The Math
of Soccer
A soccer pitch is a geometry lesson with grass on it. Measure areas, use the Pythagorean theorem and coordinates, find shooting angles with trigonometry, add vectors for wind and running, time a penalty, read stats and expected goals, and explore the shapes on a ball.
- 00First Principlesp.2
- 01The pitchp.4
- 02Pythagorasp.5
- 03Coordinatesp.6
- 04Angles & trigonometryp.7
- 05Vectorsp.9
- 06Speed, stats & xGp.10
- 07Geometry of the ballp.12
- xAlgebra Level Upp.13
- ✦Self-check · Make it · 5 Questions · Side Questp.15
You'll need: a calculator, a measuring tape, a protractor, some cones or markers, and a ball.
Principles
A first principle is a basic truth you can build on. The maths of soccer rests on four: the pitch is a coordinate plane, right triangles are everywhere, motion has size and direction and numbers describe chances, not certainties.
What do I already know?
What is necessary?
Everything in this module is built from four bricks. Rate yourself on each one, then check the example to see if your rating holds up.
What is the objective?
By the end, you’ll be able to find a pitch’s area and diagonal, use the Pythagorean theorem and coordinates for passes, find shooting angles with trigonometry, add vectors, calculate the time for a penalty, read stats and xG, and describe the geometry of a ball.
- 01Find area and perimeter.
- 02Use a² + b² = c².
- 03Find distance on a grid.
- 04Use tan to find an angle.
- 05Add two vectors.
- 06Calculate time from speed.
The list on the left is ours. Now make it yours. What do you want out of this?
How much do you know about the math of soccer right now? Shade the boxes in pencil. You'll come back at the end (p.9) and shade it again in pen.
Why start this way? When you know what you already have (1), what you need (2) and where you're going (3), you learn faster and you notice the gaps. Every module on the Soccer Path starts with these same three questions.
The pitch
Every pitch is a rectangle with measured markings.
The Laws of the Game
For adult international matches, a pitch must be 100–110 m long and 64–75 m wide. FIFA recommends 105 m × 68 m for big tournaments.
| Mark | Size |
|---|---|
| Goal | 7.32 m × 2.44 m |
| Penalty mark | 11 m from goal |
| Centre circle | 9.15 m radius |
Area and perimeter
Smaller players, smaller pitches
Youth games use smaller pitches and fewer players, so everyone touches the ball more. Many young players start with small-sided games such as 4 v 4 or 7 v 7.
Area is in square metres (m²); perimeter is in metres (m). Always write the unit.
Biggest and smallest
Find the area of the smallest (100 × 64) and largest (110 × 75) international pitches. How much bigger is the largest?How many 105 × 68 pitches would fit in 1 km² (1,000,000 m²)?
Pythagoras
The Pythagorean theorem finds distances across the pitch.
Right triangles
In a right triangle, the longest side (the hypotenuse, c) is opposite the right angle: a² + b² = c².
A pitch’s corners are right angles, so its diagonal is a hypotenuse.
Corner to corner
Forward and across
A player passes 40 m up the pitch and 30 m across. The ball travels √(30² + 40²) = √2,500 = 50 m. The numbers 3, 4, 5 (and 6, 8, 10 or 30, 40, 50) are a famous right-triangle family.
c is always the longest side, opposite the right angle. To find a shorter side, subtract: a² = c² − b².
Your pass
A pass goes 16 m forward and 12 m across. How far does the ball travel? Check with a tape measure and cones in a park.A ball travels 25 m on a pass that goes 7 m across. How far forward did it go?
Coordinates
Coordinates turn a pitch into a map you can calculate on.
Coordinates
Put one corner at (0, 0), with x along the touchline and y across. Every player has coordinates, like (30, 25).
The distance between two points is a Pythagorean problem: d = √((x₂ − x₁)² + (y₂ − y₁)²).
Pass between players
Corner flag to near post
The goal sits in the middle of a 68 m goal line. Each post is 3.66 m from the centre, so the corner flag is (68 − 7.32) ÷ 2 = 30.34 m from the near post.
Subtract coordinates in the same order (x₂ − x₁ and y₂ − y₁). Squaring makes any negative positive.
Map a play
Draw a pitch on grid paper. Place 3 players and find the distance of each pass between them.Distance from (0, 0) to (12, 16)?
Angles
The shooting angle decides how much goal is open.
How much goal can you see?
From the penalty mark, the goal posts and the shooter make a triangle. The shooting angle is the angle at the shooter between the lines to each post.
Closer to the goal and nearer the middle means a wider angle.
Angles in a triangle
Why wingers cut inside
A shot from near the touchline sees almost no goal. Players “cut inside” toward the middle to widen the angle before they shoot.
Measure from the shooter’s position. The angle shrinks as you move wider or farther away.
Measure it
Draw a goal to scale (7.32 m → 7.3 cm) and mark shooting spots. Measure each shooting angle with a protractor. Where is it widest?Two angles in a triangle are 90° and 18°. What is the third?
Trigonometry
Trigonometry turns distances into angles.
SOH CAH TOA
In a right triangle with angle θ: sin θ = opp ÷ hyp, cos θ = adj ÷ hyp, tan θ = opp ÷ adj. To find θ, use the inverse key: θ = tan⁻¹(opp ÷ adj).
Check your calculator is in degrees.
Penalty angle
A free kick
The wall stands 9.15 m from the ball and is about 1.85 m tall (made-up). The lowest straight line over it rises at tan⁻¹(1.85 ÷ 9.15) ≈ 11.4°. Real kicks curve and dip, so players aim higher and use spin (Lens 2).
Label opp, adj and hyp from the angle you want. The opposite side is across from it.
Edge of the box
Find the whole shooting angle from a spot 16.5 m straight out from the centre of the goal.A chip leaves the ground at 20°. How high is it after 9.15 m of a straight line?
Vectors
Vectors add motion in two directions at once.
Size and direction
A vector has a size (magnitude) and a direction, like “20 m/s toward goal.” Add vectors tip to tail; the arrow from start to finish is the resultant.
Speed is a size; velocity is a vector.
Kick in a crosswind
Lead the runner
A good through ball goes where the runner will be, not where they are. Coaches call it “passing into space”: adding the runner’s velocity to the ball’s path.
Only vectors at right angles can be added with plain Pythagoras. Others need a scale drawing or more trig.
Draw it
On grid paper, draw a run of 8 squares east, then 6 squares north. Draw the resultant and measure it. Does Pythagoras agree?Resultant of 8 m east and 6 m north?
Speed and time
Speed, distance and time explain the penalty duel.
Speed = distance ÷ time
A hard penalty can travel about 100 km/h. To use metres and seconds, divide by 3.6: 100 ÷ 3.6 ≈ 27.8 m/s.
Then time = distance ÷ speed.
Can the keeper react?
Decide early
With so little time to react and move, keepers often choose a side before or as the ball is kicked. They study shooters’ habits beforehand.
km/h to m/s: divide by 3.6. m/s to km/h: multiply by 3.6.
Sprint test
With a friend and a stopwatch, time a 30 m sprint. Find your speed in m/s and km/h.A 120 km/h penalty: how long to travel 11 m?
Stats and xG
Statistics describe chances, not certainties.
Rates and averages
- Pass completion: 380 of 420 passes = about 90%.
- Goals per game: Canada’s men scored 9 goals in 5 games at the 2026 World Cup, 1.8 per game.
- Christine Sinclair scored 190 international goals in 331 games, about 0.57 per game.
A chance’s probability
xG gives each shot a probability of scoring, based on thousands of similar shots: a penalty might be about 0.75, a long shot 0.03.
In the 2026 World Cup final, Spain’s 20 shots added up to about 1.94 xG; Argentina’s 2 shots, 0.2. Spain won 1–0.
Probability
If a player scores about 3 penalties in 4 (made-up rate 0.75), the chance of scoring 5 in a row is 0.755 ≈ 0.24: about 1 in 4. Shootouts are tense for a reason.
xG values are estimates from models, and different companies give different numbers for the same shot.
Add up xG
A team has shots worth 0.3, 0.1, 0.05 and 0.6 xG. What is their total? How many goals would you “expect”?Why can a team win with lower xG?
Geometry of the ball
A soccer ball is a lesson in 3-D geometry.
Pentagons and hexagons
The 1970 World Cup ball, the Telstar, had 12 black pentagons and 20 white hexagons: 32 panels. The shape is a truncated icosahedron.
Newer balls use fewer panels: the 2026 ball, Trionda, has just 4.
Euler’s formula
The ball’s size
A size-5 ball has a circumference of 68–70 cm. Since C = πd, a 70 cm ball has a diameter of 70 ÷ π ≈ 22.3 cm.
Euler’s V − E + F = 2 works for shapes without holes, like a ball, a cube or a tetrahedron.
Measure your ball
Wrap a string around your ball and measure it. Find its diameter. Is it a size 3, 4 or 5?Why divide edges by 2?
Solve for the missing wins
Turn a league table into an equation, then solve it.
Win 3, draw 1, lose 0
Most leagues give 3 points for a win and 1 for a draw. With W wins and D draws, a team’s points are P = 3W + D.
| Wins | Draws | Points |
|---|---|---|
| 10 | 4 | 34 |
| W | 5 | 50 |
How many wins?
Points as the season goes
A team that wins every game gains 3 points per game: after g more games it has P = 30 + 3g (made-up). The 30 is the starting value; the 3 is the slope, the steepness of the line.
Do the same thing to both sides until the unknown is alone. Then check: does 3 × 15 + 5 really equal 50?
Catching the leader
Team A has 30 points and wins every game: P = 30 + 3g. Team B has 42 points and draws every game: P = 42 + g. After how many games are they level?How many wins give 70 points with 10 draws?
Flight paths and pitches
When the unknown is squared, there can be two answers.
A quadratic path
A ball kicked straight up at 15 m/s has a height of about h = 15t − 5t2 metres after t seconds (gravity rounded to 10 m/s², no air drag).
| t (s) | 0 | 1 | 1.5 | 2 | 3 |
|---|---|---|---|---|---|
| h (m) | 0 | 10 | 11.25 | 10 | 0 |
When does it land?
Factor to solve
A pitch is 37 m longer than it is wide, with an area of 7,140 m²: w(w + 37) = 7,140, so w² + 37w − 7,140 = 0. Factor: (w + 105)(w − 68) = 0, so w = 68 m (not −105). The pitch is 68 m by 105 m, the size FIFA recommends.
A quadratic often has two solutions. Keep only the ones that make sense: a width or a time can’t be negative, but a ball can pass the same height twice.
Head height
Using h = 15t − 5t², when is the ball 10 m up? Write the equation and solve it.A small-sided field is twice as long as it is wide, with an area of 800 m². How wide is it?
Quick self-check
Six questions, covering every stop. Show your thinking. Answers are upside down at the bottom; no peeking till you're done.
1 · Find the area of a 100 m × 64 m pitch.
2 · A pass goes 12 m across and 16 m forward. How long is it?
3 · Find the distance from (0, 0) to (6, 8).
4 · Two angles in a triangle are 90° and 40°. Find the third.
5 · Convert 90 km/h to m/s.
6 · What does an xG of 0.1 mean?
Re-shade your knowledge meter
How much do you know about the math of soccer now? Flip back to p.3 and shade the meter again in pen (or shade this one). Did any of the four bricks on p.2 move from "Not yet" to "Got it"?
Measure a mini pitch
Use Pythagoras on real grass.
1 · Plan
With a friend and an adult, mark a mini pitch in a park with cones: 40 m by 30 m (or smaller). Predict its diagonal, then measure it with a tape.
- Safe, open ground
- Right-angle corners (use 3-4-5)
- Prediction written first
- Shooting spots marked
2 · Results
| Measure | Value |
|---|---|
| Length | |
| Width | |
| Area | |
| Diagonal (predicted) | |
| Diagonal (measured) |
3 · Must-haves
- Units on every number
- Error explained
- Two shooting angles
- Sketch to scale
4 · What I found
How close was your prediction? Why might it be off?
Choose a park or field away from roads, with an adult nearby. Watch for holes, ice and other people. Check the weather first.
5 Questions
I Still Have
This module didn't answer everything. Good. Write 5 new questions it didn't answer, one of each type below. There are no wrong questions, only ones nobody's asked yet.
★ Star the one question you'd most like answered. You'll need it on the next page.
1What I'll find out
Break your big question into 2 or 3 smaller ones you could actually answer.
2Where I'll look
3How I'll share it
Who I'll share it with:
Done by: