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A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section?
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A sphere has the same volume as a cylinder whose height is equal to th...
Yes, a sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section.
The volume of a sphere is given by the formula:
V = (4/3)πr^3
Where V is the volume of the sphere, π is a mathematical constant (approximately equal to 3.14), and r is the radius of the sphere (half the diameter).
The volume of a cylinder is given by the formula:
V = πr^2h
Where V is the volume of the cylinder, π is a mathematical constant (approximately equal to 3.14), r is the radius of the base of the cylinder (half the diameter of the cross section), and h is the height of the cylinder.
If we set the height of the cylinder equal to the diameter of its cross section (h = 2r), we can see that the two formulas are equivalent, and therefore a sphere with a given radius will have the same volume as a cylinder with the same radius and height.
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A sphere has the same volume as a cylinder whose height is equal to th...
Volume of a Sphere and Cylinder
To prove that a sphere has the same volume as a cylinder whose height is equal to the diameter of its cross-section, we need to compare the volume formulas of a sphere and a cylinder.
Volume of a Sphere
- The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.
Volume of a Cylinder
- The volume of a cylinder is given by the formula V = πr²h, where r is the radius of the base and h is the height of the cylinder.
Equal Volume
- Let's consider a cylinder with a diameter d, which means the radius of the cylinder is d/2.
- According to the given condition, the height of the cylinder is equal to the diameter, so h = d.
- Substituting the values of r and h in the cylinder volume formula, we get V = π(d/2)² * d = (π/4)d³.
- Now, let's compare the volume of the cylinder with the volume of a sphere. Since the volume of a sphere is (4/3)πr³ and the radius of the sphere is also d/2, the volume of the sphere is (4/3)π(d/2)³ = (π/6)d³.
- Therefore, we can see that the volume of the sphere and the cylinder are equal when the height of the cylinder is equal to the diameter of its cross-section.
To prove that a sphere has the same volume as a cylinder whose height is equal to the diameter of its cross-section, we need to compare the volume formulas of a sphere and a cylinder.
Volume of a Sphere
- The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.
Volume of a Cylinder
- The volume of a cylinder is given by the formula V = πr²h, where r is the radius of the base and h is the height of the cylinder.
Equal Volume
- Let's consider a cylinder with a diameter d, which means the radius of the cylinder is d/2.
- According to the given condition, the height of the cylinder is equal to the diameter, so h = d.
- Substituting the values of r and h in the cylinder volume formula, we get V = π(d/2)² * d = (π/4)d³.
- Now, let's compare the volume of the cylinder with the volume of a sphere. Since the volume of a sphere is (4/3)πr³ and the radius of the sphere is also d/2, the volume of the sphere is (4/3)π(d/2)³ = (π/6)d³.
- Therefore, we can see that the volume of the sphere and the cylinder are equal when the height of the cylinder is equal to the diameter of its cross-section.
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A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section?
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A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section?.
A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section?.
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