# Slant Height calculator

Created by Vishnuvardhan Shakthibala
Reviewed by Anna Szczepanek, PhD and Jack Bowater
Last updated: Jun 05, 2023

Do you want to calculate the slant height of a right circular cone or a right-angled pyramid? If you said "Yes," then our slant height calculator is the right match.

• What is a slant height?
• How to calculate slant height for a right circular cone and a right-angled pyramid?
• How to use our calculator effectively.

## What is slant height?

The slant height is the side of a right triangle opposite to the right angle. It is also known as the hypotenuse.

For right-angled pyramids, the slant height is the shortest distance measured from the apex to the boundary points located at the base. Particularly, the slant height connects the midpoint of the base to the pyramid's apex.

Some examples of slant height are in the figures below.

⚠️ For a right polygon-based pyramid with height $h$ and base width $2b$, the slant height is not equivalent to the lateral edge. In terms of equations, we have:

• Slant height: $l=h^2+b^2$; and
• Lateral edge $d=h^2+2b^2$.

As you can see, the lateral edge is greater than the slant height.

## How to calculate the slant height?

Now, let us try to understand how to calculate slant height for different geometric shapes. You can use the following steps to calculate the slant height for different shapes:

1. Find the height $h$ and base length $b$ of the shape for which you want to calculate the slant height $l$.
2. Use the Pythagorean theorem to find the slant height:
$\footnotesize \qquad l=\sqrt{h^2+b^2}$
• You can apply this procedure to different geometric shapes:

• Right circular cone – You need height $h$ and base radius $r$ to compute the slant height; and

• Right angle pyramid – You need height $h$ and base length $b$ to compute the slant height.

The only restriction you need to keep in mind is that the opposite sides of the slant height need to be at $90$ degrees to each other (at right angles).

Cool! Now you know how to calculate the slant height! If you are curious to learn different ways to calculate the slant height of a cone, please check out our slant height of a cone calculator.

## The angles of elevation and depression

Let us consider a hypothetical situation: two persons (P and Q) are in the park. Person $P$ has climbed up the tree, and Person $Q$ is on the ground at a distance of $b$ from the tree. You can find schematics of this problem in the figure below.

Now the stage is set to solve several problems which involve slant height and other new parameters (you will see them soon 😉).

First problem – We want to calculate how high $h$ Person P is in the tree. To do that, we need to know:

• The distance between the Person P and the tree $b$;
• The angle from the horizontal $\theta$ Person P is looking at the person Q; and
• The distance $t$ between Person P's eye-level and the ground.
1. By using simple trigonometry, we can find out the height $(h-t)$ between Person Q and Person P's eye level:
$\qquad \footnotesize (h-t) = b\tan{\theta}$
1. Rearranging the above equation, we can obtain the height Person Q is above the ground:
$\qquad \footnotesize h= b\tan{\theta}+t$

Angle $\theta$ is called the elevation angle. You notice that Person Q looks down at Person P at the angle $\phi$ from the horizontal, known as the angle of depression. To know about the angle of depression, please check out our angle of depression calculator. In the figure above, the angles of elevation and depression are equal, but this is not always the case.

Second problem – We want to determine the distance between Persons P and Q. You can see from the above figure that we have a right-angle triangle. Therefore, this problem involves slant height $l$. As a result, this problem is essentially a slant height (distance between P and Q) calculation problem.

1. Find at least two out of the following three parameters: $(h-t)$, $b$, and $\theta$.
2. Depending on what is given, you can calculate slant height $l$, or the distance between P and Q, in 3 different cases using the following equations:
• Given height $(h-t)$ and angle of elevation $\theta$: $l= {(h-t)}/{\sin{\theta}}$
• Given base $b$ and angle of elevation $\theta$: $l= {b}/{\cos{\theta}}$
• Given height $(h-t)$ and base $b$: make use of Pythagoras theorem to obtain slant height: $l=\sqrt{(h-t)^2+b^2}$.

## How to use our slant height calculator?

1. Choose the type of shape you want to find the slant height of using the Choose the shape option.
2. Insert the base $b$ or base radius $r$ and height $h$ into the respective fields.
3. Our calculator will give you the slant height $l$ right away.

## FAQ

### How do you calculate the slant height of a right circular cone?

To calculate the slant height of a right circular cone, follow the steps below:

1. Obtain the radius of the circular base r and the height h of the cone.
2. Square both values and add them together.
3. Find the square root of the resulting value.

### What is the angle of elevation?

The angle of elevation, or elevation angle, is defined as the angle between the line connecting the observer and the observed object and the horizontal.

### What is the elevation angle from point (1,5) to point (5,8)?

The elevation angle is 36.86 degrees. Use the following steps to calculate this:

1. Find the distance l between the points (1,5) and (5,8) using Pythagoras: l = √((x₂-x₁)²+(y₂-y₁)²) = √((5-1)²+(8-5)²) = 5. This is our slant height.

2. Construct a right-angle triangle by keeping slant height as its hypotenuse.

3. Calculate height: h = (y₂-y₁) = 3.

4. Take arcsin of the ratio of height h = 3 and hypotenuse l = 5 to obtain the angle of elevation as 36.86 degrees.

### How do you find a triangle's height with its slant height and base?

To calculate the height of a triangle given its slant height and base, follow the instructions below:

1. Make sure the triangle is a right-angle triangle – the height, and the base are perpendicular to each other.
2. Use Pythagoras theorem to calculate the height: height=√((slant height)²-(base)²).
Vishnuvardhan Shakthibala
Shape type
Right circular cone ft
Height (h)
ft
Slant height (l)
ft
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