Surface Area Calculator – Calculate Surface Area for 9 Shapes: Sphere, Cone, Cube, Cylinder & More

  • Adam
  • November 4, 2025

Free online surface area calculator for 9 3D shapes: sphere, cone, cube, cylinder, rectangular tank, capsule, spherical cap, conical frustum, and square pyramid. Step-by-step solutions with formulas.

Surface Area Calculator

Calculate Surface Area for 9 Common 3D Shapes with Complete Formulas and Step-by-Step Solutions

Quick Navigation

Sphere Surface Area

Enter radius to calculate the total surface area of a sphere.

Sphere Calculator
Surface Area
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Sphere Formula
SA = 4πr²

Where r = radius
π ≈ 3.14159

A sphere is a perfectly round 3D object. All surface area formula requires only the radius.

Cone Surface Area

Enter base radius and height to calculate cone surface area (with or without base).

Cone Calculator
Surface Area
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Cone Formula
Lateral SA = πr√(r² + h²)
With Base = πr√(r² + h²) + πr²

Where r = radius, h = height

Lateral surface area excludes the base. Check "Include base" for total surface area.

Cube Surface Area

Enter edge length to calculate the total surface area of a cube.

Cube Calculator
Surface Area
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Cube Formula
SA = 6a²

Where a = edge length

A cube has 6 equal square faces

Simplest 3D shape formula. All edges equal length, all faces are identical squares.

Cylinder Surface Area

Enter base radius and height to calculate cylinder surface area.

Cylinder Calculator
Surface Area
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Cylinder Formula
Base SA = 2πr²
Lateral SA = 2πrh
Total SA = 2πr² + 2πrh = 2πr(r + h)

Where r = radius, h = height

Includes both circular bases and curved lateral surface.

Rectangular Tank Surface Area

Enter length, width, and height to calculate surface area of rectangular tank.

Rectangular Tank Calculator
Surface Area
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Rectangular Tank Formula
SA = 2lw + 2lh + 2wh
or SA = 2(lw + lh + wh)

Where l = length, w = width, h = height

Sum of areas of all 6 rectangular faces.

Capsule Surface Area

Enter radius and height to calculate capsule surface area.

Capsule Calculator
Surface Area
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Capsule Formula
SA = 4πr² + 2πrh

Where r = radius, h = height

(Sphere + cylinder lateral)

Capsule is sphere on both ends with cylindrical center.

Spherical Cap Surface Area

Enter two of three values to calculate spherical cap surface area.

Spherical Cap Calculator
Surface Area
-
Spherical Cap Formula
SA = 2πRh

Where R = sphere radius
h = cap height

Alternative: SA = π(r² + h²)

Curved portion of sphere cut by a plane.

Conical Frustum Surface Area

Enter top radius, bottom radius, and height for frustum surface area.

Conical Frustum Calculator
Surface Area
-
Conical Frustum Formula
SA = π(r² + R²) + π(r + R)s

Where r = top radius
R = bottom radius
s = slant height = √[h² + (R-r)²]

Cone with top cut off. Includes two circular bases and lateral surface.

Square Pyramid Surface Area

Enter base edge length and height to calculate pyramid surface area.

Square Pyramid Calculator
Surface Area
-
Square Pyramid Formula
Base SA = a²
Lateral SA = 2a√(h² + (a/2)²)
Total SA = a² + 2a√(h² + (a/2)²)

Where a = base edge, h = height

Square base with 4 triangular faces meeting at apex.

Understanding Surface Area

Surface area measures the total area of all external surfaces of a 3D object. Expressed in square units (m², ft², cm²), surface area differs fundamentally from volume. Where volume measures interior space, surface area measures only the outer covering—what you would paint, wrap, or coat.

Surface area calculations are essential across industries. In manufacturing, surface area determines how much material is needed for coating or finishing. In engineering, surface area affects heat transfer and structural properties. In packaging, surface area determines box dimensions and material costs. In environmental science, surface area affects reaction rates and heat absorption.

Each shape requires different formulas because their surfaces have different geometries. A cube's surface consists of flat squares; a sphere has no flat surfaces at all. Understanding these differences enables accurate calculations for any 3D object.

Key Principle: Surface area is purely external measurement of 3D objects. It combines areas of all surfaces—bases, sides, curved surfaces, or triangular faces depending on shape.

Key Features & Capabilities

9 Shapes Covered Sphere, cone, cube, cylinder, rectangular tank, capsule, cap, frustum, pyramid
Flexible Inputs Some shapes offer options (include/exclude base, lateral only)
Instant Calculations Real-time surface area computation
Clear Formulas Every calculation shows exact formula used
Step-by-Step Detailed breakdown of calculation process
Mobile Friendly Fully responsive on all devices

How to Use This Calculator

General Steps

  1. Select Your Shape: Click the tab for your shape (sphere, cone, cube, etc.)
  2. Measure Dimensions: Gather all required measurements (radius, height, length, width)
  3. Enter Values: Input measurements into the form fields
  4. Click Calculate: Press Calculate button
  5. Review Results: See surface area with complete calculation breakdown

Important Notes

  • All Positive: All dimensions must be positive numbers
  • Consistency: Use same unit system throughout (all meters, all feet, etc.)
  • Optional Bases: Some shapes (cone) let you include/exclude bases
  • Two of Three: Spherical cap requires any two of: base radius, sphere radius, height
  • Measurements Matter: Accurate dimensions ensure accurate surface area

Complete Formulas Reference

Sphere
SA = 4πr²
Cone
Lateral = πr√(r² + h²)
Total = πr√(r² + h²) + πr²
Cube
SA = 6a²
Cylinder
SA = 2πr(r + h)
Rectangular Tank
SA = 2(lw + lh + wh)
Capsule
SA = 4πr² + 2πrh
Spherical Cap
SA = 2πRh
Conical Frustum
SA = π(r² + R²) + π(r + R)s
where s = √[h² + (R-r)²]
Square Pyramid
SA = a² + 2a√(h² + (a/2)²)

Shape Guide

Understanding surface area components for each shape:

Sphere Perfect round 3D object. No edges, corners, or flat surfaces. Calculated from radius alone.
Cone Circular base tapering to point (apex). Lateral surface is curved. Base can be included/excluded.
Cube Six equal square faces. Simplest formula: just 6 times one face area (6a²).
Cylinder Two circular bases connected by curved lateral surface. Like a can or pipe.
Rectangular Tank Six rectangular faces (may include some squares). All opposite faces equal in area.
Capsule Sphere on each end with cylindrical middle. Combination of sphere and cylinder formulas.
Spherical Cap Portion of sphere cut by a plane. Only the curved portion, not including base.
Conical Frustum Cone with top cut off. Two circular bases of different sizes plus curved lateral surface.
Square Pyramid Square base with 4 triangular faces meeting at apex. Like Egyptian pyramid.

Real-World Applications

Manufacturing & Fabrication

Surface area determines material requirements for coating, painting, or finishing. Manufacturers calculate surface area to estimate how much paint, protective coating, or fabric is needed for production.

Packaging & Shipping

Box and container design depends on surface area calculations. Surface area determines cardboard material needed, shipping weight, and packaging costs. Cylindrical tanks and spherical containers require precise surface area calculations.

Heat Transfer & HVAC

Surface area affects how quickly objects heat or cool. HVAC systems calculate ductwork surface area. Heat exchangers depend on large surface areas. Insulation effectiveness relates to surface area exposed to environment.

Construction & Architecture

Building designs require surface area calculations for exterior walls, roofing, flooring. Domes and curved structures use surface area formulas. Material estimates for construction projects depend on accurate surface area measurements.

Materials Science & Chemistry

Particle surface area affects reaction rates and absorption. Larger surface area means faster chemical reactions. Catalysts depend on maximizing surface area exposure. Filtration systems use surface area to capacity calculations.

Worked Examples

Example 1: Sphere (Chocolate Truffles)

Problem: Sphere-shaped chocolate truffle with radius 0.325 inches. Find surface area for coating.

Solution:
SA = 4πr² = 4 × π × 0.325²
SA = 4 × π × 0.1056
SA ≈ 1.327 in²
Coating needed: approximately 1.33 square inches per truffle

Example 2: Cylinder (Fish Tank)

Problem: Cylindrical fish tank: radius 3.5 feet, height 5.5 feet. Find surface area to be painted.

Solution:
SA = 2πr(r + h) = 2π(3.5)(3.5 + 5.5)
SA = 2π(3.5)(9)
SA = 2π(31.5)
SA ≈ 197.92 ft²
Paint needed for approximately 198 square feet

Example 3: Rectangular Tank (Box)

Problem: Rectangular box: length 3 feet, width 4 feet, height 5 feet. Find wrapping paper needed.

Solution:
SA = 2(lw + lh + wh)
SA = 2((3×4) + (3×5) + (4×5))
SA = 2(12 + 15 + 20)
SA = 2(47) = 94 ft²
Wrapping paper needed: 94 square feet

Example 4: Cube (Rubik's Cube)

Problem: Rubik's Cube with 4-inch edge length. Find total surface area.

Solution:
SA = 6a² = 6 × 4²
SA = 6 × 16
SA = 96 in²
Total surface area of cube: 96 square inches

Example 5: Cone (Party Hat)

Problem: Conical party hat: base radius 1 foot, height 5 feet. Find material needed (lateral surface only).

Solution:
Slant height = √(1² + 5²) = √26 ≈ 5.099
Lateral SA = πr√(r² + h²) = π(1)(5.099)
Lateral SA ≈ 16.02 ft²
Material needed: approximately 16 square feet

Frequently Asked Questions

❓ What's the difference between surface area and volume?
Surface area measures only the outer covering (2D), measured in square units. Volume measures interior space (3D), measured in cubic units. Surface area determines coating needs; volume determines container capacity.
❓ Why is lateral surface area different from total?
Lateral surface area excludes the bases (top and bottom), measuring only the curved or slanted sides. Total surface area includes all surfaces. For open containers, lateral SA may be all you need.
❓ What's a slant height?
The straight-line distance along the slanted surface from base to apex. For a cone: slant height = √(radius² + height²). Different from vertical height.
❓ Can surface area be the same for different shapes?
Yes. Different shapes can have identical surface areas but different volumes. Surface area doesn't determine capacity or interior space—only external covering.
❓ Why do spheres have smallest surface area?
Among shapes with equal volume, spheres have minimum surface area. This is why bubbles and water droplets form spheres—minimizing surface tension requires minimum surface area.
❓ What real-world uses require surface area calculations?
Painting, coating, wrapping, packaging materials, heat transfer systems, HVAC design, container manufacturing, chemical reactions, insulation, and any project involving covering or finishing 3D objects.

Calculate Surface Area Instantly

Whether you're designing packaging, estimating materials, planning construction, or solving geometry problems, this comprehensive surface area calculator handles 9 shapes with instant accuracy and complete step-by-step analysis. Fast, reliable, completely free.

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