Question

problème 1: détermine laire de la figure suivante sachant que son périmètre mesure 24π cm.

Explanation:

Step1: Analyze the perimeter

The figure's perimeter is made of 4 semicircles, which is equivalent to 2 full circles. Let the radius of each semicircle be \( r \). The circumference of a circle is \( 2\pi r \), so the perimeter \( P = 2\times2\pi r = 4\pi r \). We know \( P = 24\pi \), so \( 4\pi r = 24\pi \).

Step2: Solve for radius \( r \)

Divide both sides by \( 4\pi \): \( r=\frac{24\pi}{4\pi}=6 \) cm. The diameter \( d = 2r = 12 \) cm, which is also the side length of the square in the middle.

Step3: Calculate the area of the figure

The area of the figure is the area of the square plus the area of 2 full circles (since 4 semicircles make 2 circles). The area of the square \( A_{square}=d^2 = 12^2 = 144 \) \( cm^2 \). The area of one circle is \( \pi r^2=\pi\times6^2 = 36\pi \), so the area of 2 circles is \( 2\times36\pi = 72\pi \). Then the total area \( A = 144 + 72\pi \) \( cm^2 \). Wait, no—wait, actually, looking at the figure, the 4 semicircles: each semicircle has radius \( r \), and the square has side \( 2r \). Wait, maybe I made a mistake earlier. Let's re - examine. The perimeter: each of the four curved parts is a semicircle. So the perimeter is \( 4\times\frac{1}{2}\times2\pi r=4\pi r \). Given \( 4\pi r = 24\pi \), so \( r = 6 \). Then the square has side length \( 2r=12 \). Now, the area of the figure: the square plus the area of the four semicircles. The four semicircles make two full circles. So area of square: \( (2r)^2=(12)^2 = 144 \). Area of two circles: \( 2\times\pi r^2=2\times\pi\times6^2 = 72\pi \). Wait, but maybe the figure is composed of a square and four semicircles, but actually, when you look at the figure, the four semicircles: each has diameter equal to the side of the square. So radius \( r=\frac{\text{side of square}}{2} \). Let the side of the square be \( s \), then the radius of each semicircle is \( \frac{s}{2} \). The perimeter is \( 4\times(\frac{1}{2}\times2\pi\times\frac{s}{2})=4\times(\frac{\pi s}{2}) = 2\pi s \). Wait, this is a contradiction with the earlier. Wait, the original problem says the perimeter is \( 24\pi \). Let's start over.

Let's assume the figure is a square with four semicircular "bulges" on each side. So each side of the square has a semicircle attached to it. So the perimeter of the figure is the sum of the lengths of the four semicircles. Each semicircle has a diameter equal to the side length of the square, let's call the side length of the square \( s \). The length of one semicircle is \( \frac{1}{2}\times\pi\times s \) (since the circumference of a full circle is \( \pi d=\pi s \), so semicircle is \( \frac{\pi s}{2} \)). There are four semicircles, so the perimeter \( P = 4\times\frac{\pi s}{2}=2\pi s \). We know \( P = 24\pi \), so \( 2\pi s=24\pi \), which gives \( s = 12 \) cm. Then the radius of each semicircle is \( r=\frac{s}{2}=6 \) cm.

Now, the area of the figure: the area of the square plus the area of the four semicircles. The four semicircles make two full circles (because \( 4\times\frac{1}{2}=2 \)). The area of the square is \( s^2 = 12^2=144 \) \( cm^2 \). The area of one circle is \( \pi r^2=\pi\times6^2 = 36\pi \), so the area of two circles is \( 2\times36\pi = 72\pi \). Therefore, the total area \( A=144 + 72\pi \)? Wait, no, wait. Wait, if the four semicircles: each has radius \( r = 6 \), and the square has side \( s = 12 \). Wait, another way: the figure can be seen as a square with side \( 2r \) (where \( r \) is the radius of the semicircles) and four semicircles with radius \( r \). So the area is \( (2r)^2+4\times\frac{1}{2}\times\pi r^2=4r^2 + 2\pi r^2 \). We found \( r = 6 \), so \( 4\times36+2\pi\times36=144 + 72\pi \). But let's check the perimeter again. If \( r = 6 \), the diameter of each semicircle is \( 12 \), so the side of the square is \( 12 \). The length of each semicircle is \( \frac{1}{2}\times2\pi r=\pi r = 6\pi \). Four semicircles: \( 4\times6\pi = 24\pi \), which matches the given perimeter. So that's correct.

Wait, but maybe the figure is composed of a square and two circles? Wait, four semicircles: two on the top and bottom, two on the left and right. So top and bottom semicircles: each has diameter equal to the side of the square, so together they make a full circle. Left and right semicircles: each has diameter equal to the side of the square, so together they make a full circle. So total of two full circles. And the square in the middle. So area is square area plus two circle areas. Square area: \( s^2 \), circle area: \( \pi r^2 \), and \( s = 2r \). From perimeter: four semicircles, each with length \( \pi r \), so \( 4\pi r=24\pi \), so \( r = 6 \), \( s = 12 \). So square area: \( 12\times12 = 144 \), two circles: \( 2\times\pi\times6^2=72\pi \), so total area \( 144 + 72\pi \) \( cm^2 \). If we take \( \pi\approx3.14 \), \( 72\pi\approx226.08 \), \( 144 + 226.08 = 370.08 \), but maybe we can leave it in terms of \( \pi \). Wait, but maybe I made a mistake in the figure analysis. Let's look at the figure again: it's a square with four semicircles, one on each side. So the perimeter is the sum of the four semicircles, which is \( 4\times\frac{1}{2}\times2\pi r=4\pi r \) (if \( r \) is the radius of the semicircles). Given \( 4\pi r = 24\pi \), so \( r = 6 \). Then the side of the square is \( 2r = 12 \). The area of the figure is the area of the square plus the area of the four semicircles. The four semicircles have a total area equal to the area of two full circles (since \( 4\times\frac{1}{2}=2 \)). So area of square: \( 12\times12 = 144 \), area of two circles: \( 2\times\pi\times6^2=72\pi \), so total area \( A = 144 + 72\pi \) \( cm^2 \). If we want a numerical value, \( 72\pi\approx72\times3.14 = 226.08 \), so \( 144+226.08 = 370.08 \) \( cm^2 \). But let's confirm the perimeter again. Each semicircle has radius \( 6 \), so the length of one semicircle is \( \pi r=6\pi \). Four semicircles: \( 4\times6\pi = 24\pi \), which matches the given perimeter. So the area calculation is correct.

Answer:

The area of the figure is \( \boldsymbol{144 + 72\pi} \) \( cm^2 \) (or approximately \( \boldsymbol{370.08} \) \( cm^2 \) if \( \pi\approx3.14 \)).