The idea in brief. A circle's radius can be three shapes away from the length you are given. Read the length DOWN through the shapes, one shape at a time, and halve only at the end.
Lesson video, with the working
Shapes inside shapes
9 min 27 s
The film teaches 3 questions in turn. Each question shows first; its working is written as the film explains it.
Here is a rectangle. It is 40 cm long and 14 cm wide. Two squares are drawn inside it, one at each end. A quarter circle is drawn at every corner of each square. The part outside those 8 quarter circles is shaded. Find the area of the shaded part. (Take π = 22/7.)
Area of shaded = 560 − 308 = 252 cm²
Ans: 252 cm²
Example 1
Four identical semicircles sit in a row. They stand on line PQ. Each diameter, d, is 28 cm. Each one overlaps the next, and the 3 overlaps A, B and C are shaded. The unshaded part is 620 cm² more than the shaded part. Take π = 22/7 and find the area of the shaded part.
Area of shaded = 1 unit = 612 ÷ 3 = 204 cm²
Ans: 204 cm²
Example 2
A rectangle is 45 cm long and 12 cm wide. Three squares sit in a row inside it. One semicircle stands on the top side of each square, one on the bottom. They meet at the middle. All 6 semicircles are shaded. Find the area that is NOT shaded. (Take π = 3.14.)
Area of unshaded = 540 − 339.12 = 200.88 cm²
Ans: 200.88 cm²
How to solve it
Before any pi, read the lengths down through the shapes. The rectangle is 14 wide, and that 14 is the side of each square.
On that side 2 quarter circles meet, so 14 is a whole diameter. The radius is 14 divided by 2, which is 7.
Piece the 4 quarter circles in one square into 1 whole circle: 22 over 7 times 7 times 7 is 154 square centimetres.
Two squares give 2 circles: 2 times 154 is 308 square centimetres. The rectangle is 40 times 14, which is 560.
The shaded part is the rectangle with the 2 circles taken out: 560 minus 308 is 252 square centimetres.
When semicircles overlap, adding them all counts each orange overlap twice. And always read the question again: shaded, or the part that is not shaded.
Read the question again. It wants the part that is not shaded. So take the orange off the whole rectangle.
Practice questions with worked solutions
Question 1 is free to try. The other 4 open in the free trial, each with its full working.
Question 1 of 5 · Intermediate · free to tryThe figure shows a square of side 28 cm with a circle drawn inside it…The figure shows a square of…
The figure shows a square of side 28 cm with a circle drawn inside it, touching all four sides. Inside the circle, four identical quarter circles are drawn, each with its centre at a point where the circle touches the square, and each with a radius of half the radius of the circle. The part of the circle outside the four quarter circles is shaded. Find the area of the shaded part. (Take π = 22/7.)
four quarter circles = one circle = 22/7 × 7 × 7 = 154
shaded = 616 − 154 = 462
Answer: 462 cm²
For parents
How do you solve circles inside squares shaded area questions in P6?
These questions put circles, semicircles or quarter circles inside squares, rectangles or larger semicircles, and ask for a shaded or unshaded part.
What your child learns here
These questions put circles, semicircles or quarter circles inside squares, rectangles or larger semicircles, and ask for a shaded or unshaded part. They apply the Primary 6 outcome on composite figures. The lengths must be read through two or three shapes before any area is found.
Carry a length through nested shapes: the side of a square is the diameter of the circle that fits inside it.
Halve more than once when a long line is split into equal diameters.
Spot when four quarter circles inside a square make one whole circle.
Find a shaded part by subtracting the inner shapes from the shape that contains them.
Read carefully whether the question wants the shaded part or the part left unshaded.
Does the answer make sense?
A shaded part made by taking inner shapes away must be smaller than the outer shape holding them. A circle that just fits in a square covers a little over three quarters of it, so your child should find the four corners left over a little under a quarter of that square.
How to practise these questions
Intermediate questions train halving a length more than once to reach the radius of a smaller circle, and seeing one length shared by a frame, a square and a circle. Challenge questions go three shapes deep, ask for the region that is not shaded, or count layers where areas overlap. If shapes inside shapes are hard to separate, read the written step-by-step solution together.
Questions parents ask
How do you find the shaded area when circles are inside a square? Work out the square's area, then the area of the circles inside it, and subtract. Before that, use the figure to get the radius: a circle that fits exactly inside a square has a diameter equal to the side.
What does it mean when a question asks for the unshaded part? It asks for the region that is not coloured in. The same figure can have either part asked for, so read the last line of the question before deciding what to subtract from what.
Do these questions need anything beyond the P6 syllabus? No. They use the Primary 6 outcomes on circle area and on composite figures made from squares, rectangles, triangles, semicircles and quarter circles. The extra step is reading lengths through several shapes, not new content.
