Use circle vocabulary, tangent radii and semicircle angles
Distinguish radius, diameter, chord, arc, sector and segment · Use perpendicularity at tangent contact · Use a right angle standing on a diameter
Section C4.6 · Core · Circle vocabulary and relationships
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Circle definitions
- Triangle and straight-line angles
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Distinguish radius, diameter, chord, arc, sector and segment
- Use perpendicularity at tangent contact
- Use a right angle standing on a diameter
Name the right part
A radius joins the center to a circle point; a diameter is a chord through the center, of length 2r. A chord joins two circle points. An arc is part of the boundary; a sector is the region between two radii and an arc; a segment is the region between a chord and an arc.
A tangent line meets a circle at one contact point; a secant meets it at two distinct points. At tangent contact T, the radius OT is perpendicular to the tangent. Use the stated or constructed contact point, not a line that merely looks close in a sketch.
Angle standing on a diameter
If AB is a diameter and C is a different point on the circle, angle ACB is 90°. This is the angle in a semicircle; it is at the circumference, not the central straight angle AOB.
The result follows by joining C to the center: equal radii make two isosceles triangles whose angles, with the 180° central angle, force the angle at C to total 90°. Apply a triangle sum to calculate the other angles. No formal proof format is required, but name the diameter reason.
Sources for this lesson
Pause and explain
Does every angle at a circle's circumference equal 90°?
Worked example
AB is a diameter, C is on the circle and angle CAB is 38°. Find ACB and ABC.
Show the worked solution
- ACB = 90° because it stands on diameter AB.
- ABC = 180−90−38 = 52°.
Answer ACB = 90°; ABC = 52°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A circle has diameter 14 cm. Find the radius; state the angle between a tangent and the contact radius.
Give me a hint
Use half the diameter and the tangent theorem.
Compare my reasoning
- Radius = 14/2 = 7 cm.
- The contact angle is 90°.
Radius 7 cm; tangent/contact-radius angle 90°.
Look for these in your work
- Identify the quantities
- Name the tangent-radius reason
AB is a diameter and C is a circle point other than A/B. If CAB = 27°, find ACB and ABC.
Give me a hint
Use the diameter before the triangle sum.
Compare my reasoning
- ACB = 90°.
- ABC = 180−90−27 = 63°.
ACB = 90°; ABC = 63°.
Look for these in your work
- Use the stated diameter
- Check the triangle total
A line through a circle point T makes an exact 86° angle with OT. Is it tangent there? What if a sketch only appears perpendicular?
Give me a hint
The contact condition is exact, not visual.
Compare my reasoning
- An exact tangent contact needs 90°, so an exact 86° excludes tangency.
- Appearance alone is not proof; use given conditions or a valid perpendicular construction.
No at an exact 86°. A rough sketch does not establish perpendicularity.
Look for these in your work
- Use the exact contact theorem
- Distinguish evidence from appearance
Common mistakes
- An angle standing on an arbitrary chord need not be right.
- A tangent is perpendicular at its actual contact point.
What can you explain now?
Does every angle at a circle's circumference equal 90°?
Compare with the explanation
No; the right-angle result requires a diameter as the subtended chord.
Different chords can subtend different circumference angles.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does every angle at a circle's circumference equal 90°?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.