Primary 6 Maths | Composite Circle Shapes: Teacher Prep Guide | From circle area to shaded regions in 45 minutes

2026-11-08

Tutor lesson-prep guides · Primary Maths

Circle area on its own is easy. What breaks pupils is a semicircle glued to a rectangle, because they multiply the wrong number. This chapter teaches one move: mark which arc belongs to which circle and what the radius is, then decide add or subtract.

Objectives — Pupils can: (1) get the area and arc length of a semicircle or quarter circle from a radius or diameter; (2) remember that a semicircle perimeter includes the diameter; (3) find the four shaded corners around an inscribed circle by subtracting the circle from the square; (4) tell a ring (big circle minus small circle) from a running track (rectangle plus two semicircles).

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Circumference	A circle of radius 7 cm, circumference with pi as 3.14?	43.96 cm
Area of a circle	Same circle, area?	153.86 square cm
Diameter and radius	Diameter 20 cm, radius?	10 cm
Taking a fraction of a shape	One quarter of a whole circle	Area divided by 4
Area of a square	Side 14 cm	196 square cm

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-7 min	Starter: three circles from one centre	Draw radii 3, 6 and 10 and ask which number decides the area	Say it is the radius
7-16 min	Semicircle area and perimeter	Radius 7: area 76.93, arc 21.98, perimeter adds the diameter 14	Write 35.98 and circle the diameter
16-25 min	Quarter circle	Radius 10: area 78.5, arc 15.7	Say whether dividing by 4 applies to area or arc
25-33 min	Shaded corners	Square of side 14 with an inscribed circle of radius 7: 196 minus 153.86 = 42.14	Write the subtraction and name what is removed
33-41 min	Rings and running tracks	Ring 3.14 x (100 minus 36); track is a rectangle plus two semicircles that make one circle	Say which part is added and which is subtracted
41-45 min	Exit check	Three quick questions, mark the radius first	Answer independently

03 Teaching script (read it out)

Scene 1: Mark the radius before anything else

The most common error is feeding the diameter into the formula. Set the rule in the first minute: the first mark on any circle question is r written on the diagram. Diameter 20 becomes 10, and circle it.
Every later line then starts from the same number, which pupils can check themselves.

Ask: what is the radius here, and where have you written it?

Scene 2: A semicircle perimeter always gains a diameter

Pupils give the arc length and stop, losing the straight edge. Use a physical picture: a semicircular flower bed needs a fence along the curve and along the straight side too.
Radius 7: arc 21.98 plus diameter 14 gives 35.98 cm. After every perimeter answer, pupils ask themselves whether the shape is closed.

Ask: how many edges does this semicircle need fenced?

Scene 3: Shaded area is big minus small

Pupils hunt for a formula for the shaded part, fail, and give up. Teach the general move: treat shaded plus empty as one big shape, then subtract the empty part you can calculate.
In a square of side 14 with an inscribed circle, the corners are 196 minus 153.86, so 42.14 square cm. Always name the big shape and the small one first.

Ask: what is the big shape, and what are we taking away?

Scene 4: Join pieces before calculating

A running track is a rectangle plus two semicircles. Calculating piece by piece invites slips. Show that the two semicircles join into one circle whose radius is half the rectangle breadth.
A 20 by 14 track: area 280 plus 153.86 is 433.86; perimeter 40 plus 43.96 is 83.96. Pupils should say the two semicircles make one circle.

Ask: what do the two semicircles make together, and what is the radius?

04 Question types and worked steps

Type	Example	Steps	Answer
Semicircle	Radius 7 cm	Whole circle area divided by 2; perimeter is arc plus diameter	76.93 square cm; 35.98 cm
Quarter circle	Radius 10 cm	Whole area divided by 4; arc is circumference divided by 4	78.5 square cm; arc 15.7 cm
Shaded corners	Square of side 14 cm with an inscribed circle of radius 7	Square minus circle	42.14 square cm
Ring and track	Ring with radii 10 and 6; track 20 by 14	Big circle minus small circle; rectangle plus one circle	200.96 square cm; 433.86 square cm

Sanity check (teach this): Reality checks: the radius is half the diameter, always check before multiplying; a semicircle area is less than the whole circle; shaded area is less than the big shape; a ring answer must be positive because the outer radius is larger; perimeter is in cm and area in square cm and the two never add; the question says to use pi as 3.14, so use 3.14 and not the calculator constant.

05 Independent practice (3 questions)

Question	Answer	What to watch
Three semicircle and quarter items	Two areas and one perimeter	Check the diameter is added back
Two shaded items	One inscribed circle, one ring	Check the big and small shapes are named first
Two track items	One area, one perimeter	Check the two semicircles are joined into one circle

06 Common mistakes and what to say

Mistake	Student writes	Say this
Diameter fed into the area formula	3.14 x 14 x 14	The formula wants the radius. You used the diameter, so your answer is four times too big.
Semicircle perimeter without the diameter	Answering 21.98 cm	The flower bed still has a straight edge. Is it closed?
Forcing a formula on the shaded part	14 x 14 divided by 2 and similar guesses	The shaded part has no formula. Find a big shape and subtract the part you can calculate.
Adding radii for a ring	3.14 x (10 + 6)	You are cutting a hole out, not adding two circles. Big circle minus small circle.

07 Homework (paper, section and question numbers)

Primary 6 maths paper drills — Section on what each paper tests: eight circle and composite items, mark the radius on every diagram first
Primary maths test papers with answers — Pick a paper with Primary 6 circle and shaded area items, fifteen minutes, then name the big shape for every item when marking

08 Exit check (last 5 minutes)

Question	Answer	Decision
Semicircle of radius 7 cm, perimeter?	21.98 + 14 = 35.98 cm	Wrong: reteach closing the shape
Square of side 14 cm with an inscribed circle, shaded corners?	196 – 153.86 = 42.14 square cm	Wrong: reteach big minus small
Track 20 by 14, perimeter?	40 + 43.96 = 83.96 cm	Wrong: reteach joining the semicircles

09 Tutor FAQ

Why insist on 3.14?

The Hong Kong primary curriculum uses pi as 3.14 and questions usually say so. Using the calculator constant changes the last digits and gets marked wrong, so the whole lesson uses 3.14.

Is 45 minutes enough?

Yes. There are only two judgements all lesson: where is the radius, and is this part added or subtracted. The rest of the time goes on drawing and checking units.

How is this different from the circle chapter?

That chapter has a single circle. Here two or three shapes overlap, so the work is split then add or subtract. Pupils who scored full marks there still lose marks here, because the failure point is splitting, not the formula.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.