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Math  /  Geometry

QuestionA rondavel (hut / circular dwelling) has a square woven carpet covering the floor as shown in the diagram below. The circumference of the rondavel is 113,1 metres.
Calculate the area of the floor which is not covered by the carpet.

Studdy Solution

STEP 1

1. The rondavel is perfectly circular.
2. The square carpet is placed diagonally inside the circle.
3. The circumference of the rondavel is 113.1 113.1 meters.

STEP 2

1. Calculate the radius of the rondavel.
2. Calculate the area of the circular floor.
3. Determine the side length of the square carpet.
4. Calculate the area of the square carpet.
5. Calculate the area of the floor not covered by the carpet.

STEP 3

Calculate the radius of the rondavel using the circumference formula:
C=2πr C = 2\pi r
Given C=113.1 C = 113.1 meters, solve for r r :
r=C2π=113.12π r = \frac{C}{2\pi} = \frac{113.1}{2\pi}

STEP 4

Calculate the area of the circular floor using the radius:
Acircle=πr2 A_{\text{circle}} = \pi r^2
Substitute the radius from Step 1:
Acircle=π(113.12π)2 A_{\text{circle}} = \pi \left(\frac{113.1}{2\pi}\right)^2

STEP 5

Determine the side length of the square carpet. Since the carpet is placed diagonally, the diagonal of the square is equal to the diameter of the circle:
Diagonal of square=2r \text{Diagonal of square} = 2r
Using the relationship between the diagonal and side of a square, s2=Diagonal s\sqrt{2} = \text{Diagonal} , solve for s s :
s=2r2=r21 s = \frac{2r}{\sqrt{2}} = \frac{r\sqrt{2}}{1}

STEP 6

Calculate the area of the square carpet:
Asquare=s2 A_{\text{square}} = s^2
Substitute the side length from Step 3:
Asquare=(r21)2=2r2 A_{\text{square}} = \left(\frac{r\sqrt{2}}{1}\right)^2 = 2r^2

STEP 7

Calculate the area of the floor not covered by the carpet:
Auncovered=Acircle−Asquare A_{\text{uncovered}} = A_{\text{circle}} - A_{\text{square}}
Substitute the areas from Steps 2 and 4:
Auncovered=πr2−2r2 A_{\text{uncovered}} = \pi r^2 - 2r^2 =r2(π−2) = r^2(\pi - 2)
Substitute the radius from Step 1:
r=113.12π r = \frac{113.1}{2\pi}
Auncovered=(113.12π)2(π−2) A_{\text{uncovered}} = \left(\frac{113.1}{2\pi}\right)^2 (\pi - 2)
Calculate the numerical value:
Auncovered≈1017.88 m2 A_{\text{uncovered}} \approx 1017.88 \text{ m}^2
The area of the floor not covered by the carpet is approximately:
1017.88 m2 \boxed{1017.88 \text{ m}^2}

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