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Semicircle


Semicircle

Half a circle. The area of a semicircle of radius r is given by

A=int_0^rint_(-sqrt(r^2-x^2))^(sqrt(r^2-x^2))dxdy
(1)
=2int_0^rsqrt(r^2-x^2)dx
(2)
=1/2pir^2.
(3)

The weighted mean of y is

<x>_2=2int_0^rxsqrt(r^2-x^2)dx
(4)
=2/3r^3.
(5)

The semicircle is the cross section of a hemisphere for any plane through the z-axis.

The perimeter of the curved boundary is given by

 s=int_(-r)^rsqrt(1+x^('2))dy.
(6)

With x=sqrt(r^2-y^2), this gives

 s=pir.
(7)

The perimeter of the semicircular lamina is then

 L=2r+pir=r(2+pi).
(8)
SemicircleCentroids

The weighted value of x of the semicircular curve is given by

<x>_1=int_(-r)^rxsqrt(1+x^('2))dy
(9)
=int_(-r)^rrdy
(10)
=2r^2,
(11)

so the geometric centroid is

 x^__1=(<x>_1)/A=(2r)/pi.
(12)

The geometric centroid of the semicircular lamina is given by

 x^__2=(<x>_2)/A=(4r)/(3pi)
(13)

(Kern and Bland 1948, p. 113).


See also

Arbelos, Arc, Circle, Disk, Hemisphere, Lens, Right Angle, Salinon, Thales' Theorem, Yin-Yang

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References

Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948.

Cite this as:

Weisstein, Eric W. "Semicircle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Semicircle.html

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