Circles: Radius or Diameter? Why the Area Comes Out Four Times Too Big

16 October 2026 · by Larry

She knows both formulas. The area of a circle is π × r × r. The distance round it is 2 × π × r. Ask her cold and she will say them without a pause.

Then the question reads: a round coaster is 14 cm across. Find its area. Take π as 22/7. She writes 22/7 × 14 × 14 = 616 cm². The arithmetic is perfect. The answer is exactly four times too big. It should be 154 cm².

That is not carelessness, and it is not a gap in the formula. One number went into the formula as the wrong kind of number: 14 is the diameter, the formula wants the radius, and the radius is 7. Circles are a Primary 6 topic on StudyLab’s own level list, and this swap is the first thing to look for when a circle answer comes back wrong.

Why it is exactly four times

The area formula uses the radius twice: r × r. Put in a number that is twice as big and the answer is not twice as big, it is 2 × 2 = four times as big. So when the working looks right and the answer is exactly four times what it should be, a diameter has gone in where a radius belongs.

The distance round uses the radius only once, so the same swap there gives exactly double: 2 × 22/7 × 14 = 88, where the right answer is 44. “Exactly four times” and “exactly double” are useful clues, because a tidy multiple like that points to one particular slip.

1. Reading “across” and “from the centre”

A radius goes from the centre to the edge. A diameter goes all the way across, through the centre, so it is two radii end to end. Questions use both, and often not the words themselves: “14 cm across”, “a width of 14 cm”, “a diameter of 14 cm”, “a radius of 7 cm”, “the centre is O and OA is 7 cm”.

The habit that fixes it is small. Before any calculation, write r = on the diagram, and if the question gave the diameter, halve it right there and write the half. Then the formula only ever sees the number next to “r =”. Have her say it aloud for each number: across, or half-across? The area formula wants half-across.

2. r × r is not r × 2

The second slip is quieter. With the radius correctly found as 7, she writes 22/7 × 7 × 2 = 44, because she has doubled the radius where the formula squares it.

What makes this one hard to spot is that 44 is the distance round that same circle. It is the right number for the wrong question, so it looks plausible. The area is 22/7 × 7 × 7 = 154.

While she is learning it, ask her to write the area as three things in a row: π × r × r, with the real numbers underneath. Writing “r²” in a hurry is where doubling and squaring get confused.

3. The way round, or the space inside?

This is the same two-drawers problem as area and perimeter, with a curved edge. The same pictures work: the fence round a circular pond is the distance round (the circumference), and the grass that fills the circle is its area.

Units give the same clue as before. The way round is in cm or m. The space inside is in cm² or m². A useful way to ask a circle question in plain words: if a wheel turns once, how far does it roll? That is the distance round the wheel. How much paint covers the lid? That is the area.

4. Half circles and quarter circles: the straight edges

Take a semicircle with a diameter of 20 cm, and take π as 3.14. The curved edge is half of 3.14 × 20, which is 31.4 cm. But the shape also has the straight edge across the bottom, 20 cm, so the perimeter is 31.4 + 20 = 51.4 cm. The area is half of 3.14 × 10 × 10, which is 157 cm². The same shape, two different jobs, two different numbers.

A quarter circle has two straight edges, and both are radii. With a radius of 14 cm and π as 22/7: the curved edge is a quarter of 2 × 22/7 × 14 = 88, so 22 cm. Add the two radii, 14 + 14, and the perimeter is 22 + 28 = 50 cm. The area is a quarter of 22/7 × 14 × 14 = 616, which is 154 cm².

That 154 is the coaster’s area again, and it is not a coincidence. A circle with twice the radius has four times the area, so a quarter of the big circle is exactly the small circle. If she has the “four times” idea, she can see why.

The usual slip is stopping at the curve. The same trap with a semicircle is in what changes from P5 to P6. The fix is the one from the area and perimeter post: trace the outside edge with a finger, once, and say each piece as she passes it: curve, straight, straight.

5. Shapes built from circle pieces

Picture a rectangle 20 cm long and 14 cm wide, with a semicircle on each of the two 14 cm ends, like a running track. Two semicircles with the same diameter join to make one whole circle with a diameter of 14 cm.

So the perimeter is the two straight sides, 20 + 20 = 40, plus the distance round one whole circle, 22/7 × 14 = 44. That gives 84 cm. The two 14 cm lines where the semicircles meet the rectangle are inside the shape, so they are not counted. The area is the rectangle plus the circle: 20 × 14 = 280, plus 154, which is 434 cm².

A second classic: a square with sides of 14 cm, with a quarter circle drawn inside from one corner. The side of the square is the radius of the quarter circle, and the diagram never says so. The part of the square left over is 196 − 154 = 42 cm². Children who cannot find the radius here are usually not weak at circles. They have not yet learned to ask, what is this length also equal to?

Which π?

Two values turn up: 22/7 and 3.14. Both are close, and neither is exact, because π is a number that never ends. They give slightly different answers: a circle with a radius of 7 cm has an area of 154 cm² with 22/7, and 153.86 cm² with 3.14.

Both are right when the question asks for them. The rule is to read which value the question tells her to use, write it in the margin before starting (π = 22/7), and use that one throughout. Switching halfway, say 22/7 in one part and 3.14 in the next, is a slip. As a convenience, 22/7 is friendly when a radius or diameter is a multiple of 7, because the 7 cancels; 3.14 suits the other numbers.

A check that needs no formula

Take any round lid, plate or cup. Measure across it with a ruler. Then wrap a piece of string once round the edge and measure the string. The string is a little more than three times the width across. That “three and a bit” is π, and it is a better thing to understand than the formula.

It also gives a check on every answer. The way round is a bit more than 3 times the diameter. The area is a bit more than 3 squares whose sides are the radius, and always less than 4 of them, because a circle fits inside the square drawn round it, and that square is 4 radius-squares. For the coaster, 3 × 49 = 147, and the real answer is 154. The 616 does not even pass the second test, because 4 × 49 is only 196. Ask her to do that estimate before the formula. It takes ten seconds, and it catches both the diameter-for-radius slip and the doubling slip, because each gives an answer outside the range.

What helps

• Write r = on the diagram first, and halve any diameter on the spot.
• Write π × r × r with the real numbers under it, so doubling and squaring cannot hide.
• Estimate first: a bit more than three radius-squares for area, a bit more than three diameters for the way round.
• Trace the outside with a finger for any perimeter, and say curve and straight pieces aloud.
• Write the value of π in the margin before starting.

What does not help

• Saying the formulas again, louder. She has them. The slip is in reading the question, not in remembering.
• Another page of the same sums. It gives the same slip twenty more chances.
• Telling her to be careful. She was careful with the arithmetic. The mistake was before the arithmetic began, which is what careless mistakes usually turn out to be.

A ten-minute check you can do at home

Five questions. Ask her to say the radius out loud first, every time.

• A coin is 2 cm across. What is its radius?
• A circle has a diameter of 10 cm. Taking π as 3.14, find its area.
• For the same circle, how far is it round?
• A semicircle has a diameter of 10 cm. What is its perimeter?
• Without a calculator or the formula: a circle has a radius of 6 cm. Is its area closer to 36, 108 or 400 cm²?

The answers are 1 cm, 78.5 cm², 31.4 cm, 25.7 cm (the curve is 15.7, plus 10 across), and 108 cm² (about three 6 × 6 squares; the exact answer with 3.14 is 113.04). Read any wrong answer as a clue. On the second question, 314 means the diameter went in as the radius, and 31.4 means she doubled instead of squaring. On the third, 62.8 means the diameter went in as the radius again. On the fourth, 15.7 means the perimeter stopped at the curve. On the fifth, 400 means she has not yet checked the answer against the square drawn round the circle, which is only 144 cm² here.

Where this fits

On StudyLab, Circles is a Primary 6 topic in the free maths practice. Each question gives the radius, tells the child to use π = 3.14, and asks for the area or the circumference. That practises the formulas, but not the step of halving a diameter, and not half or quarter circles, so for those use your child’s own school worksheets or a few circles of your own. The maths topics level by level shows what sits around it, and which level learns what, and in what order, is set by the school and the MOE Primary Mathematics syllabus on moe.gov.sg.

Related: the next step up, from flat space to solid space, is in volume and capacity: why cm³ and litres confuse children, and the pattern of a right method with the marks still lost is in right answer, but the marks still went.