Surface Area of a Cone Calculator
Calculate the total, lateral, and base surface area of a right cone
Calculate Cone Surface Area
Distance from center to edge of base
Perpendicular distance from base to apex
Surface Area Formulas
Base Area: A_base = πr²
Lateral Area: A_lateral = πrl
Total Surface Area: A_total = πr² + πrl = πr(r + l)
With height: A_total = πr(r + √(h² + r²))
Example Calculations
Example 1: Given Height
Given: radius = 3, height = 4
Slant height: l = √(4² + 3²) = 5
Surface area: π × 3 × (3 + 5) = 75.4
Example 2: Given Slant Height
Given: radius = 5, slant height = 15
Surface area: π × 5 × (5 + 15) = 314.16
Height: h = √(15² - 5²) = 14.14
Quick Reference
Key Formulas
Base Area = πr²
Lateral Area = πrl
Total Area = πr(r + l)
Relationships
l = √(h² + r²)
h = √(l² - r²)
r = c / (2π)
Cone Properties
Quick Tips
Surface area includes both base and curved surface
Slant height is always greater than perpendicular height
Use Pythagorean theorem to relate h, r, and l
Lateral area is the curved surface when cone is unfolded
Understanding Cone Surface Area
What is a Right Cone?
This calculator works with right cones, which have a circular base and a vertex (tip) directly above the center of the base. The angle between the base and the line from the base center to the vertex is 90°.
Surface Area Components
A cone's surface area consists of two parts: the circular base area (πr²) and the lateral (curved) surface area (πrl). The total surface area is their sum.
Deriving the Formula
Step 1: Base Area
The base is a circle: A_base = πr²
Step 2: Lateral Area
When unfolded, the curved surface forms a sector: A_lateral = πrl
Step 3: Total Area
Sum both areas: A_total = πr² + πrl = πr(r + l)
Key Relationships
Pythagorean Relationship:
l² = h² + r²
Relates slant height, height, and radius
Circumference to Radius:
r = c / (2π)
Convert circumference to radius
Volume Formula:
V = (1/3)πr²h
Related volume calculation