Surface Area and Volume of a Cone

A cone looks like a party hat or an ice-cream wafer. It is closely related to a cylinder -- in fact, a cone's volume is exactly one third of a cylinder with the same base and height.

What is a cone?
Key dimensions
Slant height
Net of a cone
Base area
Lateral (side) area
Total surface area
Volume
Cone vs cylinder -- the one-third relationship
Worked example 1 -- Total surface area
Worked example 2 -- Volume
Summary of formulas
Practice exercises

What is a cone?

A cone is a 3D solid with:

One circular base
A curved lateral surface that narrows to a single point called the apex (or vertex)
The apex sits directly above the centre of the base (right circular cone)

```

▲ ▲ = apex

╱│╲

╱ │ ╲ s (slant height)

╱ │h ╲

╱ │ ╲

╱─────●─────╲ ● = centre

```

Key dimensions

Symbol	Meaning
$r$	Radius of the circular base
$h$	Height -- perpendicular distance from the base to the apex
$s$	Slant height -- distance from any point on the base edge to the apex, measured along the surface

Slant height

The radius $r$ , height $h$ , and slant height $s$ form a right triangle:

```

apex

│╲

h │ ╲ s

│ ╲

│_____╲

```

By the Pythagorean theorem:

s = r^{2} + h^{2}

Always compute $s$ first if it is not given -- you need it for the lateral area.

Net of a cone

When you "unroll" a cone you get:

One circle (radius $r$ ) -- the base
One circular sector (radius $s$ , arc length $2 π r$ ) -- the lateral surface

The sector's central angle is $θ = \frac{r}{s} \cdot 360°$ .

Base area

The base is a circle:

S_{base} = π r^{2}

Lateral (side) area

The lateral surface, when unrolled, is a sector of a circle with radius $s$ and arc length $2 π r$ . Its area is:

S_{lateral} = π r s

This elegant formula comes from $\frac{1}{2} \times arc length \times radius of sector = \frac{1}{2} \cdot 2 π r \cdot s = π r s$ .

Total surface area

Add the base and the lateral surface:

S = π r^{2} + π r s

S = π r (r + s)

Compare this with the cylinder formula $S = 2 π r (r + h)$ . They have a similar structure.

Volume

The volume of a cone is:

V = \frac{1}{3} π r^{2} h

This is exactly $\frac{1}{3}$ of the volume of a cylinder with the same base radius and height.

Cone vs cylinder -- the one-third relationship

If you have a cylinder and a cone with the same radius $r$ and height $h$ , then:

V_{cone} = \frac{1}{3} V_{cylinder}

\frac{1}{3} π r^{2} h = \frac{1}{3} \cdot π r^{2} h

This means you would need to fill the cone three times and pour it into the cylinder to fill it completely. This is a classic experiment you can try with water and physical models.

Memory aid: Both the cone and the pyramid have the factor $\frac{1}{3}$ in their volume formulas. Solids that taper to a point always have $\frac{1}{3}$ of the "full" prism or cylinder volume.

Worked example 1 -- Total surface area

Problem: A cone has radius

r = 7 cm

and height

h = 24 cm

. Find the total surface area. Step 1 -- Find the slant height:

s = r^{2} + h^{2} = 7^{2} + 2 4^{2} = 49 + 576 = 625 = 25 cm

Step 2 -- Base area:

S_{base} = π r^{2} = π \cdot 49 = 49 π cm^{2}

Step 3 -- Lateral area:

S_{lateral} = π r s = π \cdot 7 \cdot 25 = 175 π cm^{2}

Step 4 -- Total surface area:

S = 49 π + 175 π = 224 π

S \approx 703.72 cm^{2}

Worked example 2 -- Volume

Problem: A conical container has diameter

d = 20 cm

and slant height

s = 26 cm

. How much water can it hold (in litres)? Step 1 -- Find the radius and height:

r = \frac{d}{2} = 10 cm

h = s^{2} - r^{2} = 2 6^{2} - 1 0^{2} = 676 - 100 = 576 = 24 cm

Step 2 -- Calculate the volume:

V = \frac{1}{3} π r^{2} h = \frac{1}{3} π \cdot 100 \cdot 24 = \frac{2 400 π}{3} = 800 π

V \approx 2 513.27 cm^{3}

Step 3 -- Convert to litres (

1 l = 1000 cm^{3}

V \approx 2.51 l

Summary of formulas

Quantity	Formula
Slant height	$s = r^{2} + h^{2}$
Base area	$S_{base} = π r^{2}$
Lateral area	$S_{lateral} = π r s$
Total surface area	$S = π r (r + s)$
Volume	$V = \frac{1}{3} π r^{2} h$

Practice exercises

Put your knowledge to the test:

Surface Area and Volume of a Cone – Formulas and Examples

Surface Area and Volume of a Cone

Table of contents

What is a cone?

Key dimensions

Slant height

Net of a cone

Base area

Lateral (side) area

Total surface area

Volume

Cone vs cylinder -- the one-third relationship

Worked example 1 -- Total surface area

Worked example 2 -- Volume

Summary of formulas

Practice exercises

Surface Area and Volume of a Cone – Formulas and Examples

Surface Area and Volume of a Cone

Table of contents

What is a cone?

Key dimensions

Slant height

Net of a cone

Base area

Lateral (side) area

Total surface area

Volume

Cone vs cylinder -- the one-third relationship

Worked example 1 -- Total surface area

Worked example 2 -- Volume

Summary of formulas

Related articles

Practice exercises