What Is a Circle?

A circle is a two-dimensional shape consisting of all points that are the same distance from a fixed point called the center.

Unlike polygons, circles:

• Have no straight sides.
• Have no vertices.
• Have no corners.
• Have a continuously curved boundary.

Circles appear throughout mathematics and in everyday objects such as wheels, clocks, coins, plates, gears, pipes, and circular tracks.

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The Center

The center is the point exactly in the middle of a circle.

Every point on the circumference is the same distance from the center.

The center is often labelled with a capital letter such as:

O

The center is important because the radius, diameter, and many other measurements of a circle are defined using it.

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The Radius

The radius is the distance from the center of a circle to any point on its circumference.

The symbol commonly used for radius is:

r

Because every point on a circle is the same distance from its center, all radii of the same circle have equal lengths.

For example, if a circle has:

r = 5 cm

then every radius drawn from the center to the circumference measures:

5 cm

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The Diameter

The diameter is a straight line segment that:

• Begins on the circumference.
• Passes through the center.
• Ends on the opposite side of the circumference.

The symbol commonly used for diameter is:

d

The diameter is always twice as long as the radius.

Therefore:

d = 2r

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Radius and Diameter

The relationship between radius and diameter is one of the most important circle relationships.

Diameter = 2 × radius

or:

d = 2r

To find the radius from the diameter:

r = d ÷ 2

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Example: Finding the Diameter

A circle has a radius of:

7 cm

Use:

d = 2r

Substitute:

d = 2 × 7

Therefore:

d = 14 cm

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Example: Finding the Radius

A circle has a diameter of:

18 cm

Use:

r = d ÷ 2

Therefore:

r = 18 ÷ 2

r = 9 cm

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The Circumference

The circumference is the distance around the outside of a circle.

For polygons, we usually call the distance around the shape the:

perimeter

For a circle, we use the special term:

circumference

Circumference is measured using units of length such as:

• Millimetres (mm).
• Centimetres (cm).
• Metres (m).
• Kilometres (km).

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What Is π?

The symbol:

π

is the Greek letter pi.

Pi describes the relationship between the circumference of any circle and its diameter.

For every circle:

circumference ÷ diameter = π

Pi is approximately:

π ≈ 3.14

A calculator gives a more accurate value beginning:

π ≈ 3.14159265...

The decimal digits continue without ending or repeating in a fixed pattern.

For many basic calculations:

π ≈ 3.14

is accurate enough unless the question gives different instructions.

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Why π Appears in Every Circle

Imagine measuring several circular objects.

For each object, measure:

• Its circumference.
• Its diameter.

Then calculate:

circumference ÷ diameter

You will obtain a value close to:

3.14

regardless of the size of the circle.

A small coin and a huge circular track have the same circumference-to-diameter ratio.

That ratio is:

π

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Calculating Circumference Using Diameter

If the diameter is known:

C = πd

where:

C = circumference

d = diameter

and:

π ≈ 3.14

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Worked Example: Circumference from Diameter

A circular plate has a diameter of:

20 cm

Use:

C = πd

Substitute:

C = π × 20

Using π ≈ 3.14:

C ≈ 3.14 × 20

C ≈ 62.8 cm

Therefore, the circumference is approximately:

62.8 cm

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Calculating Circumference Using Radius

Because:

d = 2r

we can replace the diameter in the circumference formula.

Therefore:

C = 2πr

where:

r = radius

This gives us two useful circumference formulas:

C = πd

or:

C = 2πr

The formula you choose depends on whether the problem gives you the radius or diameter.

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Exploring Radius, Diameter, and Circumference

Changing the radius changes both the diameter and circumference. The relationships remain the same for every circle.

Notice that doubling the radius also doubles:

• The diameter.
• The circumference.

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Worked Example: Circumference from Radius

A circle has a radius of:

6 cm

Use:

C = 2πr

Substitute:

C = 2 × π × 6

Using π ≈ 3.14:

C ≈ 2 × 3.14 × 6

C ≈ 37.68 cm

Therefore:

C ≈ 37.7 cm

to the nearest tenth.

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Choosing the Correct Formula

If the question gives the diameter, the simplest formula is:

C = πd

If the question gives the radius, the simplest formula is:

C = 2πr

You could always convert between radius and diameter first, but this is not necessary.

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Example: Choosing the Formula

A bicycle wheel has a radius of:

35 cm

Since the radius is given, use:

C = 2πr

C = 2 × π × 35

C ≈ 219.9 cm

The wheel travels approximately:

220 cm

during one complete rotation, assuming it rolls without slipping.

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Estimating Circumference

Before calculating exactly, it is useful to estimate.

Since:

π ≈ 3

the circumference should be a little more than:

3 × diameter

Suppose:

d = 10 cm

Estimate:

C ≈ 3 × 10

C ≈ 30 cm

A more accurate calculation gives:

C ≈ 31.4 cm

The estimate tells us that the calculated answer is reasonable.

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Example: Checking Reasonableness

Suppose a circle has:

d = 8 m

Someone calculates:

C = 2.51 m

This cannot be correct.

Since circumference is approximately three times the diameter, we expect:

C ≈ 3 × 8

≈ 24 m

The actual value is approximately:

25.1 m

Estimation helps detect calculation errors.

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Finding Diameter from Circumference

Sometimes the circumference is known and the diameter must be calculated.

Starting with:

C = πd

Divide both sides by π:

d = C ÷ π

Example

A circular object has circumference:

31.4 cm

Calculate its diameter.

d = 31.4 ÷ 3.14

d = 10 cm

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Finding Radius from Circumference

Starting with:

C = 2πr

Rearrange:

r = C ÷ (2π)

Example

A circle has circumference:

62.8 cm

Using π ≈ 3.14:

r = 62.8 ÷ (2 × 3.14)

r = 62.8 ÷ 6.28

r = 10 cm

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Exact and Approximate Answers

Sometimes circle calculations are left in terms of π.

For example:

A circle has:

r = 5 cm

Then:

C = 2π(5)

C = 10π cm

This is an exact answer.

Using π ≈ 3.14:

C ≈ 31.4 cm

This is an approximate answer.

The symbol:

≈

means:

approximately equal to

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Why We Do Not Usually Write π = 3.14

Pi is not exactly 3.14.

Therefore:

π ≈ 3.14

is more accurate than writing:

π = 3.14

When using a calculator's π button, the calculator uses many more digits and produces a more accurate result.

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Circumference and Units

Circumference measures length.

Therefore, circumference uses ordinary length units.

Examples:

25 cm

4.7 m

12 km

Do not use square units such as:

cm²

for circumference.

Square units are used for area, not circumference.

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Radius, Diameter, and Circumference Compared

Important relationships:

d = 2r

r = d ÷ 2

C = πd

C = 2πr

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Measuring Real Circles

We can investigate circles using ordinary circular objects.

For example:

• Cup.
• Jar lid.
• Plate.
• Coin.
• Bottle.
• Wheel.

The diameter can be measured with a ruler.

The circumference can be measured by wrapping:

• String.
• Flexible measuring tape.

around the object.

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Investigating π

Suppose we measure three circular objects.

Small differences can occur because of measurement error.

However, the ratio remains close to:

3.14

This provides experimental evidence for the relationship:

C = πd

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Semicircles

A semicircle is half of a circle.

The curved part of a semicircle has a length equal to half the circumference.

If:

C = 2πr

then half the circumference is:

πr

However, the total perimeter of a semicircle also includes its straight diameter.

Therefore:

Perimeter of semicircle = πr + 2r

This distinction is important.

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Example: Perimeter of a Semicircle

A semicircle has:

r = 4 cm

Curved part:

πr = 4π

Using π ≈ 3.14:

4π ≈ 12.56 cm

Diameter:

2r = 8 cm

Total perimeter:

12.56 + 8 = 20.56 cm

Therefore:

P ≈ 20.6 cm

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Real-World Circles

Circular measurements are important in many everyday situations.

Examples include:

• Wheels.
• Tires.
• Gears.
• Clocks.
• Coins.
• Plates.
• Circular tables.
• Pipes.
• Storage tanks.
• Roundabouts.
• Running tracks.
• Ferris wheels.

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Wheels and Circumference

Circumference is especially useful when studying wheels.

One complete rotation of a wheel moves the wheel forward by approximately one circumference, assuming it rolls without slipping.

Suppose a wheel has:

d = 0.6 m

Then:

C = πd

C ≈ 3.14 × 0.6

C ≈ 1.884 m

After 10 complete rotations:

distance ≈ 10 × 1.884

distance ≈ 18.84 m

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Example: Bicycle Wheel

A bicycle wheel has a diameter of:

70 cm

Circumference:

C = πd

C ≈ 3.14 × 70

C ≈ 219.8 cm

Convert to metres:

219.8 cm = 2.198 m

Therefore, one complete rotation moves the bicycle approximately:

2.20 m

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Circular Pipes

The diameter of pipes is an important engineering measurement.

A larger diameter affects:

• How much fluid can pass through.
• The size of fittings required.
• The circumference of the pipe.
• The amount of material needed around the pipe.

If a pipe has an outside diameter of:

12 cm

its outside circumference is:

C = π × 12

C ≈ 37.7 cm

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Circular Tracks

Circumference can also be used to estimate distances around circular paths.

Suppose a circular track has:

r = 50 m

Then:

C = 2πr

C = 2 × π × 50

C = 100π

C ≈ 314 m

One complete lap is approximately:

314 m

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Scaling Circles

Suppose the radius of a circle doubles.

Original radius:

r

New radius:

2r

Original circumference:

C = 2πr

New circumference:

C = 2π(2r)

C = 4πr

The new circumference is twice the original circumference.

Therefore:

Doubling the radius doubles the circumference.

Similarly:

Doubling the diameter doubles the circumference.

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A Reliable Circle Problem-Solving Strategy

When solving a circle problem:

Step 1: Identify what information is given.

Is it:

• Radius?
• Diameter?
• Circumference?

Step 2: Identify what you need to find.

Step 3: Choose the appropriate relationship.

d = 2r

r = d ÷ 2

C = πd

C = 2πr

Step 4: Substitute the known value.

Step 5: Calculate.

Step 6: Include the correct units.

Step 7: Estimate to check whether the answer is reasonable.

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Worked Problem

A circular garden has a radius of:

8 m

Find its diameter and circumference.

Diameter

d = 2r

d = 2 × 8

d = 16 m

Circumference

C = 2πr

C = 2 × π × 8

C = 16π

Using π ≈ 3.14:

C ≈ 50.24 m

Therefore:

Diameter = 16 m

Circumference ≈ 50.2 m

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Worked Real-World Problem

A circular table has a diameter of:

1.5 m

A decorative strip will be placed around its outside edge.

How much strip is required?

Use:

C = πd

C = π × 1.5

C ≈ 4.71 m

Therefore, approximately:

4.71 m

of decorative strip is required.

In a real situation, slightly more material might be purchased to allow for joining and cutting.

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Common Mistakes

Confusing Radius and Diameter

Remember:

radius = center to edge

diameter = edge to edge through the center

The diameter is twice the radius.

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Forgetting to Double the Radius

If:

r = 7 cm

then:

d = 14 cm

not 7 cm.

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Using C = πr

This is not the circumference formula.

Use:

C = πd

or:

C = 2πr

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Multiplying the Diameter by 2 Again

If the diameter is already given, use:

C = πd

You do not need to double it.

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Using the Wrong Units

Circumference is a length.

Use:

cm, m, km

not:

cm², m², km²

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Treating π as Exactly 3.14

Remember:

π ≈ 3.14

When possible, use the calculator's π button for greater accuracy.

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Rounding Too Early

Keep several digits during calculations and round the final answer.

Rounding too early can make the final answer less accurate.

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Forgetting the Straight Side of a Semicircle

Half the circumference gives only the curved part.

The complete perimeter of a semicircle also includes its diameter.

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Check Your Understanding

1. What is a circle?

2. What is the center of a circle?

3. Define radius.

4. Define diameter.

5. Define circumference.

6. What is the relationship between radius and diameter?

7. A circle has a radius of 6 cm. Find its diameter.

8. A circle has a radius of 11 m. Find its diameter.

9. A circle has a diameter of 16 cm. Find its radius.

10. A circle has a diameter of 25 m. Find its radius.

11. Approximately what value does π have?

12. Explain what π represents.

13. Write the circumference formula using diameter.

14. Write the circumference formula using radius.

15. Calculate the circumference of a circle with diameter 10 cm.

16. Calculate the circumference of a circle with diameter 24 cm.

17. Calculate the circumference of a circle with radius 5 cm.

18. Calculate the circumference of a circle with radius 12 m.

19. A circular table has diameter 2 m. Approximately how far is it around the edge?

20. A bicycle wheel has diameter 60 cm. Approximately how far does it travel during one complete rotation?

21. A circular track has radius 40 m. Find its circumference.

22. A circle has circumference 31.4 cm. Find its diameter using π ≈ 3.14.

23. A circle has circumference 62.8 cm. Find its radius.

24. Explain why circumference is measured in cm rather than cm².

25. Estimate the circumference of a circle with diameter 15 cm before calculating it accurately.

26. Explain how estimation can help identify mistakes in circumference calculations.

27. A wheel has a circumference of approximately 2 m. Approximately how far will it travel in 25 rotations?

28. A semicircle has radius 5 cm. Calculate the length of its curved edge.

29. Calculate the complete perimeter of the semicircle in Question 28.

30. Give three real-world situations in which radius, diameter, or circumference measurements are useful.

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Key Terms

• Circle – set of points in a plane that are all the same distance from a fixed center.
• Center – point exactly in the middle of a circle.
• Radius – distance from the center to the circumference.
• Diameter – distance across a circle through its center.
• Circumference – distance around a circle.
• Pi (π) – ratio of a circle's circumference to its diameter.
• Semicircle – half of a circle.
• Exact value – value written without approximation, such as 10π cm.
• Approximate value – rounded numerical value, such as 31.4 cm.

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Key Takeaways

• A circle is a two-dimensional curved shape.
• The center is the point in the middle.
• The radius runs from the center to the circumference.
• The diameter passes across the circle through the center.
• The diameter is twice the radius:

d = 2r

• The radius is half the diameter:

r = d ÷ 2

• The circumference is the distance around a circle.
• Pi is approximately:

π ≈ 3.14

• Circumference can be calculated using:

C = πd

or:

C = 2πr

• Circumference uses units of length, not square units.
• Circumference is slightly more than three times the diameter.
• Estimation is useful for checking whether an answer is reasonable.
• Radius, diameter, circumference, and π are used extensively in real-world measurement and design.

The four relationships worth remembering are:

d = 2r

r = d ÷ 2

C = πd

C = 2πr