Send me the photo of ICSE class 9 maths project cutting a circle into equal sections of Central angle to find area of circle
Answers
Answer:
Area of a Circle by Cutting into Sectors
Here is a way to find the formula for the area of a circle:
Cut a circle into equal sectors (12 in this example)
Divide just one of the sectors into two equal parts. We now have thirteen sectors – number them 1 to 13:
Rearrange the 13 sectors like this:
Which resembles a rectangle:
What are the (approximate) height and width of the rectangle?
The height is the circle's radius: just look at sectors 1 and 13 above. When they were in the circle they were "radius" high.
The width (actually one "bumpy" edge) is half of the curved parts around the circle ... in other words it is about half the circumference of the circle.
We know that:
Circumference = 2 × π × radius
And so the width is about:
Half the Circumference = π × radius
And so we have (approximately):
radius
π × radius
Now we just multply the width by the height to find the area of the rectangle:
Area = (π × radius) × (radius)
= π × radius2
Note: The rectangle and the "bumpy edged shape" made by the sectors are not an exact match.
But we could get a better result if we divided the circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).
And the more we divided the circle up, the closer we get to being exactly right.
Conclusion
Area of Circle = π r2
Area of a Circle by Cutting into Sectors
Here is a way to find the formula for the area of a circle:
Cut a circle into equal sectors (12 in this example)
Divide just one of the sectors into two equal parts. We now have thirteen sectors – number them 1 to 13:
Rearrange the 13 sectors like this:
Which resembles a rectangle:
What are the (approximate) height and width of the rectangle?
The height is the circle's radius: just look at sectors 1 and 13 above. When they were in the circle they were "radius" high.
The width (actually one "bumpy" edge) is half of the curved parts around the circle ... in other words it is about half the circumference of the circle.
We know that:
Circumference = 2 × π × radius
And so the width is about:
Half the Circumference = π × radius
And so we have (approximately):
radius
π × radius
Now we just multply the width by the height to find the area of the rectangle:
Area = (π × radius) × (radius)
= π × radius2
Note: The rectangle and the "bumpy edged shape" made by the sectors are not an exact match.
But we could get a better result if we divided the circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).
And the more we divided the circle up, the closer we get to being exactly right.
Conclusion
Area of Circle = π r2
As performed in real lab:
Materials required:
Coloured paper, compass, scale, a pair of scissors, gum, colours
Procedure:
Draw a circle of radius r = 4 cm (say) on the paper.
Divide the circle into 16 equal parts. [Fig (a)]
Cut all the 16 parts and arrange them to get the [Fig (b)].
Take any part from any side and further divide it into 2 parts. [Fig (c)]
To complete the shape of rectangle arrange these two smaller parts at the corners of the shape obtained in [Fig (b)].
Find the length and the breadth of the rectangle so formed [Fig (d)].
Find the area of the rectangle.
Answer:
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Step-by-step explanation:
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