The surface area of a cone tells us how much total area covers the cone from all sides, including its curved part and circular base. Whether you’re working on geometry questions, designing objects like funnels or lampshades, or understanding real-life structures, this formula helps you determine the full outer covering of a cone. It’s commonly used in exams, construction planning, packaging design, and mathematical modelling.
Surface Area of a Cone Formula Overview
| Formula | Variables & Meaning | When It Is Used |
|---|---|---|
| TSA = πr(l + r) | r = base radius, l = slant height, π = 3.1416 | Used to calculate the total surface area of a cone |
What is the Surface Area of a Cone in Maths?
In mathematics, the surface area of a cone represents the combined area of its curved surface and its circular base. The curved surface area (CSA) is calculated using πrl, while the area of its base is πr². When these two are added, we get the total surface area expressed as πr(l + r).
A cone is a three-dimensional solid with a circular base and a pointed top (vertex). Its total surface area includes:
- Curved Surface Area (CSA) → πrl
- Base Area → πr²
So the formula becomes:
Total Surface Area (TSA) = πr(l + r)
If the slant height is not provided, it can be found using the Pythagoras theorem:
l = √(r² + h²),
where h is the perpendicular height of the cone.
This formula is essential in geometry, architecture, engineering, and competitive exams such as CUET, SSC, JEE, and school-level tests.
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 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 units are used for surface area?
Surface area is measured in square units such as cm² or m².
Q5. Where is this formula used in daily life?
It is used in designing tents, funnels, lampshades, ice-cream cones, and similar shapes.
Q6. Is TSA always greater than CSA?
Yes, because TSA includes both the curved surface area and the base area.