Cone Height Formula
A cone is a three-dimensional shape, formed by using a set of line segments or the lines which connect at a common point, called the apex or vertex, to all the points of a circular base(which does not contain the apex). We can also define the cone as a pyramid with a circular cross-section, unlike a pyramid that has a triangular cross-section. Let us study the cone height formula using solved examples at the end of the page.
What Is Cone Height Formula?
The cone height formula helps in calculating the distance from the vertex of the cone to the cone's base. The height of the cone can be calculated using either the volume of cube and radius or with slant height and radius of the cone.
Cone Height Formula
Cone Height Formula for Cone can be expressed as,
Formula 1: h = 3V/πr2
where,
- V = Volume of the cone
- r = Radius of the cone
This formula is derived from the formula of the volume of a cone.
Formula 2: h = √l2 - r2
where,
- l = Slant height of the cone
- r = Radius of the cone
This formula is derived using the Pythagoras theorem.
Let us see the applications of the cone height formula in the following section.
Examples Using Cone Height Formula
Example 1: A birthday cap is in conical shape having a volume of 20 units3 and its base radius is 5 units. What is the height of the cap?
Solution:
To find: The height of a cone.
Given:
volume = 20 units3
Radius = 5 units
Using cone height formula,
h = 3V/πr2
= (3 × 20)/π × 52
= (60)/ (π × 25)
= 0.76 units
Answer: The height of a cone is 0.76 units
Example 2: What is the height of the cone with the radius = 3 units and volume = 50 cubic units?
Solution:
To find: The height of a cone.
Given:
volume = 50 cubic units
Radius = 3 units
Using cone height formula,
h = 3V/πr2
= (3 × 50)/π × 32
= (150)/ (π × 9)
= 5.305 units
Answer: The height of a cone is 5.305 units.
Example 3: Determine the height of the cone with the radius = 5 units and slant height = 13 units?
Solution:
To find: The height of a cone.
Given:
Slant height = 13 units
Radius = 5 units
Using cone height formula,
h = √l2 - r2
= √(13)2 - (5)2
= √169-25
= √144
= 12 units
Answer: The height of a cone is 12 units.
FAQs on Cone Height Formula
What Is Cone Height Formula in Geometry?
The cone height formula calculates the height of the cone. The height of the cone using cone height formulas are, h = 3V/πr2 and h = √l2 - r2, where V = Volume of the cone, r = Radius of the cone, and l = Slant height of the cone.
How To Use Cone Height Formula?
To determine the height of the cone, we use the cone formula in the following way
- Step 1: Check for the given parameters, volume, and radius or slant height and radius.
- Step 2: Put the values in the appropriate formula, h = 3V/πr2 or h = √l2 - r2
What Is r in Cone Height Formula?
In the cone height formula, either h = 3V/πr2 or h = √l2 - r2, r represents the radius of the cone.
What Is Cone Height Formula Using Slant Height?
The cone height formula using slant height is √l2 - r2, where l is the slant height and r is the radius of the cone. This formula is derived using the Pythagoras theorem.
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