The surface area of a cone helps us find the total outer covering of a cone-shaped object. From ice cream cones to funnels and lampshades, cones appear everywhere in daily life. Knowing how to calculate their surface area is important in geometry, design, engineering, and even competitive exams. The formula combines both the curved surface area and the base area, giving a complete measurement of how much material the cone covers.
Surface Area of a Cone Formula Overview
| Formula | Variables & Meaning | When It Is Used |
|---|---|---|
| TSA = πr(l + r) | r = radius, l = slant height, π = 3.1416 | Used to find the total outer surface area of a cone |
What is the Surface Area of a Cone?
In mathematics, the surface area of a cone refers to the total outer covering of the cone, including both its curved surface and its circular base. The cone has two main areas:
- Curved Surface Area (CSA) = πrl
- Base Area = πr²
When we add these two parts, we get the Total Surface Area (TSA):
TSA = πr(l + r)
A cone is a 3D shape with a circular base and a pointed top known as the vertex. If the slant height (l) is not directly given, it can be calculated using the Pythagoras theorem:
l = √(r² + h²)
where h is the perpendicular height of the cone.
The surface area formula is widely used in geometry, construction, designing conical objects, and exams like CUET, SSC, Banking, and JEE.
Examples to Calculate Surface Area of a Cone
Example 1: Radius = 7 cm, Slant Height = 10 cm
Step 1: TSA = πr(l + r)
Step 2: = 3.14 × 7 × (10 + 7)
Step 3: = 3.14 × 7 × 17
Result: 373.94 cm²
So, the total surface area of the cone is 373.94 cm².
Example 2: Radius = 5 m, Slant Height = 13 m
Step 1: TSA = πr(l + r)
Step 2: = 3.14 × 5 × (13 + 5)
Step 3: = 3.14 × 5 × 18
Result: 282.6 m²
So, the total surface area of the cone is 282.6 m².
FAQs about Surface Area of a Cone Formula
Q1. What is the total surface area of a cone?
It is the sum of the curved surface area and the circular base area.
Q2. What is the curved surface area formula of a cone?
The curved surface area (CSA) of a cone is πrl.
Q3. How do you find the slant height of a cone?
Use the formula l = √(r² + h²), where h is the perpendicular height.
Q4. What are the units of surface area?
Surface area is measured in square units like cm², m², etc.
Q5. Where is this formula used in real life?
It is used in making cones, funnels, lampshades, tents, and other conical designs.
Q6. Is TSA always greater than CSA?
Yes, because TSA includes the curved surface plus the base area.