# 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/πr^{2}

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 = √l^{2} - r^{2}

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.

**Break down tough concepts through simple visuals.**

## Examples Using Cone Height Formula

**Example 1:** A birthday cap is in conical shape having a volume of 20 units^{3} 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 units^{3}

Radius = 5 units

Using cone height formula,

h = 3V/πr^{2}

= (3 × 20)/π × 5^{2}

= (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/πr^{2}

= (3 × 50)/π × 3^{2}

= (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 = √l^{2} - r^{2}

= √(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/πr^{2} and h = √l^{2} - r^{2}, 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/πr
^{2}or h = √l^{2}- r^{2}

### What Is r in Cone Height Formula?

In the cone height formula, either h = 3V/πr^{2} or h = √l^{2} - r^{2}, r represents the radius of the cone.

### What Is Cone Height Formula Using Slant Height?

The cone height formula using slant height is √l^{2} - r^{2}, where l is the slant height and r is the radius of the cone. This formula is derived using the Pythagoras theorem.

visual curriculum