Slant Height - Definition, Formula, and Examples

Have you ever considered about how long the "slope" is on the side of a birthday hat or a tent? That is called the slant height in geometry. The regular height (altitude) tells us how tall something is from the ground to the top, and the slant height tells us how long the diagonal side is. Understanding what is slant height is essential for calculating how much paper you need to wrap a cone or how much canvas is needed for a pyramid-shaped tent.

What is Slant Height?

The slant height definition is the distance from the highest point (the apex) of a 3D object to a point on the base's edge, measured along the surface.

Picture yourself at the summit of a pyramid.

• Vertical Height: If you dropped the stone straight down from the top, the vertical height (h) is the distance from the top of the stone to the floor.

• Slant Height: If you slid down the edge of the pyramid to the bottom, the Slant Height (l) is the distance you travelled.

Slant Height vs. Vertical Height  

Let’s understand the difference between vertical height and slant height:

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Slant Height Formula

The Pythagorean Theorem, a little bit of "math magic," helps us find the slant height. This is because the vertical height, the base radius (or half-side), and the slant height all make up a perfect right triangle. 

1. For a Cone

If you know the radius (r) and the vertical height (h), the formula is:

l = \sqrt{r^2 + h^2}

2. For a Square Pyramid

If you know the side length of the base (s) and the vertical height (h), the formula is:

l = \sqrt{\left(\frac{s}{2}\right)^2 + h^2}

Slant Height Examples

Let's look at some examples of slant height to see how these figures function.

Example 1: The Ice Cream Cone

The vertical height of an ice cream cone is 12 cm, while the radius is 5 cm. What is the slant height of the cone?

• Step 1: Identify the numbers. r = 5, h = 12.

• Step 2: Plug them into the formula:
  l = \sqrt{5^2 + 12^2}

• Step 3: Solve:
  l = \sqrt{25 + 144} = \sqrt{169}

• Answer: l = 13 \text{ cm}.

Example 2: The Camping Tent

The base of a square pyramid tent is 6 meters long and the height is 4 meters. What is the slant height?

• Step 1: Find half the base side. s/2 = 6 / 2 = 3 \text{ m}.

• Step 2: Identify h = 4 \text{ m}.

• Step 3: Solve:
  l = \sqrt{3^2 + 4^2}

• Step 4:
  l = \sqrt{9 + 16} = \sqrt{25}

• Answer: 

• l = 5 \text{ meters}.

Read More - Area of a Square - Formulas, How to Find, Derivation, Examples

How To Understand the Slant Height Formula Clearly?

We can picture a right triangle inside a cone or pyramid when we look at it from the side. The triangle is made up of the vertical height (h), the base radius or half-side (r or s/2), and the slant height (l).

For a cone:

• r = radius of the base
  
  

• h = vertical height
  
  

• l = slant height
  
  

So we apply the Pythagorean Theorem:
l² = h² + r²

For a square pyramid:

 l² = h² + (s/2)²

Here, s is the side of the square base.

Example with a Non-Perfect Square

A cone has radius 7 cm and height 10 cm.
Find the slant height.

l = √(7² + 10²)
l = √(49 + 100)
l = √149

Since 149 is not a perfect square, the answer remains √149 cm
(approximately 12.2 cm if rounded).

This shows that slant height does not always give whole numbers.

Connection to Surface Area

Slant height is especially important when calculating curved or lateral surface area.

For a cone:
Curved Surface Area = πrl

For a square pyramid:
Lateral Surface Area = 2sl

Without slant height, these formulas cannot be used.

Read More - Arc Formula: How to Calculate the Length of an Arc

Advanced Note (For Higher Classes)

In regular pyramids with different base shapes (like pentagon or hexagon), slant height is found using the base apothem (distance from center to midpoint of a side). The same right triangle rule applies, but instead of s/2, we use the apothem.

Practice Problems

1. A cone has height 9 cm and radius 12 cm. Find l.
   
   

2. A square pyramid has base side 8 m and height 6 m. Find l.
   
   

3. If a cone has l = 13 cm and r = 5 cm, find h.
   
   

Answers

1. l = √(81 + 144) = √225 = 15 cm
   
   

2. l = √(4² + 6²) = √(16 + 36) = √52
   
   

3. h = √(13² − 5²) = √(169 − 25) = √144 = 12 cm

Tip: Always find the right triangle first. The formula is easy to use once you see it. 

Why is Slant Height Important for Students?

For students, learning slant height isn't just about passing an exam; it's also about knowing how the world works:

1. Surface Area: You cannot find the Lateral Surface Area (the area of the sides) without the slant height. For a cone, that formula is Area = \pi r l.

2. Architecture: Architects need slant height to figure out the area of a sloped roof.

3. Manufacturing: Engineers need to know the correct slant height in order to make things like funnels, paper cups, or even rocket nose cones.