From a rectangle $ABCD$ of area $768 \text{ sq cm}$, a semicircular part with diameter $AB$ and area $72\pi \text{ sq cm}$ is removed. The perimeter of the leftover portion, in cm, is

We have a rectangle ABCD with area 768 sq cm. From this rectangle, we remove a semicircular portion whose diameter is AB (one of the sides of the rectangle). The semicircle has an area of 72π sq cm.

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The area of a semicircle is given by: Area = $\tfrac{\pi r^2}{2}$

Given that the semicircle's area is 72π sq cm:

$\tfrac{\pi r^2}{2} = 72\pi$

$\tfrac{r^2}{2} = 72$

$r^2 = 144$

$r = 12$ cm

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Since the semicircle's diameter is AB, we have:

Diameter = $2r = 2 \times 12 = 24$ cm

Therefore: AB = 24 cm

Using the rectangle's area:

Area of rectangle = length × width = AB × BC

$768 = 24 \times BC$

$BC = \tfrac{768}{24} = 32$ cm

Since ABCD is a rectangle: AD = BC = 32 cm and CD = AB = 24 cm

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After removing the semicircle, we have a leftover shape with:

Straight sides: Three complete sides of the original rectangle

BC = 32 cm

CD = 24 cm

AD = 32 cm

Curved side: Half the circumference of the removed circle

Half circumference = $\pi r = \pi \times 12 = 12\pi$ cm

When we remove the semicircle, we are left with the curved boundary of that semicircle as part of our new perimeter.

Total perimeter:

Perimeter = BC + CD + AD + curved part

$= 32 + 24 + 32 + 12\pi$

$= 88 + 12\pi$ cm

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The perimeter of the leftover portion is 88 + 12π cm.

When a curved portion is removed from a shape, the curved boundary becomes part of the new perimeter. The key is finding unknown dimensions first, then calculating the required measurement.