GEOMETRY3D GeometryMathematics Calculator
🏛️

Hemisphere — Half a Sphere

Compute curved surface area (2πr²), base area (πr²), total surface area (3πr²), and volume ((2/3)πr³) from the radius. Domes, bowls, and igloos are hemispheres.

Concept Fundamentals
2πr²
Curved SA
πr²
Base Area
3πr²
Total SA
(2/3)πr³
Volume

Did our AI summary help? Let us know.

Curved SA = 2πr² (half of sphere 4πr²) Total SA = 3πr² = curved dome + flat base Volume = (2/3)πr³ — exactly half of full sphere Open bowl uses curved SA only; closed dome includes base Reverse: r = √(A_total/(3π)) or r = ∛(3V/(2π))

Key quantities
2πr²
Curved SA
Key relation
πr²
Base Area
Key relation
3πr²
Total SA
Key relation
(2/3)πr³
Volume
Key relation

Ready to run the numbers?

Why: Hemispheres appear in domes, planetariums, bowls, igloos, and half-sphere structures. Understanding hemisphere geometry is essential for architecture, manufacturing, and heat transfer calculations.

How: Enter the radius. Curved SA is half of a sphere's 4πr², so 2πr². Base area is a circle πr². Total SA = curved + base = 3πr². Volume is half of sphere volume: (2/3)πr³.

Curved SA = 2πr² (half of sphere 4πr²)Total SA = 3πr² = curved dome + flat base
Sources:Wolfram MathWorld — HemisphereKhan Academy — 3D Geometry

Run the calculator when you are ready.

Hemisphere CalculatorEnter radius to compute curved SA, base area, total SA, and volume
🏛️
3D GEOMETRYHemisphere

Hemisphere Surface Area & Volume

Curved 2πr², base πr², total 3πr², volume (2/3)πr³.

Sphere →Cone →

🏛️ Sample Examples — Click to Load

Radius

hemisphere_calc.sh
CALCULATED
$ hemisphere --radius=5
Curved SA
157.0796 cm²
Base Area
78.5398 cm²
Total SA
235.6194 cm²
Volume
261.7994 cm³
Share:
Hemisphere r = 5 cm
235.62 cm²
Curved: 157.08Base: 78.54Vol: 261.80 cm³

3D Visualization

Hemisphere VisualizationA diagram showing a hemisphere with its radius r, curved surface (dome), and circular base.r

Property Radar

Property Comparison

Breakdown

📐 Calculation Breakdown

INPUT
Radius
5 cm
SURFACE
Curved SA (Dome)
157.0796 cm²
2\text{pi} r^{2}
Base Area (Lid)
78.5398 cm²
\text{pi} r^{2}
Total SA
235.6194 cm²
3\text{pi} r^{2}
VOLUME
Volume
261.7994 cm³
(2/3)\text{pi} r^{3}

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

🧮 Fascinating Math Facts

🏛️

Planetarium domes use hemisphere geometry for projection surface area

— Architecture

🥗

Salad bowls are often hemispheres—SA determines glaze coverage

— Manufacturing

🧊

Igloos approximate hemispheres—volume for air, SA for heat loss

— Engineering

📐

Volume is exactly half of full sphere with same radius

— Calculus

📋 Key Takeaways

  • • Curved SA = 2πr² (half of sphere's 4πr²)
  • • Base area = πr² (circle)
  • • Total SA = 3πr² = curved + base
  • • Volume = (2/3)πr³ (half of sphere's (4/3)πr³)

💡 Did You Know?

🏛️Planetarium domes use hemisphere geometry for projection surface areaSource: Architecture
🥗Salad bowls are often hemispheres — SA determines glaze coverageSource: Manufacturing
🧊Igloos approximate hemispheres — volume for air, SA for heat lossSource: Engineering
🌌Total SA = 3πr² because dome (2πr²) + lid (πr²)Source: Geometry
🏀Half a basketball is a hemisphere — same formulas applySource: Sports
📐Volume is exactly half of full sphere with same radiusSource: Calculus

📖 How Hemisphere Calculations Work

Hemisphere = half sphere. Curved part = half of 4πr² = 2πr². Flat base = circle πr².

Formulas

Acurved=2πr2,Abase=πr2,Atotal=3πr2,V=23πr3A_{curved} = 2\pi r^2, \quad A_{base} = \pi r^2, \quad A_{total} = 3\pi r^2, \quad V = \frac{2}{3}\pi r^3

🎯 Expert Tips

Open Bowl

Need only dome? Use curved SA = 2πr² (no base).

Reverse Calc

From total SA: r = √(A_total/(3π)). From volume: r = ∛(3V/(2π)).

Units

Radius in cm → area in cm², volume in cm³.

Dome Design

Curved SA for roofing material; volume for air capacity.

⚖️ Comparison

FeatureThis CalculatorManualBasic Online
All 4 properties✅⚠️⚠️
Charts & viz✅❌❌
AI analysis✅❌❌
Step-by-step✅✅❌

❓ FAQ

Why is total SA 3πr² not 2πr²?

Curved dome = 2πr². Flat base = πr². Total = 2πr² + πr² = 3πr².

Open bowl — which area?

Curved SA only: 2πr². No base included.

Volume vs sphere?

Hemisphere volume = ½ × (4/3)πr³ = (2/3)πr³.

Find radius from volume?

r = ∛(3V/(2π)).

Find radius from total SA?

r = √(A_total/(3π)).

Units?

Radius in m → area m², volume m³. Keep consistent.

📊 Stats

2πr²
Curved
πr²
Base
3πr²
Total
2/3 πr³
Volume

⚠️ Disclaimer: Results are mathematically precise. Verify for construction and engineering.

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