Confusion regarding area of this figure

In summary, the conversation discusses the area of a figure made up of two parallel lines with semicircles on either end. The correct method to find the area is by treating the figure as a two-dimensional surface rather than a three-dimensional space with curvature. Adding semicircles in this manner leads to incorrect results.
  • #1
Prem1998
148
13
figure.png
Maybe, it's a useless question. The figure which I'm talking about consists of two parallel lines each of length 'b' and are separated by a distance 2r. Their ends on one side is closed by a semicircle which in pointing inwards and decreases the area and the ends on the other side are joined by another semicircle of the same radius 'r' but this time it is bulging outwards and contributes to the area. So, we have a closed figure. Finding its area is simple because the amount of area decreased by the first semicircle gets added again by the second semi-circle. So, the area remains the same as that of a rectangle with length 2r and breadth b. A=2rb.
But I get a different answer by this reasoning:
The given figure can be thought to be made up of an infinite number of semicircular arcs from top to bottom. The figure is filled with semicircles. So, the area of this figure can be thought to be the sum of the lengths of these infinite number of semicircles. The length of each elementary semicircle, i.e. pi*r is constant. And, these semicircles are distributed over a length 'b'. So, the area of the figure = pi*r*b, which is wrong. But, what is wrong with this reasoning?
 
Physics news on Phys.org
  • #2
The fact it is not three dimensional. Your adding semicircles, but actually you only have flat straights. And you stretched these straights to a semicircle. If this were correct, you could create even larger numbers by choosing longer curves than a semicircle.

In space as a surface of a cylinder you were right, because there is no shorter way than going around. But this one is flat.
 
  • #3
fresh_42 said:
If this were correct, you could create even larger numbers by choosing longer curves than a semicircle.
I know it's not correct and that this reasoning would lead to contradictions.. But, what is wrong with the points I've made to prove that the area should be pi*r*b. I mean, consider a rectangle of length 'pi*r' and breadth 'b'. This rectangle also has an infinite number of 'pi*r lengths' distributed over a breadth 'b'. But, this rectangle has an area pr*r*b. Even if the number of semicircular arcs in the figure is infinite, but a breadth 'b' should be able to accommodate the same number of lines both in the this rectangle of length 'pi*r' and the figure that I'm talking about. After all, both the semicircular arcs in the figure and the straight lines in this rectangle have breadth equal to that of a point, i.e zero (according to definition).
 
  • #4
Prem1998 said:
But, what is wrong with the points I've made to prove that the area should be pi*r*b.
fresh_42 said:
The fact it is not three dimensional.

You are calculating in a space with curvature.

Untitled.png


You extended the green line which you should calculate with, to become the read line. Why the red one? Why not the blue one?
 
  • #5
fresh_42 said:
The fact it is not three dimensional.
How does this lead to the conclusion that I can't add semicircles? Where is it proved that I can't add curve lengths in two dimensions?
 

Related to Confusion regarding area of this figure

What is the formula for finding the area of a figure?

The formula for finding the area of a figure depends on the shape of the figure. For a rectangle, the formula is length x width. For a triangle, the formula is 1/2 x base x height. For a circle, the formula is π x radius^2. It is important to identify the shape of the figure before using the correct formula to find its area.

What is the difference between perimeter and area?

Perimeter is the distance around the outside of a figure, while area is the measure of the space inside the figure. Perimeter is measured in linear units (such as inches or meters) while area is measured in square units (such as square inches or square meters).

Do all sides of a figure have to be the same length for it to have a specific area?

No, the length of the sides of a figure do not have to be the same for it to have a specific area. The area of a figure is determined by its shape and dimensions, not the length of its sides. For example, a rectangle with sides of 6 inches and 8 inches will have the same area as a rectangle with sides of 3 inches and 16 inches.

How do I find the area of a figure with irregular shapes?

For irregular shapes, the best way to find the area is to divide the shape into smaller, simpler shapes (such as triangles, rectangles, or circles) and then use the appropriate formula to find the area of each smaller shape. Finally, add up the areas of the smaller shapes to find the total area of the irregular shape.

What is the importance of knowing how to find the area of a figure?

Knowing how to find the area of a figure is important in many real-world applications, such as calculating the amount of material needed for a construction project or determining the size of a room for furniture. It is also a fundamental concept in math and geometry, and can help with problem-solving and critical thinking skills.

Similar threads

I The Basic Area Problem (introduction to the topic of integrals)
    • Calculus
Replies
24
Views
2K
Solving Semi-circle Inside Triangle Geometry Problem
    • Computing and Technology
Replies
13
Views
2K
MHB Solving a Primary Math Problem: Find the Areas of A & C
    • General Math
Replies
10
Views
1K
Electric Field from a Charged Semicircular Rod
    • Introductory Physics Homework Help
Replies
2
Views
2K
How do I find the circumference and Area of this figure?
    • Precalculus Mathematics Homework Help
Replies
24
Views
407
I How to evaluate the enclosed area of this implicit curve?
    • Calculus
Replies
2
Views
2K
B Having trouble deriving the volume of an elliptical ring toroid
    • Calculus
Replies
4
Views
462
Which shape provides the largest area for a given perimeter?
    • Calculus
Replies
3
Views
2K
MHB How Can You Calculate the Area of Shaded Regions in Complex Geometric Figures?
    • General Math
Replies
2
Views
1K
I Area enclosed by a function involving 2 power towers
    • Calculus
Replies
3
Views
879

Hot Threads

Recent Insights

Back
Top