GEOMETRY3D GeometryMathematics Calculator
🍦

Find volume, slant height, and surface area of any right circular cone

Enter radius and height to instantly compute volume (⅓πr²h), slant height √(r²+h²), lateral SA (πrs), and total surface area.

Concept Fundamentals
V = (1/3)πr²h
Volume
s = √(r² + h²)
Slant Height
A_L = πrs
Lateral SA
πr² + πrs
Total SA

Did our AI summary help? Let us know.

Cone volume = ⅓ × cylinder volume with same r and h Slant height s = √(r² + h²) from Pythagoras Lateral SA = πrs unwraps to a circular sector Three cones fill one cylinder Right cone has apex above center of base

Key quantities
V = (1/3)πr²h
Volume
Key relation
s = √(r² + h²)
Slant Height
Key relation
A_L = πrs
Lateral SA
Key relation
πr² + πrs
Total SA
Key relation

Ready to run the numbers?

Why: Right circular cones appear everywhere—ice cream cones, traffic cones, volcanoes, megaphones. Volume and surface area are essential for manufacturing, packaging, and structural design.

How: The calculator uses the Pythagorean theorem for slant height, then applies standard cone formulas. Volume is one-third of a cylinder with the same base and height (Cavalieri's principle).

Cone volume = ⅓ × cylinder volume with same r and hSlant height s = √(r² + h²) from Pythagoras
Sources:Wolfram MathWorld — ConeKhan Academy — Geometry

Run the calculator when you are ready.

Calculate Cone PropertiesEnter radius and height to get all measurements
🍦
3D GEOMETRYRight Circular Cone

Right Circular Cone Calculator

Volume, slant height, lateral SA, total SA, base area.

Slant Height →Hemisphere →

🍦 Sample Examples — Click to Load

Dimensions

cone_calc.sh
CALCULATED
$ cone --radius=3 --height=4
Volume
37.6991
Total SA
75.3982
Slant Height
5
Lateral SA
47.1239
Base Area
28.2743
Share:
Right Circular Cone
r = 3, h = 4
37.7
SA: 75.40Slant: 5.00Lateral: 47.12

3D Visualization

Cone Diagramrhh = 4r = 3

Property Radar

Property Comparison

Breakdown

📐 Calculation Breakdown

INPUT
Radius
3
Height
4
RESULTS
Slant Height
5
√(r^{2} + h^{2})
Base Area
28.2743
\text{pi} r^{2}
Lateral SA
47.1239
\text{pi} ext{rs}
Total SA
75.3982
\text{pi} r^{2} + \text{pi} ext{rs}
VOLUME
Volume
37.6991
(1/3)\text{pi} r^{2}h

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🍦

Ice cream cone volume determines how much ice cream fits.

— Food

🚧

Traffic cones use conical shape for stackability and visibility.

— Safety

📐

Cone volume = ⅓ cylinder — proven by Cavalieri's principle.

— Geometry

🌋

Volcanoes often approximate conical shapes in geology.

— Geology

📋 Key Takeaways

  • • Volume = (1/3)πr²h — one-third of cylinder with same base and height
  • • Slant height s = √(r² + h²) — Pythagorean theorem
  • • Lateral SA = πrs — curved surface unwraps to sector
  • • Total SA = πr² + πrs = πr(r + s)

💡 Did You Know?

🍦Ice cream cone volume determines how much ice cream fitsSource: Food
🚧Traffic cones use conical shape for stackability and visibilitySource: Safety
📐Cone volume = ⅓ cylinder — proven by CavalieriSource: Geometry
🌋Volcanoes often approximate conical shapesSource: Geology
📢Megaphones use conical shape to direct soundSource: Acoustics
🎉Party hat lateral SA = πrs for paper amountSource: Crafts

📖 How Cone Calculations Work

Right circular cone: apex above center of circular base. Slant height from Pythagoras.

Formulas

s=r2+h2s = \sqrt{r^2 + h^2}, V=13πr2hV = \frac{1}{3}\pi r^2 h, AL=πrsA_L = \pi r s

🎯 Expert Tips

Cone vs Cylinder

Same r and h: cone volume = ⅓ cylinder. Fit 3 cones in one cylinder.

Height vs Slant

Volume uses perpendicular height h. Lateral SA uses slant height s.

Oblique Cone

This calculator is for right cones — apex above center. Oblique cones need different formulas.

Units

r, h in cm → area cm², volume cm³. Keep consistent.

⚖️ Comparison

FeatureThis CalculatorManualBasic Online
All 5 properties✅⚠️⚠️
Charts & viz✅❌❌
AI analysis✅❌❌
Step-by-step✅✅❌

❓ FAQ

Height vs slant height?

Height h is perpendicular apex-to-base. Slant s is along surface — always longer. s = √(r² + h²).

What is oblique cone?

Apex not above center. Different formulas. This calculator is for right circular cones only.

Cone vs cylinder volume?

Cone = ⅓ × cylinder with same r and h. Three cones fill one cylinder.

Find r or h from volume?

V = (1/3)πr²h. Given V and h: r = √(3V/(πh)). Given V and r: h = 3V/(πr²).

Lateral SA formula?

A_L = πrs. The curved surface unwraps to a circular sector.

Units?

r, h in same unit. Area in unit², volume in unit³.

Why is cone volume ⅓ of cylinder?

Cavalieri principle: same cross-sectional area at every height. Cone fills ⅓ of cylinder with same base and height.

What is a right circular cone?

Apex directly above center of circular base. Axis perpendicular to base.

📊 Stats

⅓
Cylinder vol
πrs
Lateral
√(r²+h²)
Slant
πr²
Base

⚠️ Disclaimer: Results are mathematically precise. Verify for engineering applications.

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