The idea in brief. Two regions that carry the same piece differ by exactly what their WHOLE SHAPES differ by. Write that down before a single number. In the lesson. Write the two regions before you write a number, and the difference is the difference of the two whole shapes.
Lesson video, with the working
The difference of two regions
4 min 44 s
The film teaches 2 questions in turn. Each question shows first; its working is written as the film explains it.
A square ABCD of side 20 cm. A circle of radius 10 cm has its centre at the corner B. Region X is the part of the square outside the circle. Region Y is the part of the circle outside the square. Find the difference between the areas of X and Y. (Take π = 3.14.)
Area of X − Area of Y = 400 − 314 = 86 cm²
Ans: 86 cm²
Example 1
A square ABCD of side 18 cm. A quarter circle of radius 18 cm is drawn from corner A, and the diagonal AC is drawn. X is the quarter circle outside triangle ABC, and Y is the triangle outside the quarter circle. Find the difference between X and Y. (Take π = 3.14.)
Area of X − Area of Y = 254.34 − 162 = 92.34 cm²
Ans: 92.34 cm²
How to solve it
Name the shared middle piece S. It is in the square and in the circle at once.
Write each orange region as its whole shape with S taken out: X is the square minus S, and Y is the circle minus S.
Take one from the other. S goes from both, so the area of X minus the area of Y is the square minus the circle.
Work out only the two whole shapes: 20 times 20 is 400 square centimetres, and 3.14 times 10 times 10 is 314 square centimetres.
Subtract: 400 minus 314 is 86 square centimetres.
Write the two regions before you write a number, and the difference is the difference of the two whole shapes.
Practice questions with worked solutions
Question 1 is free to try. The other 5 open in the free trial, each with its full working.
Question 1 of 6 · Intermediate · free to tryThe figure shows a semicircle of diameter 40 cm overlapping a square. Region…The figure shows a semicircle of…
The figure shows a semicircle of diameter 40 cm overlapping a square. Region X is the part of the semicircle that lies outside the square, and region Y is the part of the square that lies outside the semicircle. The area of X is 228 cm² more than the area of Y. Find the length of one side of the square. (Take π = 3.14.)
How do you find the overlapping circles shaded area difference in P6?
Two shapes overlap, and the question asks how much bigger one shaded region is than the other.
What your child learns here
Two shapes overlap, and the question asks how much bigger one shaded region is than the other. The overlap belongs to both shapes, so it cancels, and only the two whole shapes are needed. This applies the Primary 6 composite figures outcome to an exam-style problem.
Name the shared piece and write each shaded region as a whole shape minus that piece.
See that the shared piece cancels when one region is subtracted from the other.
Find the difference using only the areas of the two whole shapes.
Run the method backwards when the difference is given, to find a missing side or radius.
Recognise that a region with no measurable edge is a signal to use the difference method.
Does the answer make sense?
When your child finds how much bigger one shaded region is than the other, the bigger region belongs to the shape with the bigger whole area. The answer must also be less than that bigger whole area, so comparing the two whole shapes first shows which region wins and how large it can be.
How to practise these questions
The Intermediate questions train the idea that the shared piece cancels when one region is taken from the other. Challenge questions apply this to two curved shapes, include information the answer never uses, or work back from a given difference. When the difference of two regions seems impossible, read the written step-by-step solution.
Questions parents ask
How do you find the difference between two shaded areas that overlap? Write each shaded region as its whole shape minus the overlap. When one is subtracted from the other, the overlap cancels, so the difference is just the difference of the two whole shapes' areas.
Do we need to find the area of the overlapping part? No. The overlap is inside both shapes, so it is removed from both and does not change the difference. In many of these figures it cannot be found with Primary 6 methods at all.
Is this method in the P6 syllabus? The syllabus lists finding the area of composite figures made up of squares, rectangles, triangles, semicircles and quarter circles. The cancelling idea is a way of applying that outcome, explained step by step in the lesson notes on this page.
