Area of multiple circles inside a rectangle

In summary, the conversation discusses a figure with six identical circles inside a rectangle. The circles have a radius of 24 cm and are arranged in a pattern where they touch at least two other circles and the top and bottom rows touch the rectangle's sides. The centers of the circles form a single triangle's perimeter. The total area of the six circles is approximately 57.495% of the rectangle's area, which can be found by calculating the rectangle's area and the circles' area.
  • #1
Help seeker
15
0
Figure shows six identical circles inside a rectangle.
ScreenShot_20210317093706.png

The radius of each circle is 24 cm. The radius of the circles is the greatest possible radius so that the circles fit inside the rectangle. The six circles form the pattern shown in Figure so that
• each circle touches at least two other circles
• the circle in the top row of the pattern and the circles in the bottom row of the pattern touch at least one side of the rectangle
• the centres of the circles all lie on the perimeter of a single triangle.

Find the total area of the $six$ $circles$ $as$ $a$ $percentage$ $of$ $the$ $area$ $of$ $the$ $rectangle$.
 
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  • #2
Height of triangle = √(96²-48²) ≈ 83.1384387633 cm
Area of rectangle ≈ 144×(83.1384387633+48) ≈ 18883.9351819 cm²
Area of 6 circles ≈ 10857.34422 cm²
Area of 6 circles as a percentage of area of rectangle ≈ 57.49513603 %
 
  • #3
phymat said:
Height of triangle = √(96²-48²) ≈ 83.1384387633 cm
Area of rectangle ≈ 144×(83.1384387633+48) ≈ 18883.9351819 cm²
Area of 6 circles ≈ 10857.3442\(\displaystyle {\color{red}2}\) cm²
Area of 6 circles as a percentage of area of rectangle ≈ 57.49513\(\displaystyle {\color{red}603}\) %
Minor oversight or calculator algorithm difference.
Those might have been
Area of 6 circles ≈ 10857.3442108
and
Area of 6 circles as a percentage of area of rectangle ≈ 57.4951359778
 
  • #4
Tnx
Solved
 

What is the formula for finding the area of multiple circles inside a rectangle?

The formula for finding the area of multiple circles inside a rectangle is to first calculate the area of each individual circle using the formula A = πr^2, where r is the radius of the circle. Then, add up the areas of all the circles and subtract it from the area of the rectangle.

How do you determine the radius of each circle when given the diameter?

To determine the radius of each circle when given the diameter, simply divide the diameter by 2. This is because the diameter is the distance across the circle, while the radius is the distance from the center to the edge of the circle.

What is the importance of finding the area of multiple circles inside a rectangle?

Finding the area of multiple circles inside a rectangle can be useful in real-world applications, such as calculating the total area of circular objects in a rectangular space. It can also be used in geometry and mathematics to practice applying formulas and solving complex problems.

Can the formula for finding the area of multiple circles inside a rectangle be applied to other shapes?

Yes, the formula can be applied to other shapes as long as the shapes are circular and fit inside a rectangle. The key is to calculate the area of each individual circle and then add them together.

Are there any limitations to using the formula for finding the area of multiple circles inside a rectangle?

One limitation is that the circles must be fully contained within the rectangle. If any part of a circle extends beyond the boundaries of the rectangle, it cannot be included in the calculation. Additionally, the formula assumes that the circles do not overlap each other.

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