Question 1

How do I calculate the volume of a cone?

Accepted Answer

Use V = (1/3) x pi x r^2 x h, where r is the base radius and h is the perpendicular height. For example, a cone with radius 3 cm and height 7 cm has volume = (1/3) x pi x 9 x 7 = 65.97 cubic cm. The volume is exactly one-third of a cylinder with the same dimensions.

Question 2

What is slant height?

Accepted Answer

Slant height is the distance from the edge of the base to the apex, measured along the surface of the cone. It forms the hypotenuse of a right triangle with the radius and height as the other two sides. Calculate it using l = sqrt(r^2 + h^2). Slant height is needed for surface area calculations.

Question 3

Why is a cone's volume one-third of a cylinder?

Accepted Answer

This result, first proved rigorously by Archimedes, comes from calculus (or Cavalieri's principle). If you imagine filling a cone with water and pouring it into a cylinder of the same base and height, you would need exactly three cones to fill the cylinder. This 1/3 factor applies to all pyramidal shapes relative to their prismatic counterparts.

Question 4

What are conic sections?

Accepted Answer

Conic sections are curves obtained by intersecting a cone with a plane at different angles. A horizontal cut gives a circle, a tilted cut gives an ellipse, a cut parallel to the side gives a parabola, and a steeper cut gives a hyperbola. These curves are fundamental in physics, astronomy (planetary orbits), and engineering.

Question 5

How do I find the height of a cone from its volume and radius?

Accepted Answer

Rearrange the volume formula: h = 3V / (pi x r^2). For a cone with volume 100 cubic cm and radius 4 cm: h = 3 x 100 / (pi x 16) = 300 / 50.27 = 5.968 cm.

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