Radius diameter circumference and pi

A piece of string follows the complete curved boundary of a circular lid once. Which measure does the string approximate?

Name the centre, the straight lengths and the curved boundary

A circle is the set of points at the same distance from a fixed centre O

A radius joins the centre to any point on the circle. All radii of one circle have equal length

A diameter joins two points on the circle along a straight line through the centre

A straight segment joining two boundary points is a chord. It is a diameter only if it passes through the centre

Circumference is the length of the whole curved boundary, not a straight segment

Radius, diameter and circumference all measure length, so they use length units such as centimetres or metres. The area inside the circle is a different quantity

A diameter contains two radii

The centre divides every diameter into two equal radii

Double a radius to obtain the diameter. Halve a diameter to obtain the radius

For example, radius five centimetres gives diameter ten centimetres

A diameter of nine centimetres gives a radius of four and a half centimetres

The relationship stays the same when the circle is larger, smaller or drawn in a different position

Double the centre-to-boundary length

A circle has a radius of 5 cm

Calculate the diameter in cm. Give an exact whole number or decimal

The diameter is twice the radius, while circumference measures the curved boundary

The stated value is a radius, so double it

The diameter contains two radii in a straight line through the centre

Find the full distance through the centre

A whole-number diameter can have a half-unit radius

A circle has a diameter of 9 m

Calculate the radius in m. Give an exact whole number or decimal

The diameter is twice the radius, while circumference measures the curved boundary

The stated value is a diameter, so halve it

Check by doubling the radius to recover the stated diameter

Halve the diameter rather than doubling it

Double a decimal radius exactly

A circle has a radius of 3.75 m

Calculate the diameter in m. Give an exact whole number or decimal

The diameter is twice the radius, while circumference measures the curved boundary

The stated value is a radius, so double it

The diameter contains two radii in a straight line through the centre

Keep the same unit when doubling

Divide a decimal diameter by two

A circle has a diameter of 7.5 cm

Calculate the radius in cm. Give an exact whole number or decimal

The diameter is twice the radius, while circumference measures the curved boundary

The stated value is a diameter, so halve it

Check by doubling the radius to recover the stated diameter

Halve a decimal diameter without rounding the radius

Compare boundary length with distance through the centre

Draw several circles with a compass or choose circular objects of different sizes

Measure each diameter through the centre, not along a shorter chord

Wrap non-stretching string around each boundary once, mark the meeting point, straighten it and measure the marked length

Use the same length unit for diameter and circumference, then divide circumference by diameter

Repeat a measurement if the string slips, stretches or overlaps. A ruler and string give approximations rather than exact lengths

Different-sized circles should give ratios close to the same value, even though their individual lengths differ

Interpret a modelled measurement table

These modelled readings illustrate measurements recorded to the nearest tenth of a centimetre; they are not observations collected from physical objects

The ratios are close to one common value. Their small differences are consistent with rounded measurements

The centimetre units cancel in C divided by d, so the ratio has no length unit

More precise measurement can improve the estimates but does not change the circle relationship

Diameters and circumferences for three modelled circles were recorded to the nearest tenth of a centimetre. Their circumference-to-diameter ratios are 3.15, 3.1375 and approximately 3.1417. Which interpretation is justified?

Name the constant ratio pi

For every circle, the exact circumference divided by its diameter is the same constant, called pi

Pi is a number, not a length or an angle. Its decimal digits continue without a repeating pattern

Common approximations include 3.14 and the fraction twenty-two sevenths, but neither is exactly pi

Multiply both sides of the ratio relationship by the positive diameter

Since diameter is twice radius, the equivalent radius form is

Move the radius from two to four units, then try the other settings. Compare the diameter and circumference readouts with the circumference-to-diameter ratio. The displayed decimal readouts are rounded

What happens when the radius doubles?

Keep pi in an exact circumference

If the radius is four centimetres, the diameter is eight centimetres

The expression eight pi centimetres is exact; replacing pi by a rounded decimal would make an approximation

If the diameter is ten centimetres, use it directly without doubling

In the coefficient entry, pi and the length unit are already printed. Enter only the number multiplying pi

The coefficient can be a decimal. A diameter of seven and a half centimetres gives an exact circumference of seven and a half pi centimetres

Double a radius before multiplying by pi

A circle has a radius of 4 cm

Find the exact circumference by completing the coefficient below

Enter only the coefficient, not pi or a unit

The diameter is twice the radius, while circumference measures the curved boundary

Multiply the diameter by pi; do not double a diameter again

Keeping pi in the answer preserves the exact circumference

The missing coefficient is the number multiplying pi, so do not enter pi again

Construct the exact circumference coefficient from a radius

Use a diameter once in C = pi times diameter

A circle has a diameter of 10 cm

Find the exact circumference by completing the coefficient below

Enter only the coefficient, not pi or a unit

The diameter is twice the radius, while circumference measures the curved boundary

Multiply the diameter by pi; do not double a diameter again

Keeping pi in the answer preserves the exact circumference

The missing coefficient is the number multiplying pi, so do not enter pi again

Do not double an already supplied diameter

A decimal radius still gives an exact pi expression

A circle has a radius of 2.5 m

Find the exact circumference by completing the coefficient below

Enter only the coefficient, not pi or a unit

The diameter is twice the radius, while circumference measures the curved boundary

Multiply the diameter by pi; do not double a diameter again

Keeping pi in the answer preserves the exact circumference

The missing coefficient is the number multiplying pi, so do not enter pi again

Retain a decimal coefficient when necessary

Keep the diameter value as the exact coefficient

A circle has a diameter of 7.5 cm

Find the exact circumference by completing the coefficient below

Enter only the coefficient, not pi or a unit

The diameter is twice the radius, while circumference measures the curved boundary

Multiply the diameter by pi; do not double a diameter again

Keeping pi in the answer preserves the exact circumference

The missing coefficient is the number multiplying pi, so do not enter pi again

Separate the coefficient from the printed pi symbol

Round only when the question requests an approximation

For a decimal circumference, use the calculator pi value and carry it through the multiplication

Do not replace pi by 3.14 unless the question specifically tells you to use that approximation

To two decimal places, inspect the third decimal digit and round the hundredths accordingly

Write exactly the requested number of decimal places. A result of 37.70 has two decimal places, while 37.7 has one

Keep the distinction between an exact expression such as eight pi and a rounded approximation such as twenty-five point one three

Convert an exact pi expression to a requested decimal

A circle has a radius of 4 cm

Calculate the circumference in cm. Use the calculator pi value and give exactly two decimal places

The diameter is twice the radius, while circumference measures the curved boundary

Multiply the diameter by pi; do not double a diameter again

Use the calculator pi value, not the early approximation 3.14

Inspect the third decimal digit to round to hundredths, using the full calculator value

Round only the final result and write exactly two decimal places

Retain the trailing zero required by two decimal places

Multiply a decimal diameter by the calculator pi value

A circle has a diameter of 8.5 cm

Calculate the circumference in cm. Use the calculator pi value and give exactly two decimal places

The diameter is twice the radius, while circumference measures the curved boundary

Multiply the diameter by pi; do not double a diameter again

Use the calculator pi value, not the early approximation 3.14

Inspect the third decimal digit to round to hundredths, using the full calculator value

Round only the final result and write exactly two decimal places

Round the final circumference rather than pi first

Divide by pi to undo the circumference relationship

Divide the exact circumference by pi to recover the diameter

Then halve the diameter if the radius is requested

For an exact circumference of eighteen pi centimetres, the pi factors cancel

Do not replace pi by a decimal before cancelling it in an exact calculation

Cancel pi and then halve the recovered diameter

A circle has the following exact circumference

Calculate the radius in cm. Give an exact whole number or decimal

The diameter is twice the radius, while circumference measures the curved boundary

Divide the exact circumference by pi to recover the diameter

Halve that diameter to obtain the radius

An odd exact coefficient gives a half-unit radius

Recover a decimal diameter without approximate division

A circle has the following exact circumference

Calculate the diameter in cm. Give an exact whole number or decimal

The diameter is twice the radius, while circumference measures the curved boundary

Divide the exact circumference by pi to recover the diameter

The diameter spans the whole circle through its centre

Divide by pi but not by an extra factor of two

Which statement about pi and circle circumference is correct?

Identify the given measure before choosing the relationship

Radius reaches from centre to boundary, diameter passes through the centre across the circle, and circumference goes around the boundary

The circumference-to-diameter ratio is constant even when the circle size changes

Retain pi for an exact circumference. For a requested decimal, use the calculator pi value and round only the final result

All three circle measures use length units. These relationships concern circumference, not the area inside the circle