THE circular crown area is determined by the difference between the area of the larger circle and the area of the smaller circle.
Crown area = πR² – πr²
Crown area = π. (R² - r²)
See below a list of exercises on circular crown area, all solved step by step.
Index
- Exercises on circular crown area
- Resolution of question 1
- Resolution of question 2
- Resolution of question 3
- Resolution of question 4
Exercises on circular crown area
Question 1. Determine the area of a circular crown bounded by two concentric circles of radius 10 cm and 7 cm.
Question 2. Calculate the area of the region colored green in the figure below:
Question 3. In a park with a circular shape, you want to build a walking path around it. The park's current diameter is 42 meters and the track area will be 88π m². Determine the width of the walking path.
Question 4. Determine the area of a circular crown formed by an inscribed circle and a circumscribed circle in a square with a diagonal equal to 6 m.
Resolution of question 1
We have R = 10 and r = 7. Applying these values to the formula for the circular crown area, we have to:
Crown area = π. (10² – 7²)
⇒ Crown area = π. (100 – 49)
⇒ Crown area = π. 51
Considering π = 3.14, we have that:
Crown area = 160.14
Therefore, the area of the circular crown is equal to 160.14 cm².
Resolution of question 2
From the illustration, we have two circles with the same center, with radii r = 5 and R = 8, and the green area is the area of a circular crown.
Applying these values to the formula for the circular crown area, we have to:
Crown area = π. (8² – 5²)
⇒ Crown area = π. (64 – 25)
⇒ Crown area = π. 39
Considering π = 3.14, we have that:
Crown area = 122.46
Therefore, the area of the circular crown is equal to 122.46 cm².
Resolution of question 3
From the information given, we built a representative design:
From the illustration, we can see that the width of the track corresponds to the radius of the larger circle minus the radius of the smaller circle, ie:
Width = R - r
We know that the diameter of the green park (circle) is equal to 42 meters, so r = 21 m. Thus:
Width = R – 21
However, we need to find the value of R. We know that the crown area is 88π m², so let's substitute this value into the crown area formula.
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Crown area = π. (R² - r²)
⇒ 88π = π. (R² - 21²)
⇒ 88 = R² - 21²
⇒ R² = 88 + 21²
⇒ R² = 88 + 441
⇒ R² = 529
⇒ R = 23
Now, we determine the width of the walking path:
Width = R - 21 = 23 - 21 = 2
Therefore, the width of the track is equal to 2 meters.
Resolution of question 4
From the information given, we built a representative design:
Note that the radius of the larger circle is half the diagonal of the square, ie:
R = d/2
As d = 6 ⇒ R = 6/2 ⇒R = 3.
The radius of the smaller circle corresponds to half the measure of the L side of the square:
r = L/2
However, we do not know the square side measurement and we need to determine it first.
Fur Pythagorean theorem, it can be seen that the diagonal and the side of the square are related as follows:
d = L√2
Since d = 6 ⇒6 = L√2 ⇒L = 6/√2.
Therefore:
r = 6/2√2 ⇒ r = 3/√2.
We can already calculate the area of the circular crown:
Crown area = π. (R² - r²)
⇒ Crown area = π. (3² – (3/√2)²)
⇒ Crown area = π. (9 – 9/2)
⇒ Crown area = π. 9/2
Considering π = 3.14, we have that:
Crown area = 14.13
Therefore, the area of the circular crown is equal to 14.13 m².
To download this circular crown area list in PDF, click here!
You may also be interested:
- Exercises on Equation of the Circumference
- Circumference Length Exercises
- elements of the circle
- Difference between circumference, circle and sphere
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