Radius of a Cone Calculator
Calculate the radius of a cone using various formulas with step-by-step solutions
Calculate Cone Radius
Cone Calculation Results
Cone Components Diagram
Cone Radius Formulas
Primary Method
From Volume
From Areas
Example Calculations
From Height & Slant
Given: h = 12 cm, l = 15 cm
Formula: r = √(l² - h²)
Calculation: r = √(225 - 144) = √81
Result: r = 9 cm
From Base Area
Given: AB = 78.54 cm²
Formula: r = √(AB / π)
Calculation: r = √(78.54 / 3.14159)
Result: r = 5 cm
Cone Facts
Slant height must be greater than height
Radius determines the base area (πr²)
Volume = (1/3)πr²h
Lateral area = πrl
Understanding Cone Radius Calculations
What is the Radius of a Cone?
The radius of a cone is the distance from the center of the circular base to any point on the circumference of the base. It's a fundamental measurement that determines all other cone properties.
Key Relationships
- •The radius is perpendicular to the height at the base center
- •Radius, height, and slant height form a right triangle
- •Pythagorean theorem: l² = r² + h²
- •The radius determines the cone's base area and volume
Multiple Calculation Methods
Real-World Applications
- •Engineering: Funnel and pipe design
- •Architecture: Conical roofs and structures
- •Manufacturing: Container and packaging design
- •Mathematics: Volume and surface area calculations
Important Note: Radius vs Height Relationship
The height and radius of a cone are NOT proportional to each other. You cannot predict one based solely on the other without additional information like slant height, volume, or surface area.
Frequently Asked Questions
How can I calculate the radius of a cone?
The simplest formula is: r = √(l² - h²), where:
- Square the slant height (l²)
- Square the height (h²)
- Subtract height squared from slant height squared
- Find the square root of the result
What is the radius of a cone with base area of 34 cm²?
The radius is 3.29 cm. Using the formula r = √(AB / π):
- Divide the base area by π: 34 ÷ 3.14159 = 10.826
- Take the square root: √10.826 = 3.29 cm
Why must the slant height be greater than the height?
In a cone, the radius, height, and slant height form a right triangle where the slant height is the hypotenuse. By the Pythagorean theorem, the hypotenuse must always be longer than either of the other two sides, so l > h and l > r.