Math for Web Design you own this product

Precision layout and animation

Paul McFedries

MEAP began August 2025
Last updated November 2025
Publication in Spring 2026 (estimated)

ISBN 9781633434820
350 pages (estimated)

Included with a Manning Online subscription

printed in black & white

catalog / Programming Languages and Styles / Python

table of content

1 Web dev math fundamentals

1.1 Why bother learning about front-end math

1.2 Why math matters in front-end development

1.2.1 The connection between mathematics and web design

1.2.2 Real-world examples of math in UI/UX

1.2.3 How developers use math in animations, layouts, and interactions

1.3 The role of mathematics in CSS and JavaScript

1.3.1 How CSS uses math

1.3.2 JavaScript’s built-in mathematical capabilities

1.3.3 Comparing CSS and JavaScript math approaches

1.4 The minimum math you need to know

1.4.1 Arithmetic and arithmetic expressions

1.4.2 Algebra

1.4.3 Ratios

1.4.4 Exponents and nth roots

1.4.5 Linear equations

1.4.6 Inequalities and comparison expressions

1.4.7 Geometry

1.4.8 Trigonometry

1.4.9 Coordinate systems

1.5 Summary

2 Math basics for JavaScript

2.1 Building JavaScript expressions

2.1.1 JavaScript arithmetic operators

2.1.2 JavaScript arithmetic assignment operators

2.1.3 JavaScript comparison operators

2.1.4 JavaScript logical expressions

2.1.5 Falsy values, default values, and nullish coalescing

2.1.6 Operator precedence

2.1.7 Using parentheses to avoid ambiguity

2.2 Number types in JavaScript

2.2.1 Integer and floating-point numbers

2.2.2 Exponential notation

2.2.3 Binary, octal, and hexadecimal representations

2.2.4 NaN and Infinity in JavaScript

2.2.5 NaN

2.2.6 Infinity and -Infinity

2.2.7 Number coercion

2.3 Precision issues and floating-point arithmetic

2.3.1 Why 0.1 + 0.2 !== 0.3

2.3.2 When to be concerned about precision

2.4 Working with BigInt values

2.4.1 BigInt to the rescue

2.4.2 Mathematical operations with BigInt

2.4.3 Limitations of BigInt in JavaScript

2.4.4 When to avoid BigInt in web development

2.5 Modulo logic and wrapping values

2.5.1 JavaScript’s remainder operator (%) and modulo

2.5.2 Wrapping logic in carousels, sliders, and cyclic UI

2.6 Rounding numbers

2.6.1 Rounding to whole numbers

2.6.2 Rounding to a specific number of decimal places

2.6.3 Implementing bankers' rounding

2.6.4 Use cases for rounding in front-end development

2.7 Generating random numbers

2.7.1 Generating random integers and floats

2.7.2 Seeding and repeatable randomness

2.7.3 Use cases in animations and UI behavior

2.8 Solutions to Exercises

2.9 Summary

3 Math basics for CSS

3.1 The math of CSS measurement units

3.1.1 Unit conversion math

3.1.2 The cascading calculations of em units

3.1.3 Percentage computation rules

3.1.4 Viewport units and their responsive calculations

3.1.5 Mixing units with calc()

3.1.6 Combining units requires attention to scale

3.2 CSS math functions

3.2.1 CSS arithmetic operators

3.2.2 Evaluating expressions with calc()

3.2.3 Setting limits with min(), max(), and clamp()

3.2.4 More CSS math functions

3.3 CSS variables in math operations

3.3.1 Basic usage

3.3.2 Mixing units with variables

3.3.3 Fallbacks and missing variables

3.4 The math of CSS inheritance

3.4.1 The inheritance cascade formula

3.4.2 Which properties inherit mathematically?

3.4.3 Percentage values and the inheritance math

3.4.4 The em unit inheritance multiplier

3.4.5 Solving the heading size puzzle

3.4.6 Computed values versus inherited values

3.5 Box model math

3.5.1 The default box model formulas

3.5.2 How the box-sizing property affects the box model formulas

3.5.3 Margin math and collapsing

3.5.4 Percentages and the box model

3.5.5 Using calc() for precise box model math

3.6 Solutions to Exercises

3.7 Summary

4 CSS Grid math

4.1 The mathematical foundations of CSS Grid

4.1.1 Foundations of the grid formatting context

4.1.2 The coordinate system approach

4.1.3 The algebraic model

4.1.4 Mathematical differences from other layout systems

4.2 The Grid coordinate system

4.2.1 The grid model

4.2.2 Span notation and its mathematical meaning

4.2.3 Working smarter with negative indexing

4.3 The grid track sizing algorithm

4.3.1 The algebra of grid line placement

4.3.2 How fr units distribute available space

4.3.3 Taking the box model into account

4.3.4 The algorithm for calculating fr unit values

4.3.5 Troubleshooting: when <flex> resolves to 0px

4.4 The math of Grid’s track sizing functions and keywords

4.4.1 Controlling track sizes with minmax()

4.4.2 Understanding auto in grid tracks

4.4.3 The math of fit-content()

4.4.4 The repeat() function and its mathematical distribution

4.5 Solutions to Exercises

4.6 Summary

5 Flexbox math

5.1 The flex base size calculation

5.1.1 The base size algorithm

5.1.2 Exercises

5.2 Understanding the flex factors

5.2.1 The flex-grow calculation

5.2.2 The flex-shrink calculation

5.2.3 The flex shorthand

5.2.4 Dealing with flex-grow values between 0 and 1

5.2.5 Exercises

5.3 Taking min/max constraints into account

5.3.1 Using min-width to limit shrinking

5.3.2 Using max-width to limit growing

5.3.3 Exercises

5.4 Solutions to Exercises

5.5 Summary

6 The mathematics of responsive design

6.1 The concept of responsiveness in web design

6.1.1 Fixed vs. fluid: a visual thought experiment

6.1.2 The mathematical mindset

6.2 Media query logic

6.2.1 What grid and flexbox already give you

6.2.2 Why you might still need media queries

6.2.3 The math behind media queries

6.2.4 Using range syntax for cleaner media queries

6.2.5 Media query tips

6.3 Proportional layouts

6.3.1 How percentage-based layouts work

6.3.2 Nesting percentages

6.3.3 When to mix percentages with fixed units

6.3.4 Creating a liquid layout

6.4 Viewport-based units

6.4.1 The classic viewport units

6.4.2 Logical viewport units

6.4.3 The scrollbar problem

6.4.4 New viewport units: the sv, lv, and dv* families

6.4.5 When to use which viewport unit

6.5 Layout fluidity with CSS functions

6.5.1 Why fluid layouts matter

6.5.2 Using clamp() for fluid container widths

6.5.3 The math behind custom fluid widths

6.5.4 Calculating preferred clamp() values

6.5.5 Fluid padding and gaps

6.5.6 How to think about CSS math for layouts

6.5.7 When to bring in JavaScript

6.6 Solutions to Exercises

6.7 Summary

7 The mathematics of color

7.1 Color models and color spaces

7.1.1 The RGB color model

7.1.2 The HSL color model

7.1.3 The LAB color space

7.1.4 The OKLCH color space

7.2 Converting between color models

7.2.1 Converting RGB to grayscale

7.2.2 Converting RGB to HSL

7.2.3 Converting HSL to RGB

7.3 Creating harmonious color schemes with math

7.3.1 Complementary colors

7.3.2 Analogous colors

7.3.3 Triadic colors

7.3.4 Split-complementary colors

7.4 Specifying accessible colors with math

7.4.1 The WCAG contrast requirements (AA vs. AAA)

7.4.2 The contrast ratio formula

7.4.3 Calculating relative luminance from color

7.4.4 Calculating contrast ratios with JavaScript

7.5 Light mode and dark mode mathematics

7.5.1 Why basic inversion doesn’t work

7.5.2 A better approach: inverting perceptual lightness

7.5.3 Hue and saturation adjustments for aesthetic consistency

7.5.4 Generating a paired light/dark palette

7.5.5 CSS implementation with media queries

7.6 The math behind blend modes and opacity

7.6.1 How blending works

7.6.2 Alpha compositing

7.6.3 Blending in CSS

7.6.4 The separable blend modes

7.6.5 The non-separable blend modes

7.7 Solutions to Exercises

7.8 Summary

8 Text and typography math

8.1 The math behind the “typographic trinity”

8.1.1 General readability considerations

8.1.2 When you know the font size you want to use

8.1.3 When you know the line height you want to use

8.1.4 When you know the line length you want to use

8.1.5 Putting the formulas to work

8.2 Fluid typography with CSS math functions

8.2.1 Using vw and vh for scalable text sizes

8.2.2 Setting constraints on dynamic font scaling

8.2.3 Calculating the preferred clamp() formula for font sizes

8.2.4 Example: Hero text for a landing page

8.3 Applying a modular scale to text

8.3.1 Creating a modular scale from a scaling factor

8.3.2 Choosing a modular scale ratio

8.3.3 Implementing typographic scales with CSS variables

8.4 Solutions to Exercises

8.5 Summary

9 Trigonometry for web developers

9.1 The basics of trigonometry for web design

9.1.1 Sine, cosine, and tangent

9.1.2 Trigonometric units

9.1.3 Converting between degrees and radians

9.1.4 The inverse trig functions

9.1.5 Trig functions in CSS and JavaScript

9.2 Circular layouts and animations

9.2.1 Creating a circular menu

9.2.2 Animating circular motion

9.2.3 Creating an animated spinner

9.3 Wave effects

9.3.1 Arranging elements along a wave path

9.3.2 Animating elements with a wave effect

9.3.3 Animating an element with a pendulum effect

9.4 Directional calculations

9.5 Solutions to Exercises

9.6 Summary

10 Vectors and geometry in UI design

10.1 Mathematical foundations: points, vectors, and bounding boxes

10.2 DOM geometry: measuring and aligning elements with JavaScript

10.2.1 Calculating the center point of an element

10.2.2 Aligning one element relative to another

10.2.3 Geometric hover effects

10.2.4 Parallax scrolling with geometric precision

10.3 Manipulating vectors in JavaScript

10.3.1 Essential vector operations for DOM developers

10.3.2 Using vectors to have an element follow the mouse

10.3.3 Animating an element to a mouse click position using vectors

10.3.4 Dragging with geometric constraints

10.4 Geometric clipping and motion paths

10.4.1 Clipping elements

10.4.2 Motion paths

10.5 Solutions to Exercises

10.6 Summary

11 Bezier curves and easing functions

12 Physics-based animations

13 Interactivity and gesture-based UI math

14 Canvas-based math and drawing

Overview

1 Web dev math fundamentals

This chapter makes the case that mathematics quietly powers everyday front-end work and that learning a small, practical slice of it pays off in precision, robustness, creativity, and speed. From relative units like em/rem and animation easings to DOM measurements, much of what we style or script rests on numeric reasoning. With a bit of math literacy, layouts break less, responsive designs scale predictably, animations feel smoother, and debugging becomes a matter of logic instead of guesswork.

It shows how math underpins UI/UX decisions and behaviors: spacing, grids, type scales, color adjustments, responsive sizing with percentages and viewport units, and motion driven by timing functions and curves. You see the complementary roles of CSS and JavaScript—CSS offers declarative, built-in math (calc, min, max, clamp, proportional units) ideal for layout and typography, while JavaScript provides procedural, real-time calculations via the Math object and expressions for interaction, animation, and dynamic behavior. Core platform coordinates and measurements (viewport origin at the top-left, getBoundingClientRect, pointer positions) frame how elements are placed and updated on screen.

Finally, it outlines a concise math toolkit for front-end work: basic arithmetic and expressions; algebra with variables, functions, and formulas; ratios for aspect ratios, fractional grids, and modular typography; exponents and roots for easing and distance; linear equations for mapping inputs to outputs (scroll progress, constant-speed motion); inequalities and comparisons for constraints and breakpoints; essential geometry (points, vectors, angles, transforms, shapes); trigonometry for circular layouts, waves, rotations, and paths; and coordinate systems for positioning and responsive scaling. Mastering these fundamentals equips you to design and build interfaces that are more accurate, adaptive, and performant.

Why is the section heading so much bigger than the main heading? Math!

A type scale based on the Golden Ratio (approximately 1.618).

Centering an element within its containing block requires just a few CSS declarations, but behind the scenes the browser is jumping through many mathematical hoops to get the job done

Some basic geometric figures.

A right triangle and its sides relative to the angle theta (θ).

Using trigonometry to arrange menu items in a circle.

Summary

Learning about front-end math will help you build more precise and reliable layouts, write more performant code, and debug layout issues faster.
Mathematics in UI and UX design has practical use cases for responsive layouts, typography, colors, and spacing and alignment.
Developers use math when coding motion and animation, layouts, and user interactions.
In CSS, you use math to position and transform elements, scale layouts, build responsive elements, style text, and animate transitions.
CSS provides math functions that you can use to perform basic arithmetic right in your style declarations.
JavaScript gives you full control with dynamic, real-time math using the Math object and custom logic.
Use CSS math for layout, sizing, spacing, and typography when the values can be determined by the browser at render time.
Use JavaScript math when you need interaction, animation, or real-time adjustments based on user behavior or complex conditions.

FAQ

Why should front‑end developers learn “web dev math”?

It makes layouts more precise, responsive behavior predictable, animations smoother, debugging faster, and opens creative patterns (grids, type scales, circular layouts). A small math toolkit pays off in cleaner code and more robust UIs.

Where does math show up in everyday CSS and JS work?

Everywhere: relative units (em, rem, vw, %), Grid/Flex ratios (fr), functions like calc/min/max/clamp, easing curves, requestAnimationFrame timing, distances and angles for interactions, and DOM geometry via getBoundingClientRect().

Why is a nested h2 sometimes larger than the main h2 when using em?

em units compound. Example: main is 1.25em (≈20px), article is 1.75em of main (≈35px), and h2 is 2em of its parent. So main h2 ≈ 2 × 20 = 40px, while article h2 ≈ 2 × 35 = 70px. The nested heading looks larger because its parent’s font-size is larger.

What’s the difference between CSS math and JavaScript math?

CSS is declarative and scoped to styling: calc(), min(), max(), clamp(), unit math, and easing keywords. JS is procedural and dynamic: full Math APIs, conditionals/loops, real‑time input handling, physics-like effects, and DOM updates on the fly. Use CSS for layout/typography; JS for interaction and complex logic.

What’s the minimum math I need for solid front‑end work?

- Arithmetic and expressions ( + − * / ) - Algebra (variables, functions) - Ratios and proportions (aspect-ratio, fr, modular type scales) - Exponents/roots (easing, distances) - Linear equations (y = mx + b for mapping ranges) - Inequalities/comparisons (breakpoints, clamp bounds) - Geometry (points, lines, vectors, angles) - Trigonometry (sin/cos for circular/oscillating motion) - Coordinate systems (viewport axes, vh/vw)

How do ratios help with layout and typography?

They preserve proportions: use aspect-ratio for media (e.g., 16/9), Grid fr units for column ratios (e.g., 3fr 2fr 1fr), and modular type scales (e.g., base × 1.2 or × 1.618) for consistent rhythm.

How can I map scroll progress linearly to a UI element?

Use a linear equation: progress = scrollY / (scrollHeight − innerHeight). Progress runs 0→1 as the user scrolls; tie it to width, transforms, or opacity for scroll-linked effects.

How are inequalities used in CSS and JS?

- CSS: media queries (@media (min-width: …)), clamp() to enforce lower/upper bounds. - JS: explicit comparisons (>=, <=, >, <, ==) for breakpoints, validation, and state changes based on sizes or positions.

Where do geometry and trigonometry show up in UI work?

Geometry: element positioning, distances, box hit‑testing, canvas/SVG paths, rotations, clip-path polygons. Trig: circular/radial menus (sin/cos to place items), oscillations (sin waves), curved motion, and angle/direction calculations.

What coordinate system does the browser use, and how do viewport units relate?

The viewport origin is at the top‑left (0,0), x increases rightward, y increases downward. vh and vw are 1/100th of viewport height/width, enabling fluid sizing. Use getBoundingClientRect() and event.clientX/Y to read positions in this system.

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pdf, ePub, online

$47.99 $31.19

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pro $24.99 per month

access to all Manning books, MEAPs, liveVideos, liveProjects, and audiobooks!
choose one free eBook per month to keep
exclusive 50% discount on all purchases
renews monthly, pause or cancel renewal anytime

lite $19.99 per month

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