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Practice sheet · Soccer Path · Lens 1: Math

Pythagoras & trig

Practice for The Math of Soccer: the pitch, Pythagoras, coordinates, trigonometry, vectors and speed (pp.8–14), plus the Algebra Level Up (pp.17–18). Teams are made-up examples.
Warm-upCoreLevel Up
Name Date Time I spent
Remember

a² + b² = c² for a right triangle; c is the longest side, opposite the right angle. For a shorter side, subtract: a² = c² − b² (p.9).

SOH CAH TOA: tan θ = opposite ÷ adjacent, so θ = tan⁻¹(opp ÷ adj). Angles in a triangle add to 180°. km/h ÷ 3.6 = m/s. Set your calculator to degrees.

Warm-up

One step

Pythagoras, area, angles and speed.
1

A short pass

A pass goes 12 m forward and 9 m across. How far does the ball travel?

2

Youth pitch

A small-sided pitch is 60 m by 40 m. Find its area and its perimeter, with units.

3

Third angle

A pass, a run and a shot make a triangle. One angle is 90° and another is 27°. What is the third?

4

Fast shot

A shot travels at 90 km/h. Convert it to m/s. How long does it take to cover the 11 m from the penalty mark?

Core

Two or three steps

Sketch the triangle and label it first.
5

Corner to corner

The smallest pitch allowed for adult international matches is 100 m by 64 m (p.8). How long is its diagonal, to one decimal place?

6

Pass on the grid

Use the pitch grid from p.10. A passer is at (20, 15) and a teammate is at (44, 25). How long is the pass?

Geometry Practice · Soccer Path · Unizon1/2
Soccer Path · Unizon · PracticePythagoras & trigonometry
7

How far forward?

A ball travels 26 m on a pass that goes 10 m across. How far forward did it go?

8

Twenty metres out

A shooter stands 20 m straight out from the centre of the goal. Each post is 3.66 m to the side (p.12). Find the angle to one post, then the whole shooting angle.

9

Crosswind kick

A ball is kicked at 18 m/s straight down the pitch. A gust pushes it 4 m/s sideways (simplified). Find its resultant speed and how many degrees off line it goes.

10

Over the wall

A free kick rises in a straight line at 25° (simplified). How high is it after the 9.15 m to the wall? Would it clear a 1.85 m wall?

Level Up

Equations and quadratics

From A1–A2 (pp.17–18).
11

Missing wins

Points = 3W + D (p.17). A team has 61 points and 7 draws. How many wins?

12

Catching up

Team A has 20 points and wins every game: P = 20 + 3g. Team B has 34 points and draws every game: P = 34 + g. After how many games are they level, and on how many points?

13

Up and down

A ball kicked straight up at 20 m/s has height h = 20t − 5t² (p.18 model). When does it land? When is it 15 m up?

14

Pitch puzzle

A pitch is 30 m longer than it is wide and has an area of 4,000 m². Solve w(w + 30) = 4,000 by factoring. How long is it?

Make your own problem

Your turn to be the match analyst

  1. Pick a moment from a game: a pass, a shot, a corner, a run.
  2. Sketch it on a grid and write a distance or angle question with one value missing.
  3. Solve it, then measure it with cones and a tape in a park to check (with a friend).

My problem, and my worked answer:

Geometry Practice · Soccer Path · Unizon2/2
Soccer Path · Unizon · PracticePythagoras & trigonometry

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Answers to one decimal place; small rounding differences are fine.

Warm-up
1
15 m √(12² + 9²) = √(144 + 81) = √225 = 15. (A 3-4-5 triangle, times 3.)
2
2,400 m² · 200 m Area 60 × 40 = 2,400 m². Perimeter 2 × (60 + 40) = 200 m.
3
63° 180 − 90 − 27 = 63.
4
25 m/s · 0.44 s 90 ÷ 3.6 = 25 m/s. 11 ÷ 25 = 0.44 s: about twice a typical 0.2 s reaction time.
Core
5
≈ 118.7 m 100² + 64² = 10,000 + 4,096 = 14,096. √14,096 ≈ 118.7 m.
6
26 m Across 44 − 20 = 24; up 25 − 15 = 10. √(24² + 10²) = √676 = 26.
7
24 m √(26² − 10²) = √(676 − 100) = √576 = 24.
8
≈ 10.4° · ≈ 20.7° tan θ = 3.66 ÷ 20 = 0.183 → θ ≈ 10.4°. Both sides: ≈ 20.7°, much narrower than the 36.8° from the penalty mark (p.12).
9
≈ 18.4 m/s · ≈ 12.5° √(18² + 4²) = √340 ≈ 18.4. tan⁻¹(4 ÷ 18) ≈ 12.5° off line.
10
≈ 4.3 m · yes 9.15 × tan 25° ≈ 9.15 × 0.466 ≈ 4.3 m, well over 1.85 m (real kicks curve and dip).
Level Up
11
W = 18 3W + 7 = 61 → 3W = 54 → W = 18. Check: 54 + 7 = 61.
12
g = 7 · 41 points 20 + 3g = 34 + g → 2g = 14 → g = 7. Both have 41.
13
lands at 4 s · 15 m at 1 s and 3 s 20t − 5t² = 0 → 5t(4 − t) = 0 → t = 4. 20t − 5t² = 15 → t² − 4t + 3 = 0 → (t − 1)(t − 3) = 0.
14
w = 50 m · 80 m long w² + 30w − 4,000 = 0 → (w + 80)(w − 50) = 0. A width can’t be −80, so w = 50 and the length is 80 m.
Make your own: draw the right angle first, label opposite, adjacent and hypotenuse from the angle you want, and check the hypotenuse is the longest side.
Geometry Practice · Soccer Path · UnizonAnswer key