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Math calculator

Spherical Sector Calculator

Spherical sector V = (2/3)πr²h from sphere radius and cap height. h = r is a hemisphere. h = 2r is a sphere. Same engine as Solid Geometry. Not /spherical-cap. Not /cone. Not /sphere. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 111/111 tests · Production surface contract 3/3 · v1.18.0
Input interpretation
Enter values to calculate.
Result
—
Model
Spherical sector volume, base radius, and surface from sphere radius and height.
Scope
Euclidean 3-space
Verification
Engine tested · 111/111 tests · Production surface contract 3/3 · Specification checked · v1.18.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
42 golden · 23 boundary · 46 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.18.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

VolumeV = (2/3)πr²h
Base radiusa = √(h(2r−h))
SurfaceA = πr(2h + a)
Hemisphereh = r ⇒ V = 2/3 πr³, A = 3πr²
Sphere identityh = 2r ⇒ V = 4/3 πr³, A = 4πr²
iSame engine as /solid-geometry. The cap without the cone stays /spherical-cap. The right circular cone stays /cone. The closed sphere stays /sphere. Not a second engine.

How to use

1

Enter r and h

Sphere radius and sector height must be positive, with h ≤ 2r. Default r = 1, h = 1 (hemisphere).

2

Read V, a, and A

Unit hemisphere has V = 2π/3 and A = 3π. At h = 0.5 the sector is larger than the cap. h = 2 matches the unit sphere.

Example calculations

Common configurations with formula and result.

ϟ

Unit hemisphere

r = 1, h = 1

V = 2π/3
2π/3
ϟ

Half height

r = 1, h = 0.5

V = π/3
π/3
ϟ

Sphere identity

r = 1, h = 2

V = 4π/3
4π/3

Spherical Sector calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A spherical sector (spherical cone) is the solid of a spherical cap plus the cone from the sphere center to the cap base. With radius r and height h, 0<h≤2r, volume is (2/3)πr²h, base radius a=√(h(2r−h)), and surface πr(2h+a). h=r is a hemisphere. h=2r is the full sphere. This page is mode=spherical_sector on math.geometry.solid. Not /spherical-cap. Not /cone. Not /sphere. Not /spherical-wedge. Not /spherical-lune. Not CAS.
What it calculates
Spherical sector volume, base radius, and surface from sphere radius and height.
Inputs
  • r
  • h
Outputs
  • volume
  • surface
  • lateral
  • base
  • a
  • sphere
  • hemisphere
Formula
V = (2/3)πr²h, A = πr(2h + a)
Assumptions
  • Euclidean 3-space
  • 0 < h ≤ 2r
  • h = r is a valid hemisphere
  • h = 2r is a valid sphere
  • Not a cap page
  • Not a cone page
  • Not a sphere page
  • Not a wedge page
  • Not a lune page
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • h > 2r → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
r=1 h=1 → V=2π/3 A=3π
Validation cases

5 published on this page · 111/111 tests · Production surface contract 3/3 · View evidence

  • r=1 h=1 → V=2π/3 A=3π hemisphere=true
  • r=1 h=0.5 → V=π/3
  • r=1 h=2 → V=4π/3 A=4π sphere=true
  • r=1 h=0 → VALUE_MUST_BE_POSITIVE
  • r=1 h=3 → INVALID_INPUT
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.18.0

Other calculators in this family: Angle Between Lines Calculator, Annulus Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Annulus Calculator Annulus A = π(R² − r²) and P = 2π(R + r) with outer R > inner r. Same engine as Plane Geometry. Not /circle. Not a square. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second solid engine?

No. It is mode=spherical_sector on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this the spherical cap calculator?

No. The cap without the cone from the center stays on /calc/math/spherical-cap. At h = r both are a hemisphere. At other heights the sector is larger.

Is this the cone calculator?

No. The right circular cone stays on /calc/math/cone. This page is a cap plus the cone from the sphere center.

Is this the sphere calculator?

No. The closed sphere stays on /calc/math/sphere. Height h = 2r is a valid sector that matches the sphere volume and surface.

Is this a spherical wedge?

No. Two half-planes through a diameter stay on /calc/math/spherical-wedge.

Is this a spherical lune?

No. The curved surface 2r²θ stays on /calc/math/spherical-lune.

Where does this run?

Locally in the browser by default. REST and MCP call the same solid-geometry engine.