Cone Surface Area Calculator
Calculate the lateral, base, and total surface area of a cone: SA = πr(r + l)
Enter Radius and Slant Height
Base Area: 50.27 square units
Total Surface Area: 138.23 square units
Calculations:
Lateral SA: πrl = π(4)(7) = 28π = 87.96 square units
Base Area: πr² = π(4)² = 16π = 50.27 square units
Total SA: πr(r + l) = π(4)(4 + 7) = π(4)(11) = 44π = 138.23 square units
Cone Surface Area Formulas
Lateral Surface Area: LA = πrl
Base Area: B = πr²
Total Surface Area: SA = πr(r + l) = πrl + πr²
where: - r = radius of the circular base - l = slant height (distance from base edge to apex along the surface) - π ≈ 3.14159
How to Calculate Cone Surface Area
- Measure the radius: Find the distance from the center of the base to its edge
- Find the slant height: Measure the distance along the cone's surface from base to tip
- Calculate lateral area: Multiply π × r × l
- Calculate base area: Multiply π × r²
- Add them: Total SA = lateral area + base area
If you know the height (h) instead of slant height, use: l = √(r² + h²)
Understanding Cone Surface Area
When you "unroll" the lateral surface of a cone, it forms a sector of a circle with radius equal to the slant height. The arc length of this sector equals the circumference of the cone's base (2πr). The area of this sector is πrl, which is the lateral surface area.
Real-World Applications
Cone surface area calculations are used in: - Party hats and funnels: Material needed for construction - Ice cream cones: Wafer material calculations - Architecture: Conical roof material estimates - Industrial design: Sheet metal for conical components - Tent design: Fabric needed for tepee-style tents
Common Questions
What is the formula for cone surface area?
The total surface area of a cone is SA = πr(r + l), where r is the base radius and l is the slant height. This combines the lateral (side) area πrl and the base area πr².
What is the difference between slant height and height of a cone?
The slant height (l) is the distance along the surface from the base edge to the apex. The height (h) is the perpendicular distance from the base to the apex. They are related by l = √(r² + h²).