Calculating the Curved Surface Area of a Cone

We are given the volume and the base radius of a right circular cone and need to find its curved surface area. To do this, we first need to determine the height of the cone, then the slant height, and finally use the formula for the curved surface area.

Understanding the Given Information

• Volume of the cone (V) = 308 cm³
• Radius of the base (r) = 7 cm
• Value of \(\pi\) = \(\frac{22}{7}\)

Step 1: Find the Height (h) of the Cone

The formula for the volume of a right circular cone is given by:

\(V = \frac{1}{3}\pi r^2 h\)

We can substitute the given values into this formula to find the height \(h\):

\(308 = \frac{1}{3} \times \frac{22}{7} \times (7)^2 \times h\)

\(308 = \frac{1}{3} \times \frac{22}{7} \times 49 \times h\)

Simplify the expression:

\(308 = \frac{1}{3} \times 22 \times 7 \times h\)

\(308 = \frac{154}{3} h\)

Now, solve for \(h\):

\(h = \frac{308 \times 3}{154}\)

\(h = \frac{924}{154}\)

\(h = 6\) cm

So, the height of the cone is 6 cm.

Step 2: Find the Slant Height (l) of the Cone

In a right circular cone, the radius (\(r\)), height (\(h\)), and slant height (\(l\)) form a right-angled triangle, where the slant height is the hypotenuse. The relationship is given by the Pythagorean theorem:

\(l^2 = r^2 + h^2\)

Substitute the values of \(r\) and \(h\) we found:

\(l^2 = (7)^2 + (6)^2\)

\(l^2 = 49 + 36\)

\(l^2 = 85\)

Now, take the square root to find \(l\):

\(l = \sqrt{85}\) cm

The slant height of the cone is \(\sqrt{85}\) cm.

Step 3: Calculate the Curved Surface Area (CSA) of the Cone

The formula for the curved surface area of a right circular cone is given by:

\(CSA = \pi r l\)

Substitute the values of \(\pi\), \(r\), and \(l\):

\(CSA = \frac{22}{7} \times 7 \times \sqrt{85}\)

\(CSA = 22 \times \sqrt{85}\)

\(CSA = 22\sqrt{85}\) cm²

Thus, the curved surface area of the cone is \(22\sqrt{85}\) cm².

Revision Table: Cone Formulas

Additional Information on Right Circular Cones

A right circular cone is a cone where the apex is directly above the center of the circular base. This creates a right angle between the height and the radius of the base.

• The base of a cone is a circle.
• The height is the perpendicular distance from the apex to the base.
• The slant height is the distance from the apex to any point on the circumference of the base.
• The formulas used above are specifically for right circular cones.
• Units are important in geometric calculations. Volume is in cubic units (cm³), lengths (radius, height, slant height) are in linear units (cm), and areas (curved, total) are in square units (cm²).