Exercise 3 Page 641

Let's consider a cone with a radius of the base $r$ and height $h .$

Let's check separately how to find the surface area and the volume of this cone.

Surface Area

The surface area of a cone is the amount of material needed to build it. To determine the surface area of a cone, we can cut it and unroll it. We will get a circle with radius

r

(base), a circular sector with radius

ℓ

(the lateral surface), and arc length

2 π r .

Notice that the arc length of the circular sector is equal to the circumference of the base,

2 π r .

The area of the base is

π r^{2} .

The formula for finding the area of a circular sector is one-half the product of the radius times the arc length.

A_{lateral} = \frac{1}{2} (ℓ \cdot 2 π r) = π r ℓ

By adding the two areas mentioned above we will find the surface area of the cone.

$S = π r^{2} + π r ℓ$

Volume

To find the volume of a cone we will consider a cylinder with the same radius and height as the cone.

If we want to fill the cylinder with sand by using the cone, it will take three cones to fill the cylinder.

Therefore, the volume of the cylinder is three times the volume of the cone.

V_{cylinder} = 3 \cdot V_{cone} ⇓ V_{cone} = \frac{1}{3} V_{cylinder}

The volume of the cone is one-third the volume of the cylinder: one-third the area of the base multiplied by the height.

$V_{cone} = \frac{1}{3} π r^{2} h$