The volume of a right circular cone is 308 cm 3 and the radius of its base is 7 cm. What is the curved surface area (in cm 2) of the cone? (Take π = \(\frac{22}{7} \) )
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
| Aspect | Formula | Variables |
|---|---|---|
| Volume (V) | \(\frac{1}{3}\pi r^2 h\) | r = radius, h = height |
| Curved Surface Area (CSA) | \(\pi r l\) | r = radius, l = slant height |
| Total Surface Area (TSA) | \(\pi r (r + l)\) | r = radius, l = slant height |
| Slant Height (l) | \(\sqrt{r^2 + h^2}\) | r = radius, h = height |
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).
Important Questions from Solid Figures
A cylindrical tube, open at both ends, is made of a metal sheet which is 0.5 cm thick. Its outer radius is 4 cm and length is 2 m. How much metal (in cm 3) has been used in making the tube?
The slant height and radius of a right circular cone are in the ratio 29 ∶ 20. If its volume is 4838.4 π cm 3, then its radius is:
Six cubes, each of edge 2 cm, are joined end to end. What is the total surface area of the resulting cuboid in cm 2?
A solid cube of side 8 cm is dropped into a rectangular container of length 16 cm, breadth 8 cm and height 15 cm which is partly filled with water. If the cube is completely submerged, then the rise of water level (in cm) is:
The radii of two cylinders are in the ratio 3 : 4 and their heights are in the ratio 8 : 5. The ratio of their volumes is equal to:
