How to find the height of a cone?

Q: Q1: Can I use the Pythagorean theorem if I only know the radius and height?

Yes, you can use the Pythagorean theorem by calculating the slant height using l = r 2 + ℎ 2

Q: Q2 Is there a shortcut for finding the height using trigonometric ratios?

Yes, the formula h = r ⋅ tan ( θ ) is a quick way to find the height if you know the radius and the half-angle.

Q: Q4: How do I measure the half-angle (θ) for the trigonometric ratios method?

A4: You can measure θ using a protractor or determine it based on the given information.

Author: Noreen Niazi
Last Updated on: January 18, 2024

SECTIONS

Knowing how to locate the height of a cone is very important in geometry. For students struggling with homework or professionals facing real life problems, this complete handbook will blow off the fog of cone heights.

What is a cone?

While starting with the methodology, let’s look back at the fundamentals of a cone. A cone is a figure with a circular base, an axis of symmetry perpendicular to the level plane passing through its centre and vertex or top. The altitude of the cone is that from a vertex to center.

How to find the height of a cone?

Method 1: Pythagorean Theorem Approach

Formula:

ℎ = \sqrt{r^{2} + h^{2}}

Where:

$ℎis the height,$
$r$ is the radius of the base,
$l$ is the slant height.

Solved Examples:

Given Radius and Slant Height:

$r = 4,$ $l = 7$
$ℎ = \sqrt{4^{2} + 7^{2}} = \sqrt{16 + 49} = \sqrt{65}$

- Therefore, $ℎ \approx 8.06$
Given Radius and Height of a cone:
- $r = 3$ , $ℎ = 5$ $l = \sqrt{r^{2} + ℎ^{2}} = \sqrt{3^{2} + 5^{2}} = \sqrt{9 + 25} = \sqrt{34}$
- Therefore, $l \approx 5.83$
Given Diameter and Slant Height:
- $d = 10units,$ $l = 12$ $r = \frac{10}{2} = 5$ $ℎ = \sqrt{5^{2} + 1 2^{2}} = \sqrt{25 + 144} = \sqrt{169}$
- Therefore, $ℎ = 13$

Given radius and height of a cone calculate the slant height, volume, lateral surface area and total surface area

Let’s consider a cone with a given radius ( $r$ ) and height (h)

Slant Height ( $l$ ):
- The slant height can be calculated using the Pythagorean theorem: $l= \sqrt{r^{2} + ℎ^{2}}$
Volume ( $V$ ):
- The volume of a cone is given by the formula: $V = \frac{1}{3} π r^{2} ℎ$
Lateral Surface Area ( $A_{l}$ ):
- The lateral surface area of a cone is given by: $A_{l} = π r l.$
Total Surface Area At
- The total surface area includes the base: $A_{t} = A_{l} + π r^{2}$

Now, let’s calculate these values with a given radius and height.

Example:

Given $r = 4$ and $ℎ = 6.$

Slant Height ( $l$ ):
- Using the Pythagorean theorem: $l = \sqrt{4^{2} + 6^{2}} = \sqrt{16 + 36} = \sqrt{52}$
- Therefore, $l\approx 7.21$
Volume ( $V$ ):
- Using the volume formula: $V = \frac{1}{3} π \times 4^{2} \times 6 = \frac{1}{3} π \times 16 \times 6$
  $π \times 16 \times 6$ cubic units.
- Therefore, $V \approx 32 π cubic units.$
Lateral Surface Area (Al):
- Using the lateral surface area formula: $A_{l} = π \times 4 \times 7.21$ square units.
- Therefore, $A_{l} \approx 90.41$ $π$ square units.
Total Surface Area (At):
- Using the total surface area formula: $A_{t} = 90.41 π +π \times 4^{2}$ square units.
- Therefore, $A_{t} \approx 106.41 π$ square units.

So, for a cone with a radius of 4 units and a height of 6 units:

Slant Height l is approximately 7.21 units.
Volume $V is approximately$ $32 π cubic units.$
The Lateral Surface Area $90.41 π square units.$
The total Surface Area $106.41 π square units.$

Given r, s find h, V, L, A

Let’s consider a cone with given radius ( $r$ ) and volume ( $V$ ). We’ll calculate the height (ℎ), slant height $(s$ ), lateral surface area (Al), and total surface area ( $A_{t}$ ).
1. Height (ℎ):
  - The height can be determined using the volume formula: $ℎ = \frac{3 π}{3 r^{2}}$
    
    .
2. Slant Height (s):
  - The slant height can be calculated using the Pythagorean theorem: $s = \sqrt{r^{2} + ℎ^{2}}$
3. Lateral Surface Area ( $A_{l}$ ):
  - The lateral surface area of a cone is given by: $A_{l} = π r s$
4. Total Surface Area ( $A_{t}$ ):
  - The total surface area includes the base: $A_{t} = A_{l} + π r^{2}$
Now, let’s calculate these values with a given radius ( $r$ ) and volume ( $V$ ).
Example:
Given $r = 4$ units and $V = 32 π$ cubic units.
1. Height ( $h$ ):
  - Using the height formula: $ℎ = \frac{3 \times 32 π}{π \times 4^{2}} = \frac{96}{16}$
  - Therefore, $h = 6$ units.
2. Slant Height ( $s$ ):
  - Using the Pythagorean theorem: $s = \sqrt{4^{2} + 6^{2}} = \sqrt{16 + 36} = \sqrt{52}$
  - Therefore, $s \approx 7.21$
3. Lateral Surface Area ( $A_{l}$ ):
  - Using the lateral surface area formula:
    $A_{l} = π \times 4 \times 7.21$ square units.
  - Therefore, $A_{l} \approx 90.41 π square units.$
4. Total Surface Area ( $A_{t}$ ):
  - Using the total surface area formula:
    $A_{t} = 90.41 π + π \times 4^{2}$ square units.
  - Therefore,
    $A_{t} \approx 106.41 π$ square units.
So, for a cone with a radius of 4 units and a volume of $32 π$ cubic units:
- Height ( $h$ ) is 6 units.
- Slant Height ( $s$ ) is approximately 7.21 units.
- Lateral Surface Area ( $A_{l}$ ) is approximately $90.41 π square units.$
- Total Surface Area (At) $106.41 π$ square units.

Given r, V find h, s, L, A

Let’s consider a cone with given radius ( $r$ ) and slant height ( $s$ ). We’ll calculate the height ( $h$ ), volume ( $V$ ), lateral surface area ( $A_{l}$ ), and total surface area ( $A_{t}$ ).
1. Height ( $h$ ):
  - The height can be calculated using the Pythagorean theorem: $ℎ = \sqrt{s^{2} - r^{2}}$
2. Volume ( $V$ ):
  - The volume of a cone is given by the formula: $V = \frac{1}{3} π r^{2} ℎ$
3. Lateral Surface Area ( $A_{l}$ ):
  - The lateral surface area of a cone is given by: $A_{l} = π rs$ .
4. Total Surface Area ( $A_{t}$ ):
  - The total surface area includes the base: $A_{t} = A_{l} + π r^{2}$
Now, let’s calculate these values with a given radius ( $r$ ) and slant height ( $s$ ).
Example:
Given $r = 3$ units and $s = 5$ units.
1. Height ( $h$ ):
  - Using the Pythagorean theorem: $ℎ = \sqrt{5^{2} - 3^{2}} = \sqrt{25 - 9} = \sqrt{16}$
    units.
  - Therefore, h=4 $units.$
2. Volume ( $V$ ):
  - Using the volume formula: $V = \frac{1}{3} π \times 3^{2} \times 4 = \frac{1}{3} π \times 36cubic units.$
  - Therefore, $V = 12 π$ cubic units.
3. Lateral Surface Area ( $A_{l}$ ):
  - Using the lateral surface area formula: $A_{l} = π \times 3 \times 5$ square units.
  - Therefore, $A_{l} = 15 π$ square units.
4. Total Surface Area ( $A_{t}$ ):
  - Using the total surface area formula: $_{}$ $A_{t} = 15 π + π \times 3^{2}$ square units.
  - Therefore, $A_{t} = 24 π$ square units.
So, for a cone with a radius of 3 units and a slant height of 5 units:
- Height ( $h$ ) is 4 units.
- Volume ( $V$ ) is $12 π$ cubic units.
- Lateral Surface Area ( $_{l}$ ) is $15 π$ square units.
- Total Surface Area ( $A_{t}$ ) is $24 π$ square units.

Given r, L find h, s, V, A

Let’s consider a cone with given radius ( $r$ ) and lateral surface area ( $A_{l}$ ). We’ll calculate the height ( $h$ ), slant height ( $s$ ), volume ( $V$ ), and total surface area ( $A_{t}$ ).
1. Height ( $h$ ):
  - The height can be determined using the lateral surface area and radius: $ℎ = \frac{A_{l}}{π r}$
2. Slant Height ( $s$ ):
  - The slant height can be calculated using the Pythagorean theorem: $s = \sqrt{r^{2} + ℎ^{2}}$
3. Volume ( $V$ ):
  - The volume of a cone is given by the formula: $V = \frac{1}{3} π r 2 ℎ$
4. Total Surface Area ( $A_{t}$ ):
  - The total surface area includes the base: $A t = A l + π r 2$
Now, let’s calculate these values with a given radius ( $r$ ) and lateral surface area ( $A_{l}$ ).
Example:
Given
r=5 units

$A l = 50 π$

1. Slant Height ( $s$ ):
  - Using the Pythagorean theorem: $s = \sqrt{5^{2} + 1 0^{2}} = \sqrt{25 + 100} = \sqrt{125}$
  - Therefore, $s = 5 \sqrt{5}$
2. Volume ( $V$ ):
  - Using the volume formula: $V = \frac{1}{3} π \times 5^{2} \times 10$ units.
  - Therefore, $V = \frac{250}{3} π$
3. Total Surface Area ( $A_{t}$ ):
  - Using the total surface area formula: $A_{t} = 50 π + π \times 5^{2}$ square units.
  - Therefore, $A_{t} = 75 π$ square units.
    Height (h)
- Using the height formula:h $= \frac{50 π}{π \times 5}$
- Therefore,h $= 10$
- Height ( $h$ ) is 10 units.
- Slant Height ( $s$ ) is $5 \sqrt{5}$
- Volume ( $V$ ) is $\frac{250}{3} π$
- Total Surface Area ( $A_{t}$ ) is $75 π.$

Given s, L find r, h, V, A

Let’s consider a cone with given slant height ( $s$ ) and lateral surface area ( $A_{l}$ ). We’ll calculate the radius ( $r$ ), height ( $h$ ), volume ( $V$ ), and total surface area ( $A_{t}$ ).

Radius ( $r$ ):
- The radius can be determined using the Pythagorean theorem: $r = \sqrt{s^{2} - {(\frac{π}{2 L})}^{2}}$
Height ( $h$ ):
- The height can be calculated using the lateral surface area and radius: $ℎ = \frac{A_{l}}{π r}$
Volume (V):
- The volume of a cone is given by the formula: $V = \frac{1}{3} π r^{2} ℎ$
Total Surface Area ( $A_{t}$ ):
- The total surface area includes the base: $A_{t} = A_{l} + π r^{2}$

Now, let’s calculate these values with a given slant height ( $s$ ) and lateral surface area ( $A_{l}$ ).

Example:

Given $s = 12$ units and $A_{l} = 72 π$ square units.

Radius ( $r$ ):
- Using the Pythagorean theorem: $r = \sqrt{1 2^{2} - {(\frac{72 π}{2 π})}^{2}} = \sqrt{144 - 36} = \sqrt{108}$
- Therefore
Height ( $h$ ):
- Using the height formula: $ℎ = \frac{72 π}{π \times 6 \sqrt{3}}$ units.
- Therefore, $ℎ = 2 \sqrt{3}$
Volume ( $V$ ):
- Using the volume formula: $V = \frac{1}{3} π \times (6 \sqrt{3})^{2} \times 2 \sqrt{3}$ cubic units.
- Therefore, $V = 72 π$ cubic units.
Total Surface Area ( $A_{t}$ ):
- Using the total surface area formula: $A_{t} = 72 π + π \times (6 \sqrt{3})^{2}$ square units.
- Therefore, $A_{t} = 108 π square units.$

So, for a cone with a slant height of 12 units and a lateral surface area of $72 π$ square units:

Radius ( $r$ ) is $6 \sqrt{3}$
Height ( $h$ ) is $2 \sqrt{3}$
Volume ( $V$ ) is $72 π$ cubic units.
Total Surface Area ( $A_{t}$ ) is $108 π square units.$

Method 2: Trigonometric Ratios Approach

Formula:

$h = r \cdot tan (θ)$

Where:

$h$ is the height,
$r$ is the radius of the base,
$θ$ is the half-angle at the apex.

Solved Examples:

Given Radius and Half-Angle:
- $r = 6$ units, $θ = 3 0^{\circ}$
- $tan (3 0^{\circ}) \approx 0.577$
- $h = r \cdot tan (θ) = 6 \cdot 0.577$ units
- Therefore, $h \approx 3.46$ units.
Given Diameter and Half-Angle:
- $d = 8$ units, $θ = 4 5 ∘$
- $d = \frac{d}{2} = \frac{8}{2} = 4$
- $\tan (4 5^{\circ}) = 1$
- $h = r \cdot tan (θ) = 4 \cdot 1$ units
- Therefore, $h = 4$ units.

Real-World Applications:

Now, let’s explore how finding cone heights is applied in real-world scenarios.

Word Problems:

Construction Cone:
- A traffic cone has a base radius of 18 inches. If the slant height is 24 inches, find the height.
- $ℎ = \sqrt{1 8^{2} + 2 4^{2}} = \sqrt{324 + 576} = \sqrt{900}$
- Therefore, the height is 30 inches.
Ice Cream Cone:
- An ice cream cone has a slant height of 10 cm and a radius of 6 cm. Determine its height.
- $ℎ = \sqrt{6^{2} + 1 0^{2}} = \sqrt{36 + 100} = \sqrt{136}$
- Therefore, the height is approximately 11.66 cm.
Shadow of a Flagpole:
- A flagpole casts a shadow 20 meters long. If the angle of elevation to the sun is $3 0^{\circ}$ , find the height of the flagpole.
- $ℎ = 20 \cdot \tan (3 0^{\circ}) \approx 20 \cdot 0.577$
- Therefore, the height is approximately 11.54 meters.
Mountain Slope:
- A cone-shaped mountain has a radius of 500 meters and a slant height of 800 meters. What is the height of the mountain?
- $ℎ = \sqrt{50 0^{2} + 80 0^{2}} = \sqrt{250000 + 640000} = \sqrt{890000}$
- Therefore, the height is 943.4 meters.
Rocket Nose Cone:

The nose cone of a rocket has a radius of 2 feet and a height of 5 feet. Find the slant height.
$l = \sqrt{2^{2} + 5^{2}} = \sqrt{4 + 25} = \sqrt{29}$
Therefore, the slant height is approximately 5.39 feet.

Practice Problems

Problem: Given the radius () of a cone is 8 units, and the slant height () is 10 units, find the height ().
Problem: A cone has a height () of 12 units and a volume () of $36 π$ cubic units. Calculate the radius ().
Problem: For a cone with a slant height () of 15 units and a lateral surface area () of $45 π$ square units, determine the radius ( $r$ ).
Problem: The volume () of a cone is $64 π$ cubic units, and the height ( $ℎ$ ) is 4 units. Find the slant height ().
Problem: Given the radius () is 6 units, find the volume () of the cone. (Use )
Problem: A cone has a slant height () of 8 units and a lateral surface area () of square units. Determine the radius (r $).$
Problem: The height ( $ℎ$ ) of a cone is 9 units, and the slant height () is 12 units. Calculate the total surface area ( $A_{t}$ ).
Problem: For a cone with a volume () of cubic units and a height ( $ℎ$ ) of 5 units, find the slant height ( $s$ ).
Problem: A cone has a total surface area () of $80 π$ square units, and the radius () is four units. Determine the height ().
Problem: Given the slant height ( $s$ ) is 14 units, find the lateral surface area ( $A_{l}$ ) for a cone with a radius () of 7 units.

Answer Keys of practice Problems

units
$r = 3$ units
$r = 9$ units
units
cubic units
$r = 6$ units
square units
units
$h = 5$ units
square units

Conclusion:

In this comprehensive guide, we’ve explored three methods to find the height of a cone: the Pythagorean theorem, corresponding triangles and relation of volume-base area. However, by providing 10 solved cases and five real-life challenges, we have given you the necessary tools to calculate cone heights with ease. Whatever, whether you use them for exploring geometric principles or solving practical puzzles they will come in handy if the desire is to understand cone heights.

FAQs

Q1: Can I use the Pythagorean theorem if I only know the radius and height?

Q2 Is there a shortcut for finding the height using trigonometric ratios?

Yes, the formula $h = r \cdot tan (θ)$ is a quick way to find the height if you know the radius and the half-angle.

Q3: Are these formulas applicable to any cone?

A3: Yes, these formulas work for any cone, regardless of its size or proportions.

Q4: How do I measure the half-angle (θ) for the trigonometric ratios method?

A4: You can measure using a protractor or determine it based on the given information.

Q5: Can I find the height of a cone without knowing the slant height or half-angle?

A5: No, either the slant height or the half-angle is required to calculate the height using the given methods.

How to find the height of a cone?

What is a cone?

How to find the height of a cone?

Method 1: Pythagorean Theorem Approach

Formula:

Solved Examples:

Given radius and height of a cone calculate the slant height, volume, lateral surface area and total surface area

Example:

Given r, s find h, V, L, A

Example:

Given r, V find h, s, L, A

Example:

Given r, L find h, s, V, A

Example:

Given s, L find r, h, V, A

Example:

Method 2: Trigonometric Ratios Approach

Formula:

Solved Examples:

Real-World Applications:

Word Problems:

Practice Problems

Answer Keys of practice Problems

Conclusion:

FAQs

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