# Cone Surface Area Calculator

> Find a cone’s lateral (curved) area, base area and total surface from the radius and either the vertical height or the slant height.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/cone-surface-area-calculator

The curved surface of a cone is πrl, where l is the slant height along the side. Add the circular base πr² for a closed cone.

Also called: Cone slant height calculator.

## Inputs
- Base given as (options: Radius, Diameter; default radius)
- Base radius or diameter (default 3)
- I also know the (options: Vertical height, Slant height; default height)
- Vertical height (default 4)
- Slant height (default 5)
- Length unit (options: Millimetres (mm), Centimetres (cm), Metres (m), Kilometres (km), Inches (in), Feet (ft), Yards (yd), No unit (plain numbers); default cm)

## Results
- Total surface area (with base)
- Lateral (curved) area πrl
- Base area πr²
- Slant height l
- Vertical height h
- Volume ⅓πr²h

## Formula

```
l = √(r² + h²);  Lateral = πrl;  Base = πr²;  Total = πr(r + l)
```

Where:
- r: base radius
- h: vertical height
- l: slant height

## Worked example
1. l = √(3² + 4²) = 5 cm.
2. Lateral = π × 3 × 5 = 15π ≈ 47.12389 cm².
3. Total = 15π + 9π = 24π ≈ 75.398224 cm².

**Result:** A cone with base radius 3 cm and height 4 cm has a total surface area of about 75.40 cm².

## What the result means

Use the lateral area for an open cone such as a funnel, party hat or conical roof, and the total for a closed solid cone.

## Assumptions and limits
- A right circular cone: the tip is directly above the centre of the base.
- Seams, overlaps and material thickness are not included.
- A truncated cone (frustum) needs a different formula.

## Questions

### Why does the curved area use slant height, not height?

Unrolled, the side is a sector of a circle whose radius is the slant height l, and its area works out to πrl.

### How do I get the slant height?

By Pythagoras: l = √(r² + h²), because the radius, height and slant form a right triangle.

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- [Cylinder Surface Area Calculator](https://calculators.geekmidst.com/calculators/cylinder-surface-area-calculator)
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- [Right Triangle Calculator](https://calculators.geekmidst.com/calculators/right-triangle-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/cone-surface-area-calculator