Right Circular Cone: Volume, Surface Area, and Examples
Right Circular Cone: Volume, Surface Area, and Examples
TL;DR
A right circular cone is a 3D solid with one circular base and a single apex that sits directly above the centre of that base. Its volume is ( V = \frac{1}{3}\pi r^2 h ), its curved surface area is ( CSA = \pi r l ), and its total surface area is ( TSA = \pi r(l + r) ), where ( l ) is the slant height.
What is a Right Circular Cone?
A right circular cone is a three-dimensional solid with a flat circular base and a curved surface that tapers smoothly to a single point called the apex (or vertex), where the apex sits directly above the centre of the base. "Circular" means the base is a circle; "right" means the line from the apex to the base centre — the axis — meets the base at a right angle.
- The radius (r) is the radius of the circular base.
- The height (h) is the perpendicular distance straight up from the base centre to the apex.
- The slant height (l) is the distance from any point on the rim, straight up the slanted surface, to the apex.
A right circular cone has two surfaces (one flat circular base, one curved lateral surface), one curved edge (the rim), and one vertex (the apex).
Height, Slant Height, And The Right Triangle That Links Them
The height (h) goes straight up the middle. The slant height (l) runs along the outside surface. Because the cone is right, the radius ( r ), the height ( h ), and the slant height ( l ) form a right triangle inside the cone, with ( l ) as the hypotenuse. By the Pythagorean theorem:
[ l = \sqrt{r^2 + h^2} ]
Volume of A Right Circular Cone
The volume of a right circular cone is:
[ V = \frac{1}{3}\pi r^2 h ]
Surface Area of a Right Circular Cone
Curved (lateral) surface area (CSA):
[ CSA = \pi r l ]
Total surface area (TSA):
[ TSA = \pi r(l + r) ]
| Quantity | Formula | Units |
|---|---|---|
| Slant height | ( l = \sqrt{r^2 + h^2} ) | length |
| Volume | ( V = \frac{1}{3}\pi r^2 h ) | cubic |
| Curved surface area | ( CSA = \pi r l ) | square |
| Total surface area | ( TSA = \pi r (l + r) ) | square |
Examples of the Right Circular Cone
For consistency, every example below uses centimetres and takes ( \pi \approx 3.14 ).
Example 1
A right circular cone has radius 3 cm and height 4 cm. Find its slant height.
[ l = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm} ]
Example 2
A right circular cone has radius 7 cm and height 9 cm. A student finds the total surface area using the height instead of the slant height. Find the correct total surface area.
Correct slant height: [ l = \sqrt{7^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4 \text{ cm} ]
[ TSA = 3.14 \times 7 \times (11.4 + 7) = 404.4 \text{ cm}^2 ]
Example 3
Find the volume of a right circular cone with radius 6 cm and height 10 cm.
[ V = \frac{1}{3} \times 3.14 \times (6^2) \times 10 \approx 376.8 \text{ cm}^3 ]
Example 4
Find the curved surface area of a right circular cone with radius 5 cm and slant height 13 cm.
[ CSA = 3.14 \times 5 \times 13 \approx 204.1 \text{ cm}^2 ]
Example 5
A right circular cone has radius 8 cm and slant height 17 cm. Find its total surface area.
[ TSA = 3.14 \times 8 \times (17 + 8) = 628 \text{ cm}^2 ]
Example 6
A right circular cone has volume 100π cm³ and radius 5 cm. Find its height.
[ h = \frac{100 \times 3}{25} = 12 \text{ cm} ]
Conclusion
- A right circular cone has a circular base and an apex directly above the base centre, so its axis is perpendicular to the base.
- Volume is ( \frac{1}{3}\pi r^2 h ) — exactly one-third of the matching cylinder.
- Curved surface area is ( \pi r l ); total surface area is ( \pi r(l + r) ).