To find the overlap of two shapes, add their individual areas and subtract the total area they cover together. The sum counts the shared region twice. The combined area counts it once, so the difference is the overlap.
If two rectangular cards have areas of 96 cm² and 60 cm² and together cover 132 cm², the overlap is 24 cm². Shade that region on a sketch before writing the calculation.
- Find the area of each whole shape.
- Identify the area covered by at least one shape, with the overlap counted once.
- Subtract the combined area from the sum of the two areas.
CHAPTER 1 OF 4 · Back to contents
Separate the picture into three regions
Imagine a red rectangle partly covering a blue rectangle. Separate the picture into red only, blue only and the shared region. The red rectangle includes red only plus the shared region. The blue rectangle includes blue only plus the shared region.
When you add the two rectangle areas, the shared part appears in both totals. The combined outline covers the same three parts once each. Subtracting that outline’s area leaves one copy of the shared part.
CHAPTER 2 OF 4 · Back to contents
Worked example | Find the area covered by both cards
The red card measures 12 cm by 8 cm. The blue card measures 10 cm by 6 cm. Together they cover 132 cm². Find the area where the cards overlap.
Red card area = 12 × 8 = 96 cm². Blue card area = 10 × 6 = 60 cm². Their sum is 156 cm². The combined area is 132 cm², so the overlap is 156 − 132 = 24 cm².
Check by separating the parts. Red only is 96 − 24 = 72 cm². Blue only is 60 − 24 = 36 cm². Including the overlap gives 72 + 36 + 24 = 132 cm².
CHAPTER 3 OF 4 · Back to contents
Use the overlap area to find a length
Suppose the diagram also shows that the overlapping region is a rectangle 6 cm high. Its width is 24 ÷ 6 = 4 cm. Use the height of the overlapping region from the diagram. The height of an entire card may be different.
Now try two shapes with areas of 48 cm² and 35 cm² that cover 70 cm² together. Their overlap is 48 + 35 − 70 = 13 cm². Ask your child to colour the shared area and explain where it appears in the calculation.
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
What if the shapes just touch at an edge?
The shared edge has no area. The total covered area is the sum of the two individual areas.
Could the overlap be bigger than either shape?
It can be at most the area of the smaller shape, since the overlap must fit inside both shapes.
Can I use the same calculation for a rectangle overlapping a triangle?
Yes. Calculate the individual areas with their own formulas. Then subtract the combined area from their sum to find the overlap.
Contents · Official syllabus and next practice
Official syllabus and next practice
Read the official MOE Primary Mathematics syllabus
For another primary worked example, read how to find an original price after two discounts.