Question

1. how is the volume of a cone calculated?

a. volume = \\( \frac { 1 } { 4 } \times \text { base area } \times \text { height } \\)
b. volume = \\( \pi \times \text { radius } \times \text { height } \\)
c. volume = \\( \frac { 1 } { 3 } \times \pi \times r ^ { 2 } \times h \\)
d. volume = \\( \frac { 1 } { 2 } \times \text { base area } \times \text { height } \\)

Explanation:

Step1: Recall the formula for the volume of a cone

The volume \(V\) of a cone is given by \(V=\frac{1}{3}Bh\), where \(B\) is the base - area.

Step2: Substitute the base - area formula

Since the base of a cone is a circle and the area of a circle \(B = \pi r^{2}\) (where \(r\) is the radius of the base of the cone), substituting \(B=\pi r^{2}\) into \(V = \frac{1}{3}Bh\) gives \(V=\frac{1}{3}\pi r^{2}h\).

• Option A: \(\frac{1}{4}\times\) Base Area \(\times\) Height is incorrect.
• Option B: \(\pi\times\) Radius \(\times\) Height is the formula for the lateral surface area of a cylinder (not the volume of a cone).
• Option D: \(\frac{1}{2}\times\) Base Area \(\times\) Height is the formula for the volume of a pyramid with a triangular base (not a cone).

Answer:

C. Volume = $\frac{1}{3}\times\pi\times r^{2}\times h$