How to find the shaded area in a square with circular parts

Key takeaway: These composite-area questions are mostly bookkeeping. Once the radius relation is pinned down, the rest is subtraction and a little geometry.

Let the square have side length $1\text{ m}$.

To find the shaded region, first identify the pieces that are not shaded. From the description, the square is combined with a semicircle and a quarter circle, so the cleanest route is to compute each area and then subtract the unshaded parts from the square (or add the overlapping parts, depending on the diagram).

If the semicircle and quarter circle each have radius $\frac12\text{ m}$, then:

• Area of the square: $1^2=1\text{ m}^2$
• Area of a semicircle: $\frac12\pi\left(\frac12\right)^2=\frac{\pi}{8}$
• Area of a quarter circle: $\frac14\pi\left(\frac12\right)^2=\frac{\pi}{16}$

If those curved parts are outside the shaded region, then the shaded area is

$$1-\frac{\pi}{8}-\frac{\pi}{16}=1-\frac{3\pi}{16}$$

square metres.

If your diagram has the arcs drawn inside the square instead, the answer changes, so the exact placement matters.

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Pitfalls the pros know 👇 The big trap is assuming the semicircle and quarter circle both use the full side length as the radius. In many diagrams, the radius is only half the side length. Another issue is not noticing overlap: if two circular regions overlap, you cannot just add both areas blindly.

What if the problem changes? If the square side were $s$ instead of $1$, the same setup would use $s^2$ for the square and whatever radii the diagram gives for the circular parts. If all radii were $s/2$, the final expression would scale to $s^2-\frac{3\pi s^2}{16}$ when the curved pieces are excluded from the shaded region.

Tags: composite area, semicircle area, quarter circle area