PSLE Mathematics Learning Guide · Guide 66 · Wintour V1.0 Companion to Guides 5 and 60
Return to the PSLE Learning Guide · Parent guide: Circle Area and Circumference
A circle question does not always give the radius. Sometimes it gives the circumference. Sometimes it gives the area. Sometimes the radius is hidden inside a perimeter statement and must be recovered before anything else can be found.
The strong habit is:
identify which circle quantity is known → reverse the correct relationship → recover radius or diameter → only then calculate the next requested quantity.
This page is a narrow companion to the main circle guide and the semicircle/quarter-circle boundary guide. Current P6 revision platforms commonly include reverse circle applications such as finding area from perimeter; eduKate’s missing job is to make the reverse state explicit rather than treat it as another formula to memorise.
The MOE Primary Mathematics syllabus updated October 2025 provides the current circle framework. All examples below are original eduKate teaching material.
The two parent relationships
For radius r and diameter d:
C = 2πr = πd.
A = πr².
Reverse questions simply move the unknown.
Circumference → diameter
If C = πd, then:
d = C ÷ π.
Example: circumference = 44 cm, π=22/7.
d = 44 ÷ (22/7) = 14 cm.
Radius = 7 cm.
Circumference → radius
If C=2πr, then:
r = C ÷ (2π).
Example: C=62.8 cm, π=3.14.
r = 62.8 ÷ 6.28 = 10 cm.
Circumference → radius → area
A circle has circumference 88 cm using π=22/7. Find area.
Step 1: d=88÷(22/7)=28 cm.
Step 2: r=14 cm.
Step 3: A=22/7×14²=616 cm².
Do not substitute circumference into the area formula as if it were radius.
Area → radius
If A=πr², then:
r² = A÷π.
r = √(A÷π).
Example: A=154 cm², π=22/7.
r²=154÷(22/7)=49.
r=√49=7 cm.
Area → radius → circumference
Area=314 cm², π=3.14.
r²=314÷3.14=100.
r=10 cm.
C=2×3.14×10=62.8 cm.
Square root is justified by r²
Taking a square root is not a generic “undo area” trick. It works because the area formula contains the square of the same radius:
A/π = r².
The geometry creates a perfect square relationship.
Diameter is not radius
A circle has circumference 31.4 cm, π=3.14.
d=31.4÷3.14=10 cm.
Radius=5 cm.
Area=3.14×25=78.5 cm².
Using d=10 as the radius would make the area four times too large.
Use a size check
If circumference is about 63 cm and π≈3.14, diameter should be about 20 cm because 3.14×20≈62.8.
So a radius of 20 cm would be suspicious; that would imply diameter 40 cm and circumference about126 cm.
Keep π consistent
If the question specifies π=22/7, use it throughout the same problem unless instructed otherwise.
If it specifies 3.14, keep 3.14.
If exact form in π is required, do not replace π with a decimal unnecessarily.
Reverse questions can be exact in π
If C=18π cm:
2πr=18π.
r=9 cm.
Area=81π cm².
The π cancels because both sides contain the same factor.
Semicircle perimeter can hide the radius
For a semicircle:
P=πr+2r.
If P=36 cm and π=22/7:
(22/7)r+2r = (36/7)r.
(36/7)r=36.
r=7 cm.
This is a reverse boundary problem; Guide60 explains why the diameter term must be present.
Quarter-circle perimeter can also hide radius
P=1/2πr+2r.
If π=22/7 and P=25 cm:
(11/7+14/7)r =25/7 r.
25/7 r=25.
r=7 cm.
A composite perimeter may contain only part of a circumference
If a figure’s curved boundary is a semicircular arc of length 22 cm with π=22/7:
semicircle arc=πr.
22=(22/7)r.
r=7 cm.
Do not use C=2πr because only half the circumference is present.
Area changes with the square of radius
If radius doubles, area multiplies by 4.
If radius triples, area multiplies by 9.
This gives a fast check in reverse problems. If one circle’s area is four times another’s, its radius is twice as large.
Circumference changes directly with radius
If radius doubles, circumference doubles.
This contrast is useful:
- circumference ∝ r;
- area ∝ r².
Concentric-circle reverse reasoning
Two concentric circles have circumferences 44 cm and 88 cm using the same π.
The larger circumference is twice the smaller, so the larger radius is twice the smaller.
Radii are7 cm and14 cm if π=22/7.
The annulus area is π(14²−7²)=147π cm².
This is an extension application of the same reverse-radius idea.
Main worked workshop: sixteen original reverse-circle tasks
1.
C=44 cm, π=22/7. Find d and r.
Answer:d14,r7.
2.
C=31.4 cm, π=3.14.
Answer:d10,r5.
3.
C=75.36 cm, π=3.14.
Answer:d24,r12.
4.
C=66 cm, π=22/7. Find area.
Answer:d21,r10.5,A=346.5 cm².
5.
C=88 cm, π=22/7. Find area.
Answer:616 cm².
6.
A=154 cm², π=22/7. Find r.
Answer:7 cm.
7.
A=616 cm², π=22/7. Find r and C.
Answer:r14,C88 cm.
8.
A=314 cm², π=3.14.
Answer:r10,C62.8 cm.
9.
C=20π cm.
Answer:r10,A100π cm².
10.
A=49π cm².
Answer:r7,C14π cm.
11.
A learner divides C by π and calls the answer radius.
Repair:C/π gives diameter; halve it for radius.
12.
A learner takes A/π and stops.
Repair:A/π=r²; take square root to recover r.
13.
Semicircle perimeter36 cm, π=22/7.
Answer:r7 cm.
14.
Quarter-circle perimeter25 cm, π=22/7.
Answer:r7 cm.
15.
Semicircle arc31.4 cm, π=3.14.
Answer:πr=31.4 →r10 cm.
16.
Circle A has four times the area of Circle B. Compare radii.
Answer:2:1.
Diagnose the first weak link
Circumference divided by π reported as radius: diameter/radius confusion.
Area divided by π reported as radius: missing square-root step.
Full-circle formula used for semicircular arc: boundary-fraction error.
π switched midway: consistency error.
Correct radius but wrong next quantity: state-transition error after successful reversal.
Independent transfer check
- C=50.24 cm, π=3.14. Find r.
- C=132 cm, π=22/7. Find d and r.
- A=452.16 cm², π=3.14. Find r.
- A=1386 cm², π=22/7. Find r.
- C=16π cm. Find area in π.
- A=121π cm². Find circumference in π.
- Explain why C/π gives diameter.
- Explain why √(A/π) gives radius.
Independent-check answers
1.8 cm.
2.d42 cm,r21 cm.
3.A/π=144,r12 cm.
4.A/π=441,r21 cm.
5.r8,A64π cm².
6.r11,C22π cm.
7.C=πd, so dividing by π isolates d.
8.A/π=r²; square root reverses the squaring.
Wintour V1.0 return test
Move the known quantity from radius to diameter, circumference, area, semicircle arc and composite perimeter. A robust learner should identify the correct parent relationship before reversing it.
The learner’s final card
What circle quantity is actually given? Is it a full circumference, partial arc, area or composite perimeter? Does dividing by π give diameter or r²? Do I still need to halve or take a square root? Can I run the answer forward and recover the original quantity?
Continue through Batch 17
Use Guide 65: Combine Two Ratios. Continue with Guide 67: Tank Flow Rate and Liquid Level and Guide 68: Form a Linear Equation From a Word Relationship.
Return to the PSLE Learning Guide Mathematics route.
Sources and boundaries
MOE Primary Mathematics syllabus, updated October 2025; SEAB 2026 PSLE formats.
All examples and solutions are original eduKate teaching material. This page is subordinate to the established circle owner and isolates reverse recovery of radius or diameter.