Hemisphere Calculator
Calculate hemisphere volume V = (2/3)πr³, curved surface area 2πr², flat base area πr², and total surface area 3πr² — plus full sphere circumference C = 2πr, surface area, and volume. Enter any known value to find all others instantly with step-by-step working.
Hemisphere: dome height h = radius r · Curved SA = 2πr² · Base = πr² · Total SA = 3πr²
Full Sphere Reference (same radius)
Sphere circumference C = 2πr = great circle perimeter · SA = 4πr² · V = (4/3)πr³
The circumference of a sphere = C = 2πr is the perimeter of its great circle — the largest possible circle on the sphere's surface, passing through the center. From C alone: SA = C²/π and V = C³/(6π²).
Hemisphere Volume Formula — V = (2/3)πr³
The hemisphere volume formula is V = (2/3)πr³, where r is the radius. A hemisphere is exactly half of a full sphere, so the hemisphere volume equals half the sphere volume: hemisphere V = (1/2) × (4/3)πr³ = (2/3)πr³. This hemisphere volume calculator finds volume, surface area, and circumference from any known dimension.
Half Circle vs Hemisphere — Critical Distinction
Searches for "volume of a half circle" almost always mean hemisphere volume. But a half circle is a 2D flat shape — it has area, not volume. The 3D equivalent is the hemisphere. These are completely different:
⚠️ 2D Semicircle (Flat)
Result in cm² (area)
A flat shape — no volume
✅ 3D Hemisphere (Solid)
Result in cm³ (volume)
A solid shape — has depth
"Volume of a Half Circle" — What You Actually Want
If you searched for "volume of a half circle formula" or "semicircle volume formula", you almost certainly want the hemisphere volume: V = (2/3)πr³. A 2D semicircle has area = πr²/2 (measured in cm²). A 3D hemisphere has volume = (2/3)πr³ (measured in cm³). They share the same circular cross-section but are fundamentally different — one is flat, one is solid.
How to Find the Volume of a Hemisphere — Step-by-Step
Finding the volume of a hemisphere takes four steps. This volume of half sphere calculator handles all five entry modes — from radius, diameter, volume (inverse), curved surface area (inverse), or total surface area.
- Step 1: Identify the radius r (or convert: if given diameter d, then r = d/2)
- Step 2: Cube the radius: compute r³
- Step 3: Multiply by (2/3)π ≈ 2.0944
- Step 4: Write the unit as cubic units (cm³, m³, etc.)
Example 1: From radius — r = 5 cm
- r = 5 cm, r³ = 125 cm³
- V = (2/3) × π × 125 = 250π/3 ≈ 261.799 cm³
- Check: sphere V = (4/3)π × 125 = 500π/3 ≈ 523.599 cm³ = 2 × 261.799 ✓
Example 2: From diameter — d = 20 cm
- r = d/2 = 10 cm, r³ = 1000 cm³
- V = (2/3) × π × 1000 = 2000π/3 ≈ 2,094.4 cm³
Example 3: From volume (inverse) — V = 100 cm³, find r
- r = ∛(3V / 2π) = ∛(300 / 6.2832) = ∛(47.746) ≈ 3.628 cm
- ⚠️ Must use cube root (∛), NOT square root — common mistake!
- Verify: V = (2/3)π × 3.628³ = (2/3)π × 47.76 ≈ 100 cm³ ✓
Example 4: From curved SA (inverse) — CSA = 157.08 cm²
- r = √(CSA / 2π) = √(157.08 / 6.2832) = √(25) = 5 cm
- V = (2/3)π × 125 ≈ 261.799 cm³
Example 5: Real-world dome — Pantheon, Rome (r ≈ 21.65 m)
- r = 21.65 m, r³ = 10,145.5 m³
- V = (2/3)π × 10,145.5 ≈ 21,237 m³
- Curved SA = 2π × 21.65² = 2π × 468.7 ≈ 2,946 m²
- Total SA = 3π × 468.7 ≈ 4,418 m²
Hemisphere Surface Area — Curved SA vs Total SA
The hemisphere surface area has two distinct meanings — and confusing them is the most common mistake. The curved surface area (CSA) = 2πr² covers only the dome exterior. The total surface area (TSA) = 3πr² includes the dome plus the flat circular base.
When to Use Each:
- Curved SA (2πr²): Painting the outside of a dome, waterproofing a bowl, cost of exterior coating
- Total SA (3πr²): Building material for a complete enclosed hemisphere (walls + floor), packaging
- Note: Curved SA = (1/2) × sphere SA = (1/2) × 4πr² = 2πr² ✓
- But: Total SA ≠ (1/2) × sphere SA, because total SA adds the base circle πr²
| r = 5 cm | Formula | Value | Notes |
|---|---|---|---|
| Curved SA | 2πr² | ≈ 157.08 cm² | Dome only = sphere SA/2 |
| Flat base | πr² | ≈ 78.54 cm² | Circle at bottom |
| Total SA | 3πr² | ≈ 235.62 cm² | Dome + base |
| Sphere SA | 4πr² | ≈ 314.16 cm² | Full sphere (reference) |
Circumference of a Sphere — Formula and Meaning
The circumference of a sphere refers to the circumference of its great circle — the largest circle that can be drawn on the sphere's surface, passing through the center. The circumference of a sphere formula is C = 2πr = πd.
What Is a Great Circle?
A great circle is any circle drawn on a sphere whose center coincides with the center of the sphere. The equator of Earth is a great circle. A sphere has infinitely many circles on its surface, but "the circumference of a sphere" conventionally means the great circle — the largest one, with circumference C = 2πr. This is how to find the circumference of a sphere from radius alone.
Earth Circumference Verification:
r = 6,371 km → C = 2π × 6,371 = 40,030 km
Known equatorial circumference ≈ 40,075 km (slight difference: Earth is oblate — slightly flattened at poles, not a perfect sphere). The circumference of a sphere formula C = 2πr gives the ideal sphere result.
Elegant Relationships — Find Everything from Circumference Alone
The circumference of a sphere C = 2πr gives elegant shortcuts. Once you know C, you never need to find r first:
- r = C / (2π) — radius from circumference
- SA = C² / π — surface area directly from circumference
- V = C³ / (6π²) — volume directly from circumference
Step-by-Step from Circumference: C = 100 cm
- Radius: r = C/(2π) = 100/(2π) = 100/6.2832 ≈ 15.915 cm
- Surface area: SA = C²/π = 10,000/π ≈ 3,183.1 cm²
- Volume: V = C³/(6π²) = 1,000,000/(6 × 9.8696) ≈ 16,877 cm³
- Hemisphere V: = V/2 ≈ 8,438.5 cm³
Volume of a Half Circle — Clearing Up the Confusion
The search "volume of a half circle" and "volume of a half circle formula" are among the most common geometry searches — and they almost always mean hemisphere volume. Let's clarify once and for all:
"Volume of a Half Circle" vs "Hemisphere Volume"
2D semicircle (half circle): Area = (1/2)πr² — this is a flat shape, measured in square units (cm²). It has NO volume because it has no depth.
3D hemisphere (half sphere): Volume = (2/3)πr³ — this is a solid shape, measured in cubic units (cm³). If you want a "volume of a half circle formula," this is it: V = (2/3)πr³.
The semicircle volume formula people are searching for is the hemisphere formula V = (2/3)πr³. The "volume of a half circle" is the volume of the 3D solid swept by rotating a semicircle around its diameter — which is exactly a sphere. Half of that sphere = hemisphere = (2/3)πr³.
| Shape | Dimensions | Formula | Unit |
|---|---|---|---|
| Semicircle (2D half circle) | Flat, no depth | Area = (1/2)πr² | cm² (area) |
| Hemisphere (3D half sphere) | Solid, has depth | Volume = (2/3)πr³ | cm³ (volume) |
| Full sphere | Solid, all directions | Volume = (4/3)πr³ | cm³ (volume) |
Hemisphere vs Full Sphere — Key Relationships
The hemisphere and full sphere have elegant exact halving relationships for volume and curved surface area — but not for total surface area. Here are all the key hemisphere relationships compared to the full sphere for r = 5 cm:
| Quantity | Hemisphere (r=5) | Full Sphere (r=5) | Ratio |
|---|---|---|---|
| Volume | ≈ 261.80 cm³ | ≈ 523.60 cm³ | exactly ½ ✓ |
| Curved SA | ≈ 157.08 cm² | ≈ 314.16 cm² | exactly ½ ✓ |
| Total SA | ≈ 235.62 cm² | ≈ 314.16 cm² | ¾ (NOT ½) |
| Flat base | ≈ 78.54 cm² | — | extra πr² |
Key insight: Hemisphere volume = sphere volume / 2 ✓ and Hemisphere curved SA = sphere SA / 2 ✓. But hemisphere total SA ≠ sphere SA / 2 — because total SA includes the flat base πr², making it 3πr² rather than 2πr². Total SA = (3/4) of sphere SA.
Real-World Hemisphere and Sphere Examples
🏛️ Pantheon Dome, Rome — Hemisphere r ≈ 21.65 m
- V = (2/3)π × 21.65³ = (2/3)π × 10,145.5 ≈ 21,237 m³ of enclosed space
- Curved SA = 2π × 21.65² = 2π × 468.7 ≈ 2,946 m² of dome surface
- Diameter = 43.3 m — the Pantheon's dome diameter exactly equals the height of the building
🌍 Earth — Sphere r = 6,371 km
- Circumference of sphere: C = 2π × 6,371 ≈ 40,030 km
- Surface area: SA = 4π × 6,371² ≈ 510,064,472 km² ≈ 510 million km²
- Hemisphere volume: V = (2/3)π × 6,371³ ≈ 5.41 × 10¹¹ km³
- Full sphere volume: V = (4/3)π × 6,371³ ≈ 1.083 × 10¹² km³
🏀 Basketball — r = 12 cm (diameter ≈ 24 cm)
- Circumference: C = 2π × 12 ≈ 75.4 cm
- Surface area: SA = 4π × 144 ≈ 1,809.6 cm²
- Volume: V = (4/3)π × 1,728 ≈ 7,238.2 cm³
- Hemisphere volume: V/2 ≈ 3,619.1 cm³
Common Mistakes With Hemisphere Calculations
❌ Mistake 1 — Confusing Curved SA (2πr²) with Total SA (3πr²)
- Wrong: SA = 2πr² for total surface area
- Correct: Curved SA = 2πr² (dome only); Total SA = 3πr² (dome + base)
- For r=5: curved SA ≈ 157.08 cm², total SA ≈ 235.62 cm²
❌ Mistake 2 — Not Simplifying (1/2) × (4/3)πr³
- Wrong: writing (1/2) × (4/3)πr³ and stopping there
- Correct: (1/2) × (4/3) = 4/6 = 2/3, so V = (2/3)πr³
- Both expressions are equivalent — always simplify to V = (2/3)πr³
❌ Mistake 3 — Confusing 2D Semicircle Area with 3D Hemisphere Volume
- Wrong: "volume of half circle" = (1/2)πr² (this is area in cm², not volume)
- Correct: hemisphere volume = (2/3)πr³ (measured in cm³)
- A semicircle has AREA; a hemisphere has VOLUME — completely different
❌ Mistake 4 — Using Diameter Instead of Radius in C = 2πr
- Wrong: C = 2π × d (uses diameter, giving 2× the correct answer)
- Correct: C = 2πr where r = d/2, OR equivalently C = πd
- For d=10 cm: correct C = π × 10 ≈ 31.416 cm, not 2π × 10 ≈ 62.832 cm
❌ Mistake 5 — Using Square Root Instead of Cube Root to Find r from V
- Wrong: r = √(3V/2π) using Math.sqrt (this gives wrong radius)
- Correct: r = ∛(3V/2π) using Math.cbrt (cube root)
- For V=100 cm³: r = ∛(47.746) ≈ 3.628 cm (not √47.746 ≈ 6.91 cm)