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SPI Mathematics · Mensuration 2D

Circumference and Area of a Circle

The distance round a circle is 2πr and the space inside it is πr²; with π as 22/7, radii that are multiples of 7 give whole answers.

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Why this matters

Ten questions in the bank, five EASY and five MODERATE: wheels turning over a road, a wire bent from a circle into a square, a semicircular table top and a circle inside a square. Each is one of two formulas; the marks go on radius against diameter and on kilometres against centimetres.

Concept 1 of 4: Circumference of a circle

Roll a coin once along a table and the distance it moves is its circumference: a little more than three diameters, exactly π\pi diameters.

Definition

  • Circumference =2πr=πd= 2\pi r = \pi d.
  • Take π=227\pi = \frac{22}{7} unless told otherwise; a radius of 7, 14, 21 or 35 then gives a whole answer.
  • Circumference is proportional to radius, so a circle whose circumference is the sum of two others has a radius equal to the sum of their radii.
  • 'Circumference minus diameter' is 2r(π−1)=2r×1572r(\pi - 1) = 2r \times \frac{15}{7}.

Circumference

C=2πr=πdC = 2\pi r = \pi d

Worked example

Find the circumference of a circle of diameter 35 cm.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Mensuration 2D · Circumference and Area of a Circle

If the difference between the circumference and diameter of a circle is 60 cm, then the radius of the circle will be:

The perimeter of a semicircle

A semicircle's boundary is half the circumference plus the straight diameter: πr+2r\pi r + 2r. Leaving out the diameter gives only the curved arc.

Circumferences add by radius, areas do not

If two circumferences are added, the radii simply add: 3 cm and 4 cm give 7 cm. If two areas are added, the new radius is r12+r22\sqrt{r_1^2 + r_2^2}: 3 cm and 4 cm give 5 cm, not 7 cm.
Drill 1 more on circumference of a circle

Concept 2 of 4: Wheels: distance and revolutions

Each full turn of a wheel rolls it forward by one circumference. So the distance travelled is the number of turns times the circumference.

Definition

  • Distance == revolutions ×\times circumference.
  • Revolutions == distance ÷\div circumference.
  • Put the distance and the circumference in the same unit: 1 km =1000= 1000 m =100 000= 100\,000 cm.
  • Read whether the question asks for the radius or the diameter of the wheel.

Wheel

distance=n×2πr,n=distance2πr\text{distance} = n \times 2\pi r, \qquad n = \frac{\text{distance}}{2\pi r}
  • nnumber of revolutions

Worked example

A wheel of radius 35 cm rolls 2.2 km. How many revolutions does it make?
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mensuration 2D · Circumference and Area of a Circle

A bicycle wheel makes 5000 revolutions in moving 11 km. Find the diameter of the wheel.

Kilometres against centimetres

1 km is 100 000 cm, not 1000 cm. Turn the distance into the unit of the circumference before you divide.

Radius or diameter of the wheel

Distance per turn gives πd\pi d. Dividing by π\pi gives the diameter; halve it only if the radius is asked.
Drill 2 more on wheels: distance and revolutions

Concept 3 of 4: Area of a circle and a semicircle

The area of a circle is π\pi times the square on its radius. Half a circle has half the area; a circle drawn inside a square just touches all four sides.

Definition

  • Area =πr2= \pi r^2. If the diameter is given, halve it first.
  • Semicircle: 12πr2\frac{1}{2}\pi r^2.
  • Circle inside a square (touching all four sides): its diameter equals the side of the square.
  • Circle round a square (through its corners): its diameter equals the square's diagonal, a2a\sqrt{2}.

Area of a circle

A=πr2,Asemicircle=12πr2A = \pi r^2, \qquad A_{\text{semicircle}} = \frac{1}{2}\pi r^2

Worked example

Find the area of a circle of radius 14 cm.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Mensuration 2D · Circumference and Area of a Circle

A table top is semicircular in shape with diameter 2.8 m. Area of this table top is

Squaring the diameter

A circle of diameter 14 cm has area π×72=154\pi \times 7^2 = 154, not π×142=616\pi \times 14^2 = 616. Using the diameter makes the area four times too big.

A semicircle given the full circle's area

A semicircular top is half of πr2\pi r^2. The full circle's area is usually a printed wrong option.
Drill 2 more on area of a circle and a semicircle

Concept 4 of 4: A wire bent from one shape into another

Bending a wire changes its shape but not its length. So the circumference of the old circle is the perimeter of the new square, triangle or rectangle.

Definition

  • Length is kept: old perimeter == new perimeter.
  • Circle to square: 2πr=4a2\pi r = 4a, so a=πr2a = \frac{\pi r}{2}.
  • For the same length of wire, a circle encloses more area than a square. With π=227\pi = \frac{22}{7} the areas are always in the ratio circle : square =14:11= 14 : 11.

Same wire, new shape

2πr=4a,AcircleAsquare=4π=14112\pi r = 4a, \qquad \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{4}{\pi} = \frac{14}{11}

Worked example

A wire in the shape of a square of side 11 cm is bent into a circle. Find the radius of the circle.
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The same idea in a practice question:

Practice questionEasy

Example 4 · Mensuration 2D · Circumference and Area of a Circle

A wire is in the form of a circle of radius 42 cm. It is bent into a square. The side of the square is

Keeping the area instead of the length

When a wire is bent, its length stays the same, not the area it encloses. Equate the perimeters, never the areas.

Turning the area ratio round

Circle to square is 14 : 11 because the circle encloses more. If the question asks square to circle, the answer is 11 : 14.
Drill 1 more on a wire bent from one shape into another

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

  • Circumference of a circle

    Circumference

    C=2πr=πdC = 2\pi r = \pi d
  • Wheels: distance and revolutions

    Wheel

    distance=n×2πr,n=distance2πr\text{distance} = n \times 2\pi r, \qquad n = \frac{\text{distance}}{2\pi r}
  • Area of a circle and a semicircle

    Area of a circle

    A=πr2,Asemicircle=12πr2A = \pi r^2, \qquad A_{\text{semicircle}} = \frac{1}{2}\pi r^2
  • A wire bent from one shape into another

    Same wire, new shape

    2πr=4a,AcircleAsquare=4π=14112\pi r = 4a, \qquad \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{4}{\pi} = \frac{14}{11}

Watch out for (8)

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