Practice Maths

Parts of a Circle and Circumference

Key Ideas

Key Terms

centre
the middle point of a circle; every point on the circle is the same distance from the centre.
radius (r)
the distance from the centre to any point on the circle's edge; plural: radii. d = 2r.
diameter (d)
the distance across the circle through the centre; always twice the radius: d = 2r.
circumference (C)
the distance around a circle (its perimeter); C = πd = 2πr.
π (pi)
the ratio of circumference to diameter for any circle; π ≈ 3.14159… — an irrational constant.
chord
a straight line connecting any two points on the circle's edge; the diameter is the longest chord.
arc
a curved part of the circle's edge between two points; a minor arc is the shorter piece, a major arc is the longer.
sector
a "pie slice" region bounded by two radii and the arc between them.
tangent
a straight line that touches the circle at exactly one point; always perpendicular to the radius at that point.
Hot Tip Always check whether you are given the radius or the diameter! The diameter is twice the radius. A very common mistake is using r when you have been given d, or vice versa.

Worked Example

Question A: A circle has radius 5 cm. Find the circumference. Round to 2 decimal places.

Step 1 — Write the formula: C = 2πr

Step 2 — Substitute r = 5: C = 2 × π × 5 = 10π

Step 3 — Evaluate: C = 10 × 3.14159… ≈ 31.42 cm

Question B: A circle has circumference 50 cm. Find the diameter.

Step 1 — C = πd, so d = C ÷ π

Step 2 — d = 50 ÷ π ≈ 50 ÷ 3.14159 ≈ 15.92 cm

The Parts of a Circle

A circle is a perfectly round shape where every point on the edge is exactly the same distance from the centre. Learning the correct names for each part is essential — exams will use these terms, so you need to recognise them instantly.

  • Centre: The middle point of the circle.
  • Radius (r): The distance from the centre to any point on the circle's edge. The plural is radii.
  • Diameter (d): A straight line passing through the centre from one side of the circle to the other. The diameter is always exactly twice the radius: d = 2r.
  • Chord: A straight line connecting any two points on the circle's edge, but not necessarily through the centre. The diameter is the longest possible chord.
  • Arc: A curved piece of the circle's edge — like a slice of the perimeter. A minor arc is the shorter piece; a major arc is the longer piece.
  • Sector: A "pie slice" shape bounded by two radii and an arc — like a slice of pizza. A minor sector is the smaller slice; a major sector is the larger one.
  • Tangent: A straight line that just touches the circle at exactly one point and does not cross inside it. A tangent is always perpendicular (at 90°) to the radius at that point.

What is π and Why Does It Matter?

The number π (pronounced "pi") is one of the most famous numbers in all of mathematics. Here is where it comes from: take any circle — big or small — and measure its circumference (the distance around the edge). Then divide that circumference by the diameter. You will always get the same answer: approximately 3.14159...

This ratio never changes, no matter what size circle you use. We call this special constant π. So by definition: π = circumference ÷ diameter, which rearranges to give us the circumference formula.

π is an irrational number — its decimal digits go on forever without repeating. For most calculations, using your calculator's π button gives the best accuracy. When rounding is needed, 3.14 or 3.14159 are common approximations.

The Circumference Formula

The circumference is the total distance around a circle — its perimeter. There are two versions of the formula, and both are equally correct:

  • C = πd — use this when you are given the diameter.
  • C = 2πr — use this when you are given the radius. (Since d = 2r, these are the same formula.)

Example 1: A circle has a radius of 5 cm. Find its circumference.
C = 2πr = 2 × π × 5 = 10π ≈ 31.42 cm

Example 2: A circle has a diameter of 12 m. Find its circumference.
C = πd = π × 12 = 12π ≈ 37.70 m

Working Backwards — Finding Radius or Diameter from Circumference

Sometimes a question gives you the circumference and asks you to find the radius or diameter. Just rearrange the formula:

  • If C = πd, then d = C ÷ π
  • If C = 2πr, then r = C ÷ (2π)

Example: A circle has a circumference of 50 cm. Find its radius.
r = 50 ÷ (2π) = 25 ÷ π ≈ 7.96 cm

Key tip: Always check whether you have been given the radius or the diameter before substituting into the formula — this is the most common mistake. If you see "diameter = 10", the radius is 5, not 10. Write down r = ... or d = ... clearly at the start of every working-out.

Real-World Applications

Circumference appears everywhere in everyday life. The distance a bicycle wheel travels in one full rotation equals the circumference of the wheel. If a wheel has a diameter of 70 cm, one rotation covers π × 70 ≈ 220 cm = 2.2 m. A running track's curved ends are semicircles, so the total curved distance is π × diameter of the track. Architects use circumference when designing curved walls, arches, and fountains.

Mastery Practice

  1. Convert between radius and diameter. Fluency

    1. Radius = 7 cm. Find the diameter.
    2. Diameter = 18 m. Find the radius.
    3. Radius = 3.5 mm. Find the diameter.
    4. Diameter = 9 km. Find the radius.
    5. Radius = 12 cm. Find the diameter.
    6. Diameter = 4.8 m. Find the radius.
    7. Radius = 25 mm. Find the diameter.
    8. Diameter = 100 m. Find the radius.

  2. Find the circumference of each circle. Give answers to 2 decimal places. Fluency

    1. r = 4 cm
    2. r = 10 m
    3. r = 3.5 cm
    4. d = 14 cm
    5. d = 8 m
    6. r = 6.2 mm
    7. d = 25 cm
    8. r = 1.5 m

  3. Find the radius or diameter. Round to 2 decimal places where necessary. Fluency

    1. C = 31.42 cm. Find r.
    2. C = 62.83 m. Find d.
    3. C = 18 cm. Find r.
    4. C = 44 m. Find d.
    5. C = 100 mm. Find r.
    6. C = 25 cm. Find d.
    7. C = π cm. Find r.
    8. C = 10π m. Find d.

  4. Find the circumference. Leave your answer in terms of π. Understanding

    1. r = 6 cm
    2. d = 9 m
    3. r = 15 cm
    4. d = 7 mm
    5. r = ½ m
    6. d = 20 km

  5. Solve each problem, converting units where required. Understanding

    1. A circle has diameter 3 m. Find the circumference in cm.
    2. A circle has circumference 1 km. Find the radius in metres (to 2 dp).
    3. A wheel has radius 35 cm. How far does it travel in one full revolution? Give your answer in cm and in metres.
    4. A semicircle has diameter 10 cm. Find its perimeter (the straight edge plus the curved arc).

  6. Circumference in real-world situations. Problem Solving

    1. A circular garden has a diameter of 8 m. How much edging (in metres) is needed to go around the outside? Round to 2 decimal places.
    2. A bicycle wheel has a diameter of 66 cm. How many full revolutions does the wheel make when the bicycle travels 1 km? Round to the nearest whole number.
    3. Two circles have diameters in the ratio 1:3. If the smaller circle has circumference 12π cm, find the circumference of the larger circle in terms of π and as a decimal to 2 dp.
    4. A running track has two straight sections of length 100 m and two semicircular ends. Each semicircle has a diameter of 60 m. Find the total length of one lap of the track to 2 decimal places.

  7. Find the circumference of each circle shown. Give answers to 2 decimal places. Understanding

    1. r = 9 cm
    2. d = 14 m
    3. r = 6.5 mm

  8. Use the diagram below to answer the questions. Understanding

    O radius diameter chord arc sector
    1. The radius of this circle is 9 cm. What is the diameter?
    2. Name the part of the circle labelled “chord”. How is a chord different from a diameter?
    3. Name the curved part of the boundary between two radii.
    4. What is the name of the “pie slice” shaped region?
    5. Calculate the circumference of this circle to 2 decimal places.

  9. Complete the table. Round decimal answers to 2 d.p. Understanding

    Radius (r)Diameter (d)Circumference (C)C in terms of π
    5 cm______10π cm
    ___12 m______
    ______62.83 mm___
    7.5 cm_________
    ___3 km______

  10. Spot the error and extended response. Problem Solving

    1. A student found the circumference of a circle with diameter 8 cm as: C = 2πr = 2 × π × 8 = 50.27 cm. What mistake did they make? What is the correct answer?
    2. Complete the steps: A circle has radius 7 cm.
      Circumference = π × d = π × ___ = ___ cm (2 d.p.)
    3. A circular fountain has circumference 18.85 m.
      1. Find the diameter of the fountain (to 2 dp).
      2. Find the radius.
      3. A path of uniform width 0.8 m is built around the outside of the fountain. Find the circumference of the outer edge of the path (to 2 dp).
    4. A circular athletics track has an inner radius of 36.8 m and an outer radius of 40 m.
      1. Find the circumference of the inner edge.
      2. Find the circumference of the outer edge.
      3. How much further does an athlete running on the outer edge travel compared to the inner edge in one lap?
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