Cone Surface Area Calculator | Versa Calculator

Cone Calculator – Calculate Volume, Surface Area & Dimensions

Choose A Calculation i

Radius r = i

Height h = i

Let Pi π = i

Units i

Significant Figures i

Cone Diagram

r = radius

h = height

s = slant height

V = volume

L = lateral surface area

B = base surface area

A = total surface area

θ = half-angle or apex angle of the full cone between the center axis and any side, in degrees

φ = 2θ = aperture or opening angle or vertex angle of the full cone between opposite sides, in degrees

β = base angle of the cone, in degrees

π = pi = 3.1415926535898

√ = square root

What is a Cone Calculator?

A cone calculator is a specialized digital tool that computes various geometric properties of cones, including surface area, volume, dimensions, and angles. Our advanced calculator goes beyond basic calculations, offering eight different computation modes based on whatever parameters you already know.

Unlike many basic calculators that only handle simple radius-and-height inputs, our tool can determine all cone properties from multiple parameter combinations, making it exceptionally versatile for students, engineers, designers, and anyone working with conical shapes in real-world applications.

Cones are three-dimensional geometric shapes with a circular base that tapers smoothly to a single point called the apex or vertex.

The calculator specifically handles right circular cones, where the apex is positioned directly above the center of the base, forming a right angle with the base center.

How to Use Our Cone Calculator

Select Your Calculation Mode: Choose from eight different calculation options based on your known values.
Enter Your Known Values: Our calculator accepts:
- Radius (r) and Height (h)
- Radius (r) and Slant Height (s)
- Radius (r) and Volume (V)
- Radius (r) and Lateral Surface Area (L)
- Radius (r) and Total Surface Area (A)
- Height (h) and Slant Height (s)
- Height (h) and Volume (V)
- Slant Height (s) and Lateral Surface Area (L)
Configure Settings: Adjust π precision, measurement units, and significant figures based on your accuracy requirements.
Calculate.

The Mathematical Formulas

Surface Area Calculations

The surface area of a cone consists of two distinct components: the base area (circular) and the lateral surface area (curved side).

Base Area Formula:
B = πr²
Where r is the radius of the circular base .
Lateral Surface Area Formula:
L = πrl or L = πr√(r² + h²)
Where r is the radius, l is the slant height, and h is the vertical height .
Total Surface Area Formula:
A = L + B = πrl + πr² = πr(l + r) or A = πr(r + √(r² + h²))
This represents the complete exterior surface of the cone .

Volume Formula

Volume Calculation:
V = (1/3)πr²h
The volume of a cone is exactly one-third the volume of a cylinder with the same base and height .

Slant Height Formula

Slant Height Calculation:
l = √(r² + h²)
This derives from the Pythagorean theorem, with the slant height representing the hypotenuse of a right triangle formed by the height, radius, and slant height.

Angle Calculations

Our calculator also determines three important cone angles:

Half-angle (θ): The angle between the central axis and any side
Aperture angle (φ): The full angle at the apex between opposite sides (2θ)
Base angle (β): The angle between the slant height and the base.

Worked Examples

Example 1: Calculate all properties of a cone with radius 5 cm and height 12 cm.

Slant height: l = √(5² + 12²) = √169 = 13 cm
Lateral surface area: L = π × 5 × 13 = 65π ≈ 204.2 cm²
Base area: B = π × 5² = 25π ≈ 78.5 cm²
Total surface area: A = 65π + 25π = 90π ≈ 282.7 cm²
Volume: V = (1/3) × π × 5² × 12 = 100π ≈ 314.2 cm³

Example 2: Determine the height of a cone with a total surface area of 375 in² and radius of 5 inches.

Using the formula A = πr(r + l) = 375
π × 5 × (5 + l) = 375
Solving gives slant height l = 19.86 inches
Height h = √(l² - r²) = √(394.4 - 25) = √369.4 ≈ 19.22 inches