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 (apex).

Parts of a Cone

  • Radius (r): The distance from the center to the edge of the base.
  • Height (h): The perpendicular distance from the base to the apex.
  • Slant height (l): The distance from any point on the edge of the base to the apex.

Formula for Surface Area of a Cone

The total surface area (TSA) of a cone includes both the curved surface and the circular base:

TSA = πrl + πr2

TSA = πr (l + r)

where:

  • π ≈ 3.1416
  • r = radius
  • l = slant height

Curved Surface Area (CSA)

  • The curved (lateral) surface area of a cone is given by:

CSA = πrl

Finding the Slant Height (If Not Given)

  • If the slant height is not given, it can be calculated using the Pythagorean theorem:

l = √r2+h2

where h is the vertical height of the cone.

Units

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

Applications of Surface Area of a Cone

  • Used in designing party hats, ice cream cones, traffic cones, and water tanks.
  • Important in real-life constructions like rooftops and tents.

Example Problem

  • If a cone has a radius of 5 cm and a slant height of 12 cm, find its total surface area.
  • Solution:

CSA = π(5)(12) = 60π = 188.4 cm2

Base Area = π(5)2 = 25π = 78.5 cm2

TSA = 188.4 + 78.5 = 266.9 cm2

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!