BTT Mathematics / Primary Mathematics Learning Hub / Guide 17
A circle is controlled by one centre and one radius. Every point on the circumference is the same distance from the centre. Diameter, circumference and area are different quantities built from that structure.
The diameter is twice the radius. Circumference measures the boundary. Area measures the surface enclosed. When circles are combined with rectangles, squares or other circular parts, the difficult step is usually not the arithmetic but deciding which boundaries and regions belong to the requested figure.
Use the value of π specified by the exercise or your school instructions. Where a decimal approximation is needed in this guide, π≈3.14 is used unless a question states otherwise. The examples are original teaching material aligned to upper-primary circle and measurement reasoning; select sections according to current school scope.
Parts of a circle · Circumference · Area · Semicircles and quarter-circles · Composite figures · 24 questions · Worked answers
1. Radius, diameter and circumference are not interchangeable
A radius joins the centre to the circumference. A diameter joins two points on the circumference and passes through the centre. Therefore every diameter consists of two radii laid end to end.
If the radius is 7 cm, the diameter is 14 cm. If the diameter is 18 cm, the radius is 9 cm.
Do not read a diameter as a radius
When a diagram labels a straight line all the way across a circle through its centre, that length is the diameter. Using it directly as r in an area calculation would make the circle four times too large because area depends on r².
Circumference is the full boundary
The circumference is the distance once around the circle. Its formula can be written C=πd or C=2πr. These are equivalent because d=2r.
If r=5 cm, then d=10 cm and C≈3.14×10=31.4 cm. Using 2×3.14×5 gives the same answer.
Units reveal the quantity
Radius, diameter and circumference are lengths, so their units are centimetres, metres and so on. Area uses square units such as cm² or m².
A result of 78.5 cm² cannot be a circumference. The unit already tells us it describes a surface.
2. Circumference scales directly with diameter
Because C=πd, doubling the diameter doubles the circumference. Tripling the diameter triples the circumference.
Worked example A: Find circumference from diameter
A circle has diameter 12 cm. Using π≈3.14:
C=3.14×12=37.68 cm.
Worked example B: Find circumference from radius
A circle has radius 8 cm. Its diameter is 16 cm. Therefore C=3.14×16=50.24 cm.
Worked example C: Find diameter from circumference
A circular track has circumference 62.8 m using π≈3.14. Since C=πd, d=62.8÷3.14=20 m. The radius is therefore 10 m.
Arc length as part of a circumference
A semicircle contains half the circumference. A quarter-circle contains one quarter. If a full circle has circumference 40 cm, the curved part of a semicircle is 20 cm and the curved part of a quarter-circle is 10 cm.
Do not confuse the curved arc with the full perimeter of the semicircle or quarter-circle. Straight sides may also belong to the boundary.
3. Circle area grows with the square of the radius
The area formula is A=πr². The radius is multiplied by itself before multiplying by π.
Worked example D: Find circle area
A circle has radius 5 cm. A≈3.14×5×5=78.5 cm².
Worked example E: Diameter is given
A circle has diameter 14 cm. First find r=7 cm. Then A≈3.14×7×7=153.86 cm².
Why doubling radius quadruples area
If the radius changes from r to 2r, then area changes from πr² to π(2r)²=4πr². The new area is four times the old area.
Circumference only doubles because it depends on the first power of radius. Area changes faster because two perpendicular dimensions scale.
Worked example F: Find radius from area
A circle has area 314 cm² using π≈3.14. Then r²=314÷3.14=100, so r=10 cm.
At Primary level, this reverse step is easiest when the resulting square number is familiar.
4. Semicircles and quarter-circles are fractions of a circle
Semicircle area
A semicircle with radius 6 cm has half the area of a full radius-six circle. Full area is 3.14×36=113.04 cm². Half is 56.52 cm².
Semicircle perimeter
The perimeter includes the curved half-circumference plus the diameter. For r=6 cm, the curved part is πr≈18.84 cm. The diameter is 12 cm. Total perimeter is 30.84 cm.
A common mistake is to report only the curved part. Trace the entire outside boundary before adding.
Quarter-circle area
A quarter-circle with radius 8 cm has area one quarter of 3.14×64=200.96 cm², giving 50.24 cm².
Quarter-circle perimeter
The curved arc is one quarter of the full circumference: one quarter of 2πr, which equals πr/2. With r=8 cm, arc length≈12.56 cm. Add the two radii, 8+8, to obtain perimeter 28.56 cm.
Sector reasoning
If a sector represents one third of a circle, its area is one third of the full area and its arc is one third of the full circumference, provided the sector angle truly represents one third of a full turn.
This fraction-of-a-whole reasoning connects circles to fractions and pie charts.
5. Composite figures require a boundary map and a region map
Worked example G: Circle inside a square
A circle of radius 5 cm fits exactly inside a square, touching all four sides. The circle’s diameter is 10 cm, so the square side is 10 cm.
Square area=100 cm². Circle area≈78.5 cm². The area inside the square but outside the circle is 21.5 cm².
Worked example H: Square with four quarter-circles
A square has side 14 cm. Four quarter-circles of radius 7 cm are placed at the corners. Together the four quarter-circles make one full circle of radius 7 cm.
Total circular area≈3.14×49=153.86 cm². Square area=196 cm². The remaining central area is 42.14 cm².
Worked example I: Rectangle with two semicircular ends
A stadium-shaped figure has a rectangle 20 m long and 10 m wide, with a semicircle attached at each short end. The two semicircles together form one full circle of diameter 10 m, radius 5 m.
Area=rectangle area 200 + circle area 78.5=278.5 m².
Perimeter uses the two long rectangle sides, 20+20, plus the full circumference of the combined semicircles, 31.4. Total perimeter=71.4 m.
Worked example J: Shaded ring
A circular ring has outer radius 10 cm and inner radius 6 cm. The ring area is the larger circle minus the smaller:
3.14×100−3.14×36=314−113.04=200.96 cm².
The width of the ring is 4 cm, but multiplying circumference by four would not give the exact ring area. Use the difference of the two circle areas.
Boundary versus internal lines
When figures are joined, internal shared edges are not part of the external perimeter. When a hole is cut out, the boundary around the hole may become part of the total boundary if the problem asks for all exposed edges.
Draw a finger-trace around exactly what is being measured before calculating.
6. Practice: 24 questions
Use π≈3.14 unless stated otherwise. Keep answers covered until an attempt is complete.
Questions 1–8: Circle foundations
1. A circle has radius 9 cm. Find its diameter.
2. A circle has diameter 26 cm. Find its radius.
3. Find the circumference of a circle of diameter 10 cm.
4. Find the circumference of a circle of radius 7 cm.
5. Find the area of a circle of radius 4 cm.
6. Find the area of a circle of diameter 12 cm.
7. A circle has circumference 31.4 cm. Find its diameter.
8. A circle has area 314 cm². Find its radius.
Questions 9–16: Parts of circles
9. Find the area of a semicircle of radius 5 cm.
10. Find the perimeter of a semicircle of radius 5 cm.
11. Find the area of a quarter-circle of radius 6 cm.
12. Find the perimeter of a quarter-circle of radius 6 cm.
13. A full circle has circumference 75.36 cm. Find the curved length of one quarter of the circle.
14. A full circle has area 200.96 cm². Find the area of three quarters of the circle.
15. A semicircle has diameter 16 cm. Find its area.
16. A semicircle has diameter 16 cm. Find its perimeter.
Questions 17–24: Composite figures
17. A circle of radius 5 cm fits exactly inside a square. Find the square area.
18. For question 17, find the area inside the square but outside the circle.
19. A rectangle is 18 cm by 8 cm. A semicircle of diameter 8 cm is attached to each short end. Find the total area.
20. For question 19, find the external perimeter.
21. A ring has outer radius 9 cm and inner radius 5 cm. Find its area.
22. A square has side 20 cm. A circle of radius 10 cm is inscribed. Find the area outside the circle but inside the square.
23. Four quarter-circles of radius 4 cm are combined. What full-circle area do they make?
24. Explain why the perimeter of a semicircle is not simply half the circumference of its full circle.
7. Worked answers
Answers 1–8
1. 18 cm. Diameter=2r=18.
2. 13 cm. Radius is half the diameter.
3. 31.4 cm. C=πd≈3.14×10.
4. 43.96 cm. Diameter is fourteen, so C≈3.14×14.
5. 50.24 cm². A≈3.14×4²=3.14×16.
6. 113.04 cm². Diameter twelve gives radius six. A≈3.14×36.
7. 10 cm. d=31.4÷3.14.
8. 10 cm. r²=314÷3.14=100, so r=10.
Answers 9–16
9. 39.25 cm². Full area is 78.5 cm²; take half.
10. 25.7 cm. Curved half is πr≈15.7 cm. Add diameter ten.
11. 28.26 cm². Full area is 3.14×36=113.04; divide by four.
12. 21.42 cm. Quarter arc is one quarter of 37.68=9.42 cm. Add two radii, twelve centimetres.
13. 18.84 cm. Divide 75.36 by four.
14. 150.72 cm². Multiply 200.96 by three quarters.
15. 100.48 cm². Radius is eight. Full area is 200.96; half is 100.48.
16. 41.12 cm. Curved half is πr≈25.12 cm. Add diameter sixteen.
Answers 17–24
17. 100 cm². Diameter ten equals the square side.
18. 21.5 cm². Square area 100 minus circle area 78.5.
19. 194.24 cm². Rectangle area=18×8=144. Two semicircles form a circle of radius four, area 50.24. Total 194.24.
20. 61.12 cm. Two long sides total 36. Curved ends form one circumference of diameter eight: 25.12. Total 61.12.
21. 175.84 cm². Outer area=3.14×81=254.34. Inner area=3.14×25=78.5. Difference=175.84.
22. 86 cm². Square area 400; circle area 314; difference 86.
23. 50.24 cm². Four quarters form one full radius-four circle.
24. Half the circumference gives only the curved arc. A semicircle perimeter also includes the straight diameter.
8. Diagnose the boundary before calculating
If a learner repeatedly mixes circumference and area, ask them to trace the boundary with a finger and shade the surface. The first is one-dimensional; the second is two-dimensional.
If semicircle perimeter answers omit the diameter, ask whether a real fence around the shape could leave the straight side open. The physical model often exposes the missing boundary.
Use scale checks
Doubling a radius should double the circumference and multiply the area by four. If a computed pair does not follow that direction, revisit the formula or radius-diameter conversion.
Split composite figures deliberately
State whether the figure is being built by addition or carved by subtraction. Write the area of each component before combining. For perimeter, count only external boundary pieces.
Connect circles to fractions and data
Semicircles and quarter-circles are geometric fractions of a whole circle. Pie charts use the same whole-circle idea to represent proportions. This creates a direct route from geometry into proportional data.
Continue through the Primary Mathematics series
For area, perimeter and units, use Measurement, Units, Perimeter, Area and Volume. For non-circular composite shapes, continue to Triangle Area and Composite Figures. For circle fractions in data, use Pie Charts and Proportional Data.
Return to the BTT Primary Mathematics Learning Hub.
Original learning guide. Use the π value and rounding instruction specified by the learner’s school or exercise. Examples are not official examination questions.