Question

If the volume of a cone is equal to one-third the volume of a cylinder with the same base and height, explain why this relationship holds true.

Ask by Perkins Bowman. in South Africa
Dec 16,2024

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Answer

The volume of a cone is one-third the volume of a cylinder with the same base and height because the cone's volume formula includes a factor of one-third compared to the cylinder's formula. This relationship is due to the geometric properties of the shapes.

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Mind Expander

The relationship holds true because of the geometric principles governing the shapes. The formula for the volume of a cylinder is V = πr²h, while for a cone, it’s V = (1/3)πr²h. The factor of one-third emerges from the way a cone tapers to a point, meaning it occupies less space than a cylinder. If you imagine filling a cone with water, the cone can only hold a third of what the cylinder can with the same base and height since it’s not as wide at the top! Moreover, this volume relationship can be tested by visualizing pouring the cone's contents into the cylinder. If you had three identical cones, their combined volume would perfectly fill the cylinder to the brim! This captivating connection between these basic shapes showcases the beauty of geometry in action, where even simple mathematical principles have far-reaching implications.

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