3D Geometry: Cone / Sphere / Cylinder / Pyramid Volume & Surface Area
Pick a cone, sphere, cylinder or square-based pyramid, enter radius, height and slant height to get volume, total and lateral surface area.
Input Data
Results
At a glance:Cone V=⅓πr²h, sphere V=⁴⁄₃πr³, cylinder V=πr²h, square pyramid V=⅓s²h. Surface areas: cone TSA=πr(r+l), sphere SA=4πr², cylinder TSA=2πr(r+h), square pyramid TSA=s²+2sl, where l is the slant height.
Formula
Cone: V = ⅓πr²h; TSA = πr(r + l); l = √(r² + h²).
Sphere: V = ⁴⁄₃πr³; SA = 4πr².
Cylinder: V = πr²h; TSA = 2πr(r + h); LSA = 2πrh.
Square pyramid: V = ⅓s²h; TSA = s² + 2sl; l = √((s/2)² + h²).
$$V_{\text{cone}} = \frac{1}{3}\pi r^{2}h$$$$V_{\text{sphere}} = \frac{4}{3}\pi r^{3}$$$$V_{\text{cyl}} = \pi r^{2}h$$$$V_{\text{pyr}} = \frac{1}{3}s^{2}h$$How to Use
- Choose the solid (cone / sphere / cylinder / pyramid).
- Enter base radius r and height h; pyramid also needs base side s.
- Leave slant height l blank (auto from r, h) unless you want to override it.
- The tool returns volume, total and lateral surface area; compare with the case studies.
Formulas for the four solids (r radius, h height, s base side, l slant)
| Solid | Volume V | Total SA | Lateral SA |
|---|---|---|---|
| Cone | ⅓πr²h | πr(r+l) | πrl |
| Sphere | ⁴⁄₃πr³ | 4πr² | (no separate lateral) |
| Cylinder | πr²h | 2πr(r+h) | 2πrh |
| Square Pyramid | ⅓s²h | s²+2sl | 2sl |
Cone/pyramid slant height l follows from √(r²+h²) or √((s/2)²+h²).
Case Studies
Cone r=3, h=4
Slant height l = √(3²+4²) = 5.
V = ⅓π·9·4 = 12π ≈ 37.6991.
TSA = π·3·(3+5) = 24π ≈ 75.3982; LSA = π·3·5 = 15π ≈ 47.1239.
Sphere r=3
V = ⁴⁄₃π·27 = 36π ≈ 113.0973.
SA = 4π·9 = 36π ≈ 113.0973 (numerically equal here).
Cylinder r=3, h=4
V = π·9·4 = 36π ≈ 113.0973.
TSA = 2π·3·(3+4) = 42π ≈ 131.9469; LSA = 2π·3·4 = 24π ≈ 75.3982.
Square pyramid s=4, h=3
Slant height l = √(2²+3²) = √13 ≈ 3.6056.
V = ⅓·16·3 = 16.
TSA = 16 + 2·4·3.6056 = 44.8440; LSA = 28.8440.
Cylinder vs Cone (same r, h)
With same base and height, cone volume = ⅓ cylinder volume.
r=3, h=4: cylinder V=113.0973, cone V=37.6991 (= ⅓).
Doubling the sphere radius
r from 3 to 6: volume 36π → 288π (×8, since r³).
Surface area 36π → 144π (×4, since r²).
FAQ
Cone vs cylinder volume?
With the same base and height, the cone volume is exactly ⅓ of the cylinder (Cavalieri's principle) — a common DSE comparison.
What is the slant height l?
The slant height is the cone's generatrix or the pyramid's triangular face altitude, not the solid height h. Cone l=√(r²+h²); square pyramid l=√((s/2)²+h²).
Total vs lateral surface area?
Total surface area (TSA) includes the base(s); lateral (LSA) counts only curved/Slanted faces. A sphere has no separate base, so only total SA.
Must the pyramid base be square?
This tool assumes a square base so a single side s suffices. Other regular polygon bases change base area and face count.
Doubling the radius?
Volume scales with r³ (×8 when doubled); surface area scales with r² (×4). This is the similarity scaling law.
Does the cylinder count two bases?
TSA = 2πr(r+h) includes both top and bottom bases (2πr²) plus the lateral 2πrh. For lateral only use LSA = 2πrh.
Can I use diameter instead of radius?
Inputs are the radius r. If you have diameter d, divide by 2 (r = d/2) first.
What are the units?
Volume is the cube of the input length (e.g. cm³), surface area is the square (e.g. cm²). The tool is unitless — add units yourself.
Related Tools
References
Content review: Calculatorism Science Team. Volume and surface-area formulas for all four solids verified. Results are for reference only.