Direct answer

What are the main cone formulas?

The main cone formulas are volume $V = \frac{1}{3} π r^{2} h$ , lateral surface area $L S A = π r l$ , total surface area $T S A = π r (r + l)$ , and slant height $l = r^{2} + h^{2}$ . Surface area uses slant height, not vertical height.

Short answer

Cone formulas at a glance

Volume: $V = \frac{1}{3} π r^{2} h$ Lateral surface area: $L S A = π r l$ Total surface area: $T S A = π r (r + l)$

Definition

What is a cone?

A cone is a 3D solid with one circular base and one vertex connected by a curved side.

Diagram

Cone diagram

Cone variables: radius r, height h, and slant height l.

Quick formula table

Cone Formulas quick table

Formula	Equation	Variables
Base area	$B = π r^{2}$	r = radius
Slant height	$l = r^{2} + h^{2}$	Right cone only
Volume	$V = \frac{1}{3} π r^{2} h$	h = vertical height
Lateral surface area	$L S A = π r l$	l = slant height
Total surface area	$T S A = π r (r + l)$	Includes base

Formula explanation

How to use these formulas

A cone has one third the volume of a cylinder with the same base and height. Surface area needs slant height, not vertical height.

Calculator

Cone Calculator

Enter measurements to get the result, unit, formula, and calculation steps.

Examples

Step-by-step examples

Find slant height

Given r = 4 cm and h = 9 cm.
Use $l = r^{2} + h^{2}$ .
$l = 97 \approx 9.85 c m$ .

Find volume

Given r = 4 cm and h = 9 cm.
Use $V = \frac{1}{3} π r^{2} h$ .
$V = 48 π \approx 150.80 c m^{3}$ .

Find total surface area

Given r = 4 cm and l ≈ 9.85 cm.
Use $T S A = π r (r + l)$ .
$T S A \approx π \times 4 (4 + 9.85) \approx 174.05 c m^{2}$ .

Find height from volume

Given V = 48π cm³ and r = 4 cm.
Use $h = 3 V / (π r^{2})$ .
$h = 3 (48 π) /16 π = 9 c m$ .

Find lateral surface area

Given r = 5 cm and l = 13 cm.
Use $L S A = π r l$ .
$L S A = 65 π \approx 204.20 c m^{2}$ .

Reverse calculations

Solve backwards from known values

Task	Formula	When to use it
Find height from volume	$h = 3 V / (π r^{2})$	Use when volume and radius are known.
Find radius from volume	$r = 3 V / π h$	Use when volume and height are known.
Find height from slant height	$h = l^{2} - r^{2}$	Use for right cones.

Common mistakes

Using height instead of slant height in surface area.
Forgetting the one third in volume.
Using diameter as radius.

Practice

Practice problems with answers

1. Find volume when r = 3 cm and h = 12 cm.

$V = 36 π \approx 113.10 c m^{3}$ .

2. Find slant height when r = 5 and h = 12.

$l = 25 + 144 = 13$ .

3. Find lateral area when r = 5 cm and l = 13 cm.

$L S A = 65 π \approx 204.20 c m^{2}$ .

4. Find height when V = 100π in³ and r = 5 in.

$h = 3 V / π r^{2} = 300 π /25 π = 12 in$ .

5. Find total surface area when r = 3 and l = 5.

$T S A = π \times 3 (3 + 5) = 24 π \approx 75.40$ .

Related formulas

FAQ

Cone Formulas FAQ

What is the cone volume formula?

The formula is $V = \frac{1}{3} π r^{2} h$ .

What is slant height?

Slant height is the distance from the cone vertex to a point on the edge of the circular base.

What is the total surface area of a cone?

The formula is $T S A = π r (r + l)$ .