Surface area of cones

key notes :

Definition of a Cone

  • A cone is a three-dimensional geometric shape with a circular base and a curved surface that tapers to a single point (the apex).

Surface Area of a Cone

The total surface area of a cone includes two parts:

  • Curved Surface Area (CSA)
  • Total Surface Area (TSA)

Curved Surface Area (CSA) Formula

CSA = πrl

Where:

  • r = Radius of the base
  • l = Slant height
  • π ≈ 3.1416

Total Surface Area (TSA) Formula

TSA = πrl + πr2

  • This includes the curved surface area plus the area of the circular base.

Finding Slant Height (l)

  • The slant height l can be calculated using the Pythagorean theorem:

l = √r2 + h2

  • Where hhh is the vertical height of the cone.

Units of Measurement

  • Surface area is measured in square units (e.g., cm², m²).

Real-Life Applications

  • Used in designing ice cream cones, funnels, party hats, and architectural structures like domes.

Key Concept

  • The curved surface is a sector of a circle when unfolded.

Common Mistakes to Avoid

  • Confusing slant height lll with vertical height hhh.
  • Forgetting to add the base area when finding TSA.

Learn with an example

What is the surface area of this cone?

Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.

______  square metres

Find the radius and slant height of the cone.

radius=1/2 . diameter=1/2 . 6=3

slant height=8

Find the area of the base. Use ​𝜋 ≈ 3.14.
Area of base = ​𝜋r2
  ≈ 3.14 × 32
 ≈ 28.26

Find the lateral area.
Lateral area = ​𝜋rs
  ≈ 3.14 × 3 × 8
  ≈ 75.36

Now add the areas to find the surface area of the cone.
Surface area = area of base + lateral area
  ≈ 28.26 + 75.36
  ≈ 103.62

The surface area of the cone is about 103.62 square metres.

What is the surface area of this cone?

Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.

_______ square metres.

Find the radius and slant height of the cone.

radius=1/2 . diameter=1/2 . 2=1

slant height=3

Find the area of the base. Use ​𝜋 ≈ 3.14.
Area of base = ​𝜋r2
  ≈ 3.14 × 12
  ≈ 3.14

Find the lateral area.
Lateral area = ​𝜋rs
  ≈ 3.14 × 1 × 3
  ≈ 9.42

Now add the areas to find the surface area of the cone.
Surface area = area of base + lateral area
  ≈ 3.14 + 9.42
 ≈ 12.56

The surface area of the cone is about 12.56 square metres.

What is the surface area of this cone?

Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.

_____square centimetres

Find the radius and slant height of the cone.

radius = 5
slant height = 10


Find the area of the base. Use ​𝜋 ≈ 3.14.
Area of base = ​𝜋r2
  ≈ 3.14 × 52
  ≈ 78.5

Find the lateral area.
Lateral area = ​𝜋rs
  ≈ 3.14 × 5 × 10
 ≈ 157

Now add the areas to find the surface area of the cone.
Surface area = area of base + lateral area
  ≈ 78.5 + 157
  ≈ 235.5

The surface area of the cone is about 235.50 square centimetres.

let’s practice!