Overlapping Areas: Count the Shared Region Once

Two mats can cover one floor patch together, but the shared patch belongs to both mats. If you add the two mat areas without correcting it, that patch is counted twice. Add first, then subtract the shared region once.

Parents: ask your child to shade the shared patch and decide how many times it appears in the first addition before showing the correction.

By the end you'll be able to split a covered region into rectangles, remove a double-counted overlap, and check that each floor patch is counted once.

See the shared region

The overlap is the region shared by two shapes. If the full area of shape A and the full area of shape B are added, the overlap appears in both totals. The covered area is therefore:

Let A and B be the two whole areas, including their shared patch.

The overlap relationship

$$\text{Covered} = A + B - \text{overlap}$$

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Count the shared patch once

🤔 Predict first: If two mats overlap, how many copies of the shared patch belong in the covered floor total?

One copyTwo copiesNo copies

Worked example: two mats on one floor

Here is the complete question:

A blue mat is 8 m by 6 m. A yellow mat is 7 m by 5 m. They overlap in a rectangle 3 m by 4 m. Find the total floor area covered by at least one mat.

Find both full mat areas, then find the shared patch. The phrase “at least one” means the overlap should appear once in the final answer.

Find the three areas

$$\text{Blue} = 8 \times 6 = 48\,\text{m}^2$$

$$\text{Yellow} = 7 \times 5 = 35\,\text{m}^2$$

$$\text{Overlap} = 3 \times 4 = 12\,\text{m}^2$$

Remove the double count

$$\text{Covered area} = 48 + 35 - 12 = 71\,\text{m}^2$$

The mats cover 71 m² of floor. The subtraction removes one copy of the shared 12 m² patch, leaving one copy in the total.

Check by using non-overlapping parts

Another route is to split the floor into blue-only, shared and yellow-only regions. These pieces do not overlap, so adding them should give the same answer.

Check each floor patch once

$$\text{Blue only} = 48 - 12 = 36\,\text{m}^2$$

$$\text{Shared} = 12\,\text{m}^2$$

$$\text{Yellow only} = 35 - 12 = 23\,\text{m}^2$$

$$36 + 12 + 23 = 71\,\text{m}^2$$

The check works because every covered patch appears exactly once in the second method.

Graduated practice

Try each question before opening its solution. Sketch the shapes and shade the shared region first.

Practice 1 — start here

Two posters cover 30 cm² and 24 cm². Their overlap is 6 cm². Find the area covered by at least one poster.

Show the solution

Subtract the shared region once because it was included in both full areas.

Working

$$\text{Covered area} = 30 + 24 - 6 = 48\,\text{cm}^2$$

The covered area is 48 cm².

Practice 2 — identify the double count

Two rectangular garden beds have a combined summed area of 96 m². Their shared strip is 14 m². Find the union area and the area counted twice in the simple sum.

Show the solution

The simple sum counts the shared strip twice. Remove one copy for the union, while the amount counted twice is the shared strip itself.

Working

$$\text{Union area} = 96 - 14 = 82\,\text{m}^2$$

$$\text{Area counted twice} = 14\,\text{m}^2$$

The union area is 82 m², and 14 m² was counted twice in the simple sum.

Practice 3 — no overlap

Two floor mats have areas 18 m² and 25 m² and do not touch. Find the total area covered.

Show the solution

There is no shared region, so the overlap is zero.

Working

$$\text{Covered area} = 18 + 25 - 0 = 43\,\text{m}^2$$

The total covered area is 43 m².

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

1. Two mats have areas 48 m² and 35 m² and overlap by 12 m². What area do they cover together?

   95 m²71 m²60 m²

2. Why is the overlap subtracted?

   It was counted in both full areasIt is outside both shapesIt has no area

3. Rectangle A is 30 m², B is 24 m² and the overlap is 6 m². What is the covered area?

   60 m²54 m²48 m²

4. Areas are 54 m² and 42 m², and the covered total is 82 m². What is the overlap?

   96 m²14 m²82 m²

5. Two mats do not touch and have areas 18 m² and 25 m². What correction is needed?

   Subtract 18 m²Subtract 25 m²Subtract 0 m²

6. Which check counts every covered patch exactly once?

   Blue-only + shared + yellow-onlyBlue full + yellow full + sharedShared only

7. If the overlap grows while both full mat areas stay fixed, what happens to the covered area?

   It increases by the same overlapIt decreasesIt becomes the overlap only

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