Study Guides/Maths/Slant Height of Cone
Study Guide · Maths

How to Find the Slant Height of a Cone

When calculating the surface area of a 3D cone in Mathematics (Mensuration), you must know its Slant Height. It is a different measurement from the normal, straight height of the cone.

Question (Click to Flip)

If a cone has a radius of 3 cm and a height of 4 cm, what is its slant height?

Answer

Using the formula l = √(r² + h²): l = √(3² + 4²) l = √(9 + 16) l = √25 = 5 cm.

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Key Facts

While the Slant Height is required to find the Surface Area, it is NOT used to find the Volume. The Volume of a cone (⅓ π r² h) only requires the normal perpendicular height (h).

What is Slant Height?

Imagine an ice cream cone.

  • The perpendicular height (h) is the straight line dropping directly from the top tip (apex) down to the exact center of the circular base.
  • The Slant Height (l) is the distance measured along the outside curved surface of the cone, from the top tip down to the edge of the circular base.

The Formula

If you slice a cone in half right down the middle, you will see that the radius (r), the perpendicular height (h), and the slant height (l) form a perfect Right-Angled Triangle.

Therefore, we can easily find the slant height using the Pythagoras Theorem (Hypotenuse² = Base² + Perpendicular²):

l = √(r² + h²)

Where:

  • l = Slant Height
  • r = Radius of the base
  • h = Perpendicular height

Why do we need the Slant Height?

You cannot calculate the surface area of a cone using the normal height (h). You must use the slant height (l).

  • Curved Surface Area (CSA) = π × r × l
  • Total Surface Area (TSA) = πr(l + r)

Questions and Answers

If a cone has a radius of 3 cm and a height of 4 cm, what is its slant height?+

Using the formula l = √(r² + h²): l = √(3² + 4²) l = √(9 + 16) l = √25 = **5 cm**.

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