Learning objectives
- Recognise, explain, and apply “Ähnlichkeit”.
- Recognise, explain, and apply “Satz des Pythagoras”.
- Recognise, explain, and apply “Satz des Thales”.
- Recognise, explain, and apply “Sinus”.
- Recognise, explain, and apply “Kosinus”.
German Gymnasium Klasse 9 / Mathematics / Curriculum
Geometrie und Trigonometrie
Geometrie und Trigonometrie: structured theory, worked examples, answered practice, and a mastery checklist for German Gymnasium Klasse 9.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
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completed
Estimated active study time
209 minutes
1. Understand
Objectives, key ideas, and structured theory.
Open →2. Follow the method
Worked examples that explain every step.
Open →3. Practise
Graded tasks with hints and answer guidance.
Open →4. Consolidate
Assignments from core work to challenge.
Open →5. Check
A timed, marked test with solution guidance.
Open →
Progress is stored only in this browser and does not require AI credits.
Key ideas and checks
- I select a theorem from the diagram's givens.
- I justify every step of the solution.
- Every solution or explanation for “Geometrie und Trigonometrie” should include a method, justification, and final check.
Unit
Core theory
The essential chapter ideas in a clear sequence before practice.
Diagram, givens, target
Mark every given on the diagram and state the target clearly. Start from a relationship that connects exactly those quantities.
Property with justification
Every geometric step needs a property, theorem, or definition. A result without justification is not a complete proof.
Units and precision
Length, area, volume, and angle use different units. Round only at the end and keep sufficient intermediate precision.
Ähnlichkeit
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Satz des Pythagoras
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Satz des Thales
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Sinus
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Kosinus
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics
Worked examples
Follow the method step by step and check why every step is valid.
Worked geometry example
A right triangle has perpendicular sides 12 cm and 16 cm. Find its hypotenuse and area.
- 1Pythagoras: c² = 12² + 16².
- 2c = √(400) = 20 cm.
- 3Area = 1/2 · 12 · 16.
Hypotenuse 20 cm, area 96 cm².
Reasoning example
Before calculating, explain the key idea from “Ähnlichkeit” and which conditions must be checked.
- 1Define the idea in one clear sentence.
- 2Connect it to a representation, law, or formula.
- 3State a restriction, unit, or final check that makes the solution valid.
The answer should show not only which rule is used for “Ähnlichkeit”, but also why it is valid here.
Geometrie und Trigonometrie
Practice with answers
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
Core fluency4 minutes
A 7 cm by 6 cm rectangle: find perimeter and area.
Hint
P = 2(l+w), A = lw.
Answer guide
P = 26 cm, A = 42 cm².
Core fluency5 minutes
A triangle has base 9 cm and height 3 cm. Find its area.
Hint
A = 1/2 bh.
Answer guide
A = 1/2 · 9 · 3 = 13.5 cm².
Application6 minutes
Two angles in a triangle are 60° and 50°. Find the third.
Hint
Triangle angles sum to 180°.
Answer guide
70°.
Application7 minutes
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
Hint
Use a²+b²=c².
Answer guide
10 cm.
Reasoning8 minutes
A circle has radius 4 cm. Find circumference and area in terms of π.
Hint
C = 2πr, A = πr².
Answer guide
C = 8π cm, A = 16π cm².
Reasoning9 minutes
Scale 1:800. What real length in metres does 6 cm represent?
Hint
Multiply in cm, then convert to metres.
Answer guide
6 · 800 = 4800 cm = 48 m.
Exam style10 minutes
A cuboid is 4 cm × 3 cm × 7 cm. Find its volume.
Hint
V = lwh.
Answer guide
V = 84 cm³.
Exam style12 minutes
Points A(0,0) and B(7,5). Find AB.
Hint
AB = √(Δx²+Δy²).
Answer guide
AB = √74.
Common mistakes
- Using a formula without checking its conditions.
- Confusing linear, square, and cubic units.
- Treating a not-to-scale diagram as a measurement.
Mastery check
- I select a theorem from the diagram's givens.
- I justify every step of the solution.
- I use correct units and a final check.
Geometrie und Trigonometrie
Chapter homework
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Core8 minutes
Homework 1: A 3 cm by 2 cm rectangle: find perimeter and area.
Hint
P = 2(l+w), A = lw.
Answer guide
P = 10 cm, A = 6 cm².
Core10 minutes
Homework 2: A triangle has base 24 cm and height 4 cm. Find its area.
Hint
A = 1/2 bh.
Answer guide
A = 1/2 · 24 · 4 = 48 cm².
Stretch12 minutes
Homework 3: Two angles in a triangle are 90° and 60°. Find the third.
Hint
Triangle angles sum to 180°.
Answer guide
30°.
Stretch15 minutes
Homework 4: A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
Hint
Use a²+b²=c².
Answer guide
15 cm.
Challenge18 minutes
Homework 5: A circle has radius 5 cm. Find circumference and area in terms of π.
Hint
C = 2πr, A = πr².
Answer guide
C = 10π cm, A = 25π cm².
Challenge20 minutes
Homework 6: Scale 1:400. What real length in metres does 2 cm represent?
Hint
Multiply in cm, then convert to metres.
Answer guide
2 · 400 = 800 cm = 8 m.
50 minutes / 40 marks
Full chapter test
A timed, full-mark self-assessment with model-answer guidance.
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. A 8 cm by 2 cm rectangle: find perimeter and area.
2 marksAnswer guide
P = 20 cm, A = 16 cm².
2. A triangle has base 16 cm and height 4 cm. Find its area.
3 marksAnswer guide
A = 1/2 · 16 · 4 = 32 cm².
3. Two angles in a triangle are 70° and 60°. Find the third.
3 marksAnswer guide
50°.
4. A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
4 marksAnswer guide
15 cm.
5. A circle has radius 5 cm. Find circumference and area in terms of π.
4 marksAnswer guide
C = 10π cm, A = 25π cm².
6. Scale 1:900. What real length in metres does 2 cm represent?
4 marksAnswer guide
2 · 900 = 1800 cm = 18 m.
7. A cuboid is 5 cm × 4 cm × 9 cm. Find its volume.
5 marksAnswer guide
V = 180 cm³.
8. Points A(0,0) and B(8,6). Find AB.
5 marksAnswer guide
AB = √100.
9. Similar shapes have length scale factor 1:3. If the small area is 4 cm², find the large area.
5 marksAnswer guide
4 · 3² = 36 cm².
10. Explain which givens and theorem are needed for a complete argument about “Satz des Pythagoras”.
5 marksAnswer guide
Full response: a labelled diagram, givens, an applicable theorem with its conditions, and a logically derived conclusion.
Unit
Official sources and verification
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
Keep each unit in GR/DE: concept, German term, example, exercise, recap.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
Ähnlichkeit
Satz des Pythagoras
Satz des Thales
Sinus
Kosinus
Tangens
Rechtwinklige Dreiecke
Kreisberechnungen
Where to focus
The areas that usually create mistakes or need extra revision.
I select a theorem from the diagram's givens.
I justify every step of the solution.
I use correct units and a final check.
Sources, daily material, and resources
Where to start: textbook, daily material, PDFs, videos, and worked examples.
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
Practice by subtopic
Targeted practice before full tests so coverage is clear.
A 7 cm by 6 cm rectangle: find perimeter and area.
A triangle has base 9 cm and height 3 cm. Find its area.
Two angles in a triangle are 60° and 50°. Find the third.
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
A circle has radius 4 cm. Find circumference and area in terms of π.
Scale 1:800. What real length in metres does 6 cm represent?
A cuboid is 4 cm × 3 cm × 7 cm. Find its volume.
Points A(0,0) and B(7,5). Find AB.
Mocks and progress checks
How to measure progress in this chapter and when it enters a cumulative mock.
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
Next step
What to do after finishing the chapter and how it connects to the next unit.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.