Area of Shaded Region | Semicircles inside Circle problem | Advanced math problems | Mathematics

This problem is an application of the famous Pythagoras theorem and the basic properties of circles and semicircles. If you figure out the symmetry of the diagram, this problem is not a heavy one.

Any queries regarding the subject or videos are invited.
Instagram ID : math_problem_analysis

#maths #mathematics #circle #semicircle #area #pythagorastheorem #advancedmathproblems #calculation #problems #geometry

[Inspirator_AG]

8π, because it is one of those types of questions.

[neuraaquaria]

I drew 2 segments from the center of the larger circle to each corner of the bottom shaded circle. These segments are perpendicular because they are both inscribed angles to the diameter of the shaded circle. We know that the length of this segment is 4 using area of the larger circle. I then used properties of a right isosceles triangle to figure out the diameter of the yellow circle: 4*sqrt(2).
This means the area of the shaded semicircle is 0.5 * pi* (4* sqrt(2)/2)^2 = 4pi.
The area of both shaded regions is then 8pi

[kevinmorgan]

All you need is any two (not necessarily identical) semi circles of any size touching as yours are and symmetrical about a vertical axis as yours are. The enclosing circle (passing through the four extremities of two diameters) will have an area twice that of the two semicircle.

[firstnamelastname]

FYI:

The vertical two smaller half circles actually need not be equal in size...

*if* they are tangent, and, *if* the larger circle cuts both of them in half, and, *if* three circle centers are collinear, *then* the sum of both areas equals *half* the area of the larger circle. The proof is a bit harder.

Typical example : let larger circle have radius 5 and let two smaller (half) circles have radius 4 and 3. Any other three 'Pythagorean' circle radius examples will do.

Further, in the limit, if one smaller half circle shrinks to a point, then the other one grows to (half) the larger circle.