses it.

Transcript

I love exploring ancient mathematics because the ancient mathematicians had to make decisions, had to work things out without the tools that we have today. So how did they do it? Well the greatest of them all was Archimedes; and he found the volume of a sphere. How did he do it? Well we think it may have gone like this: he produced a cylinder and a... Read More

Key Insights

• 🔨 Ancient mathematicians like Archimedes had to solve complex problems without the modern tools we have today.
• 🔇 Archimedes used geometric shapes and experimental methods to determine the volume of a sphere.
• 🔇 The volume of a cone is one-third of the volume of a cylinder, similar to how a pyramid is one-third of the volume of a cuboid.
• 💦 Archimedes' experiment with water provided a practical way to compare the volumes of different shapes.
• 🔇 The volume of a sphere is two-thirds of the volume of the cylinder that encloses it.
• 💦 Archimedes' work exemplifies the ingenuity and problem-solving skills of ancient mathematicians.
• 🤗 Today, educational kits like KiwiCo provide hands-on learning experiences for children to explore concepts in science, engineering, and design.

Questions & Answers

Q: What geometric shapes did Archimedes use to determine the volume of a sphere?

Archimedes used a cylinder, sphere, and double napped cone in his calculations to determine the volume of a sphere. He compared the cross-sections of these shapes to find a relationship between their volumes.

Q: How did Archimedes prove his hypothesis about the volume of a sphere?

Archimedes conducted an experiment by submerging the shapes in water. He measured the levels of water before and after submerging the shapes to compare their volumes. By observing that the volumes remained the same, Archimedes concluded that the volume of the sphere is two-thirds of the volume of the enclosing cylinder.

Q: What is the relationship between the volume of a cone and a cuboid?

Just as a pyramid is one-third of the volume of the cuboid it fits into, a cone is also one-third of the volume of the cylinder it fits into. This relationship provided an important reference for Archimedes' calculations.

Q: How did Archimedes calculate the volume of a sphere?

Archimedes used the formula for the volume of a cylinder, which is pi r squared times the height. Since the cylinder and sphere have the same height and width, the volume of the sphere is two-thirds of the volume of the cylinder, resulting in the formula four over three pi r cubed.

Summary & Key Takeaways

• Archimedes used various geometric shapes to determine the volume of a sphere, including a cylinder, sphere, and double napped cone.

• By comparing the cross-sections of these shapes, Archimedes hypothesized that the volume of the sphere is equal to the volume of the cylinder minus the volume of the double napped cone.

• Archimedes conducted an experiment by dunking these shapes in water to measure their volumes, ultimately proving his hypothesis and determining the volume of a sphere.