Calculate volume of sphere, cone, cylinder, and cube instantly with our interactive 3D visualizer. Learn formulas with step-by-step solutions, explore geometric properties, and discover mathematical relationships through visualization. Free online geometry calculator and math solver.

CalcVisual is a free online volume calculator and interactive 3D geometry visualizer that combines the precision of a math solver with the clarity of real-time visualizations. Unlike static calculators that only provide numeric results, CalcVisual lets you see the formulas, manipulate 3D shapes in real-time, and discover mathematical relationships through exploration.

The core idea: when you drag the radius slider from 50 to 100, three things happen simultaneously - the sphere grows, the r³ in the formula highlights, and the calculation steps show r³ = 1,000,000. When radius doubles, a message appears: radius × 2 = volume × 8. That's when the relationship becomes obvious.

The interface uses Swiss Style principles:

No Rounded Corners - Maximum 2px border radius Explicit Grid - Visible 2px black borders define structure Academic Typography - Times New Roman serif headings Minimal Color - Pure white (#FFFFFF) and black (#000000) High Contrast - Pure black text for maximum readability Single Accent - Mathematical Blue (#0055FF) for emphasis only No Shadows - Flat design, content over decoration Border-Based Interaction - Border changes on hover, not background effects

This transforms the interface from generic "SaaS product" into a serious mathematical publication.

Sphere Volume Calculator

• Calculate volume with adjustable radius: V = (4/3)πr³
• Real-time 3D visualization
• Step-by-step formula explanations
• Surface area calculator: S = 4πr²

Cone Volume Calculator

• Calculate volume: V = (1/3)πr²h
• Surface area with slant height: S = πr(r+l)
• Compare cone vs cylinder volume (1/3 relationship)
• Interactive apex and base manipulation

Cylinder Volume Calculator

• Calculate volume: V = πr²h
• Surface area: S = 2πr(r+h)
• Real-time radius and height adjustment
• Curved surface unrolling visualization

Cube & Cuboid Volume Calculator

• Calculate volume: V = l × w × h
• Surface area: S = 2(lw + lh + wh)
• Space diagonal: d = √(l² + w² + h²)
• Interactive face highlighting

Pythagorean Theorem Calculator

• Calculate hypotenuse: c² = a² + b²
• Interactive 2D triangle visualization
• Visual proof with colored squares
• Famous Pythagorean triples detector (3-4-5, 5-12-13)

Formula Explanations

• Step-by-step derivations for each shape
• Variable definitions and units
• Real-world application examples
• Historical context (Archimedes, Euclid)

FAQ Sections

• Common geometry questions answered
• Bold keywords for easy scanning
• Step-by-step calculation guides
• Shape property comparisons

Discovery Zones

• Interactive experiments and comparisons
• "What if?" scenarios
• Mathematical relationship insights
• Real-world object comparisons

ExplorableFormula

• Step-by-step formula derivations
• Variable highlighting tied to slider manipulation
• Real-time calculation results

DiscoveryInsight

• Context-aware insights triggered by exploration
• Messages appear when you hit specific relationships
• Examples: doubling radius, finding famous triples

InteractiveSlider

• Touch-friendly sliders (44px minimum touch targets)
• Synchronized formula feedback
• Swiss styling: square thumb, 2px black border

DiscoveryZone

• Themed exploration areas
• Experiments and comparisons
• Historical context sections

FAQSection (NEW)

• Shape-specific frequently asked questions
• Bold keywords for easy scanning
• Step-by-step calculation guides
• FAQPage schema for "People Also Ask" optimization
• 20+ FAQ entries across all shapes

• 5 Geometric Shapes: Sphere, Cone, Cylinder, Cube, Pythagorean Triangle
• Real-time Parameter Adjustment: Slider controls for all dimensions
• Visible/Hidden Line Rendering: Smart solid/dashed line separation
• High-Performance Rendering: 60fps SVG rendering
• Responsive Design: Mobile-first touch-friendly interface
• SEO-Optimized Content: Educational material with formulas and context

• Node.js 16+
• npm 8+

cd geovisual
npm install
npm run dev

Browser opens at http://localhost:5173

npm run build
npm run preview

• Framework: React 18.3.1
• Build Tool: Vite 5.3.1
• Styling: Tailwind CSS 3.4.4 with custom Swiss design system
• Icons: Lucide React 0.400.0
• Math Engine: Pure JavaScript (no external math libraries)
• Typography: Times New Roman (headings), Inter (body), JetBrains Mono (math)
• Color System: Custom Swiss tokens (Black, White, Math Blue #0055FF)

geovisual/
├── src/
│   ├── components/
│   │   ├── premiumPages/
│   │   │   ├── PremiumSpherePage.jsx       # Sphere exploration
│   │   │   ├── PremiumConePage.jsx         # Cone exploration
│   │   │   ├── PremiumCylinderPage.jsx     # Cylinder exploration
│   │   │   ├── PremiumCubePage.jsx         # Cube exploration
│   │   │   └── PremiumPythagoreanPage.jsx  # Pythagorean theorem
│   │   ├── ExplorableFormula.jsx           # Formula display
│   │   ├── DiscoveryInsight.jsx            # Insight triggers
│   │   ├── DiscoveryZone.jsx               # Exploration containers
│   │   ├── InteractiveSlider.jsx           # Touch sliders
│   │   ├── PythagoreanVisualization.jsx    # 2D triangle
│   │   └── FAQSection.jsx                  # FAQ with schema (NEW)
│   ├── App.jsx                             # Main app with 3D engine
│   ├── App.css                             # Swiss typography
│   └── main.jsx                            # Entry point
├── index.html
├── vite.config.js
├── tailwind.config.js                      # Swiss design tokens
└── package.json

const Engine = {
  ROTATION: { x: -25, y: -35 }, // Optimized fixed viewing angle

  // 3D projection: map spatial points to screen coordinates
  project(x, y, z) {
    const radX = (this.ROTATION.x * Math.PI) / 180;
    const radY = (this.ROTATION.y * Math.PI) / 180;

    // Rotate around Y-axis (horizontal rotation)
    let nx = x * Math.cos(radY) + z * Math.sin(radY);
    let nz = -x * Math.sin(radY) + z * Math.cos(radY);

    // Rotate around X-axis (pitch rotation)
    let ny = y * Math.cos(radX) - nz * Math.sin(radX);
    let fz = y * Math.sin(radX) + nz * Math.cos(radX); // Projected depth Z

    return { x: nx, y: ny, z: fz };
  }
}

// Determine if face is front-facing (based on screen space cross product)
isFrontFacing(p1, p2, p3) {
  // In SVG coordinate system (Y-axis down), clockwise points with nz < 0 indicate front face
  const v1 = { x: p2.x - p1.x, y: p2.y - p1.y };
  const v2 = { x: p3.x - p1.x, y: p3.y - p1.y };
  const nz = v1.x * v2.y - v1.y * v2.x;
  return nz < 0; // Front face if normal points toward viewer
}

// Edge processing: logic is "visible edge = edge belonging to at least one front-facing face"
const edgeMap = new Map();

processedFaces.forEach(face => {
  const n = face.idxSet.length;
  for (let i = 0; i < n; i++) {
    const v1 = face.idxSet[i], v2 = face.idxSet[(i + 1) % n];
    const key = [v1, v2].sort().join('-');
    const existing = edgeMap.get(key) || { isVisible: false };
    if (face.isFront) existing.isVisible = true;
    edgeMap.set(key, existing);
  }
});

// Render visible edges as solid lines, hidden edges as dashed lines
Array.from(edgeMap.values()).forEach((edge, i) => {
  const line = (
    <line
      key={`e-${i}`}
      x1={edge.p1.x} y1={edge.p1.y}
      x2={edge.p2.x} y2={edge.p2.y}
      stroke={edge.isVisible ? "#3b82f6" : "#3b82f6"}
      strokeWidth={edge.isVisible ? 2.5 : 1}
      strokeDasharray={edge.isVisible ? "0" : "5,5"}
      opacity={edge.isVisible ? 1 : 0.3}
    />
  );
});

// 1. Face processing and depth sorting (Painter's Algorithm)
const processedFaces = faces.map((idxSet, i) => {
  const pts = idxSet.map(idx => projectedPts[idx]);
  const isFront = Engine.isFrontFacing(pts[0], pts[1], pts[2]);
  const avgZ = pts.reduce((s, p) => s + p.z, 0) / pts.length;
  return { idxSet, isFront, avgZ, id: i };
}).sort((a, b) => b.avgZ - a.avgZ); // Sort by depth (back to front)

// 2. Render faces from back to front
processedFaces.forEach((f) => {
  const pathData = f.idxSet.map((idx, j) =>
    `${j === 0 ? 'M' : 'L'} ${projectedPts[idx].x} ${projectedPts[idx].y}`
  ).join(' ') + ' Z';

  return <path
    key={`f-${f.id}`}
    d={pathData}
    fill="url(#grad)"
    fillOpacity={f.isFront ? 0.3 : 0.05}
  />;
});

const ShapeGenerator = {
  generateSphere(radius) {
    const latS = 10, lonS = 16;
    const vertices = [], faces = [];

    for (let i = 0; i <= latS; i++) {
      const lat = (i * Math.PI) / latS - Math.PI / 2;
      const y = radius * Math.sin(lat), r = radius * Math.cos(lat);
      vertices.push(...this.getCircle(r, y, lonS));
    }

    for (let i = 0; i < latS; i++) {
      for (let j = 0; j < lonS; j++) {
        const c = i*lonS+j, n = i*lonS+(j+1)%lonS;
        const bc = (i+1)*lonS+j, bn = (i+1)*lonS+(j+1)%lonS;
        faces.push([c, n, bn, bc]);
      }
    }

    return { vertices, faces };
  },

  getCircle(r, y, segments = 64) {
    return Array.from({ length: segments }).map((_, i) => {
      const ang = (i * 2 * Math.PI) / segments;
      return { x: r * Math.cos(ang), y: y, z: r * Math.sin(ang) };
    });
  }
}

• V = Volume (cubic units)
• S = Surface Area (square units)
• r = Radius (units)
• h = Height (units)
• l, w = Length, Width (units)
• l = Slant Height (units)
• a, b = Triangle legs (units)
• c = Hypotenuse (units)
• π = Pi (≈3.14159)

Traditional Calculator:

Input: radius = 5, height = 10 Output: volume = 261.8

CalcVisual Approach:

1. See the Formula - V = (4/3)πr³ displayed with highlighted variables
2. Manipulate 3D Shape - Drag sliders and watch sphere grow in real-time
3. Step-by-Step Calculation - Follow each calculation step
4. Discover Relationships - Insights trigger when conditions met (e.g., "Doubling radius increases volume 8x!")
5. Real-World Context - Compare with basketball, planets, molecules

1. Calculate - Get instant results with interactive calculators
2. Visualize - See formulas update in 3D as you adjust parameters
3. Discover - Insights appear when you hit specific mathematical relationships

// Input: Adjust radius slider
radius = 70

// Step-by-step calculation:
Step 1: r³ = 70³ = 343,000
Step 2: π = 3.14159
Step 3: (4/3) × π × r³ = 1,436,755

// Visual feedback:
- Sphere grows in 3D
- Formula highlights "r³"
- Volume updates in real-time

// Discovery insight:
"Doubling radius increases volume 8x! (2³ = 8)"

1. Highlight - Dragging a slider highlights the corresponding variable in the formula
2. Connect - Step-by-step calculations show how inputs become outputs
3. Discover - Insights trigger when specific conditions are met

<ExplorableFormula
  formula="V = (4/3)πr³"
  variables={{ r: 70 }}
  highlight="r"
  showSteps={true}
  result={1436755}
  unit="cubic units"
/>

<DiscoveryInsight
  triggerCondition={() => radius >= initialRadius * 2}
  message="Doubling radius increases volume by 8x! (2³ = 8)"
/>

Primary (Calculator & Solver Cluster):

• sphere volume calculator
• cone volume calculator
• cylinder volume calculator
• cube volume calculator
• Pythagorean theorem calculator
• math solver
• geometry solver

Secondary (Visualization Cluster):

• 3D geometry visualizer
• interactive 3D shapes
• geometry simulation
• mathematical visualization
• interactive geometry tools

Educational (Problem-Solving Cluster):

• step-by-step calculator
• volume formulas
• surface area calculator
• geometry education
• interactive math tools

✅ Structured Data (Schema.org)

• WebApplication schema
• MathSolver schemas for each calculator (Sphere, Cylinder, Cone, Pythagorean)
• FAQPage schema for "People Also Ask" optimization

✅ Meta Tags Optimization

• Title tags with calculator-focused keywords
• Descriptions emphasizing "Calculate instantly" and "Step-by-step"
• Open Graph and Twitter Card tags for social sharing

✅ Content Optimization

• Educational content with 2500+ words
• FAQ sections with bold keywords
• Formula explanations with step-by-step guides
• Real-world application examples

✅ Technical SEO

• Fast loading (Vite optimization)
• Mobile-responsive design
• Accessible (ARIA labels, keyboard navigation)
• Semantic HTML structure

Discovery Zones:

• Radius exploration with volume scaling insights
• Archimedes' 2/3 cylinder-sphere relationship
• Real-world object comparison (baseball, basketball, planets)
• Surface area-to-volume ratio analysis

Insights:

• Radius × 2 → Volume × 8
• Radius × 3 → Volume × 27
• Large objects cool slower (surface/volume ratio)

Discovery Zones:

• Why 1/3? (Cone vs Cylinder volume comparison)
• Apex angle experiments (tall vs wide)
• Slant height and Pythagorean connection
• Ice cream cone calculator

Insights:

• 3 cones = 1 cylinder
• Height ≈ radius → optimal 53° apex angle
• Large slant height → huge surface area

Discovery Zones:

• Unrolling the cylinder (curved surface → rectangle)
• Volume comparison: Cylinder vs Sphere vs Cone
• Soda can optimization puzzle (330ml target)
• Pipe flow and cross-sectional area

Insights:

• Unrolled area = 2πrh (rectangle)
• Sphere = 2/3 of cylinder volume
• Cone = 1/3 of cylinder volume
• Pipe radius × 2 → Flow × 4

Discovery Zones:

• Interactive face highlighting
• Spatial Pythagorean theorem (d = √(l²+w²+h²))
• Box packing optimization puzzles
• Surface area-to-volume ratio

Insights:

• Space diagonal formula
• Packing efficiency comparisons
• Surface/volume ratio explains cell size

Discovery Zones:

• Interactive 2D triangle with draggable sides
• Visual proofs (Rearrangement, Similar Triangles, Euclid's Windmill)
• Real-world applications (ladders, TV sizes, construction)
• Famous Pythagorean triples detector

Features:

• Clean triangle without side labels
• Three colored squares (blue a², green b², purple c²)
• Hypotenuse square rotates to align with hypotenuse
• Real-time area calculation
• Famous triple recognition

1. Top Nav - Switch between shapes (Sphere, Cone, Cylinder, Cube, Pythagorean)
2. Left Panel - Parameter controls + Mathematical Derivation
3. Center Canvas - 3D/2D visualization with real-time rendering
4. Premium Content - Scroll down for Discovery Zones

• Drag Sliders → Formula highlights + Results update
• Adjust Parameters → Insights trigger when conditions met
• Click Shapes → Face highlighting (Cube page)
• Scroll Discovery Zones → Educational content unfolds

<ExplorableFormula
  formula="V = (4/3)πr³"
  variables={{ r: 70 }}
  highlight="r"
  showSteps={true}
  result={1436755}
  unit="cubic units"
/>

<DiscoveryInsight
  triggerCondition={() => radius > 100}
  message="Discovery message"
  type="discovery"  // or 'warning', 'tip'
/>

<DiscoveryZone
  title="Explore Title"
  variant="primary"  // 'primary' | 'secondary' | 'tertiary'
>
  {/* Content */}
</DiscoveryZone>

<InteractiveSlider
  label="Radius (r)"
  value={radius}
  onChange={setRadius}
  min={10}
  max={120}
  variableName="r"
  unit=" units"
  color="math-blue"
/>

• useMemo for expensive calculations
• React.memo on pure components
• Debounded slider events (16ms frame budget)

• 44px touch targets (Apple HIG compliant)
• Reduced motion respects prefers-reduced-motion
• Overflow scrolling for long content

• Semantic HTML (button, label, summary)
• ARIA labels on interactive controls
• Keyboard navigation support

• 650+ words per shape page (Comprehensive educational articles)
• 71 total FAQ entries (14-15 FAQs per shape covering long-tail keywords)
• 4 discovery zones per shape (Interactive experiments and comparisons)
• 4 content sections per shape: Introduction, Formula, Applications, Derivation
• Historical context: Archimedes, Euclid, ancient mathematicians
• Real-world applications: Engineering, astronomy, construction, manufacturing

• Meta descriptions: All 11 pages optimized to 140-160 characters for SERP display
• Dynamic title tags: High-traffic keywords (volume of a cylinder, cone geometry formulas, etc.)
• Structured data: WebApplication, MathSolver (4 shapes), FAQPage schemas
• Long-tail keyword targeting: Question-based FAQs for "People Also Ask" optimization
• Content depth: 6.5x expansion from ~100 to ~650 words per shape

• Component modularity - Reusable UI library
• Performance - 60fps rendering
• Responsive - Mobile-first Tailwind classes

Cause: Parent container has pointer-events: none

Fix: Add pointer-events-auto to interactive children

Cause: variableName prop doesn't match formula variable

Fix: Ensure consistent naming between slider and formula

Debug: Add console.log to check condition

npm run build

Output: dist/ directory

• All meta descriptions optimized (140-160 characters)
• Educational content expanded (650+ words per shape)
• FAQ sections enhanced (71 total entries)
• Structured data deployed (WebApplication, MathSolver, FAQPage)
• High-traffic keywords integrated
• Google Rich Results Test validation
• Google Search Console re-indexing request

npm run build
# Outputs to: dist/

Create .env.production:

VITE_BASE_URL=/path/to/app

npm i -g vercel
vercel --prod

npm i -g netlify-cli
netlify deploy --prod

The interface uses Swiss Style + Digital Brutalism principles to create a serious mathematical publication aesthetic.

/* Foundation Colors */
--swiss-white: #FFFFFF        /* Primary background */
--swiss-offwhite: #FAFAFA     /* Secondary backgrounds */
--swiss-black: #000000        /* Primary text */
--swiss-charcoal: #1A1A1A     /* Secondary text */

/* Accent Colors */
--math-blue: #0055FF          /* Single accent color */
--math-blue-light: #E6F0FF    /* Accent backgrounds */

Display Headings (Serif) - Times New Roman for authority Body Text (Sans-Serif) - Inter for readability Math & Code (Monospace) - JetBrains Mono for precision

Primary card

<div className="bg-swiss-white border-2 border-swiss-black rounded-swiss-sm p-8">

Secondary card

<div className="bg-swiss-offwhite border-2 border-swiss-black rounded-swiss-sm p-8">

Formula display

<span className="font-serif-display text-display-xl font-semibold text-swiss-black">
  V = (4/3)πr³
</span>

Math numbers

<span className="font-mono-math text-display-2xl text-math-blue">
  {volume.toFixed(2)}
</span>

MIT License - Free for educational purposes

CalcVisual demonstrates interactive math visualization using pure JavaScript algorithms. No 3D libraries (Three.js, Babylon.js) are used - all projection is calculated from scratch.

Status: Production Ready (v2.1.0)

Latest Updates: February 1, 2026

Recent SEO Optimization: ✅ Keyword migration from "Dictionary" to "Calculator & Visualizer" ✅ MathSolver schemas deployed for all shapes ✅ FAQ sections with step-by-step guides ✅ Enhanced meta tags for search visibility ✅ Mobile-optimized with 44px touch targets

• Pure math engine (no external 3D libraries)
• 60fps performance with optimized rendering
• 5 MathSolver schemas (Sphere, Cylinder, Cone, Pythagorean)
• SEO-optimized with calculator-focused keywords
• Mobile-first responsive design
• Accessible (ARIA labels, keyboard navigation)

• Interactive calculators with real-time 3D visualization
• Context-aware insights triggered by exploration
• Step-by-step calculation guides for each formula
• Real-world connections (soda cans, ice cream, buildings)
• Visual proof library with interactive demonstrations
• FAQ sections with bold keywords for easy scanning

• Swiss Style design system with academic typography
• High contrast (pure black on white)
• Minimal aesthetic (no emojis, gradients, shadows)
• Sharp geometry (2px borders, 2px max corner radius)
• Single accent color (Mathematical Blue #0055FF)

• 5 Interactive Calculators (Sphere, Cone, Cylinder, Cube, Pythagorean)
• 20+ FAQ entries with step-by-step guides
• 5 MathSolver schemas for rich snippets
• 2500+ Words of educational content
• 8 Reusable Components
• 2000+ Lines of React/JavaScript code

• 3-Month: 1,000 monthly impressions, 100 visitors
• 6-Month: 10,000 monthly impressions, 1,000 visitors
• 12-Month: 50,000 monthly impressions, 5,000 visitors
• Keywords: "sphere volume calculator", "Pythagorean theorem calculator", etc.

• React Team - Excellent UI framework
• Vite Team - Lightning-fast build tool
• Tailwind CSS - Practical utility-first CSS
• Lucide - Beautiful icon library
• Explorable Explanations - Pedagogical inspiration by Bret Victor