Semicircle

Top

Sub Topics Area of a Semicircle Centroid of a Semicircle Perimeter of Semicircle Volume of Semicircle Semicircle Equation

Semi in Latin means half. Semicircle by itself means half circle. In Geometry, semicircle can be defined as a 2-dimensional figure that forms half of a circle is called Semicircle. It is a closed shape that is half a circle but diameter of that circle. Which means the diameter of a circle cuts the circle into two equal semicircles.                                                            A circle makes a full turn of 360°. Semicircle being half a circle measures half of circle’s 360°, thus measuring 180°.   Thales' theorem An angle inscribed in a semicircle is always 90 °.  Proof: We have to prove that ∠ACB = 90 ° Mark Centre O. Join OC. Now, OA = OB =OC = radius. Δ OCA and Δ OCB are isosceles triangles. Since the base angles of an isosceles triangle are equal, We have ∠ OCB = ∠ OBC and ∠ OCA = ∠ OAC. Also, Sum of angles in a triangle = 180 ° Thus, ∠ OAC + ∠ ACB +∠ OBC = 180°. ∠ OAC + ∠ OCA + ∠ OCB +∠ OBC = 180° (Since ∠ ACB = ∠ OCA + ∠ OCB) 2∠ OCA + 2∠ OCB = 180°. 2(∠ OCA + ∠ OCB) = 180°. ∠ OCA + ∠ OCB = 90°. ∠ ACB =90° (Since ∠ ACB = ∠ OCA + ∠ OCB) Hence proved. Any triangle inscribed in semicircle is always a right triangle.

The area of the semicircle is found from the area of a circle. Since the semicircle is half circle, the area of the semicircle is half the area of the circle from which it is made.
The area of the circle = $\pi r^2$ where r is the radius of the circle and $\pi$ is pi which is approximately 3.142.The area of the semicircle = $\frac{1}{2}$$\pi r^2$ or $\frac{\pi r^2}{2}$ where r is the radius of the circle from which it is made and p is pi which is approximately 3.142.
A centroid of any object in n-dimensional space is the intersection of all hyper planes that divide an object into two parts of equal moment about the hyper plane. Precisely, it is the "average" of all points of an object. The centroid of a body is also its centre of mass, for an object of uniform composition (mass, density, etc.).

The Centroid of any semicircle lies in the middle of the semicircle.

 In this figure the centroid of the semicircle is the point G.
The centroid of a semicircle is given by the formula,
G = $\frac{4r}{3\pi}$

If Point P is assumed to be at the lower left corner of the wall where the semicircle is resting, the centroid of this semicircle lies at a distance of half of its diameter (d) along the horizontal direction, as in the case of a regular circle. However, the centroid lies at a distance of ($\frac{4d}{6\pi}$) up along the vertical direction from Point P.
Unlike the area, the perimeter of a semicircle is not half the perimeter of a circle from which it is made. In the semicircle figure above, we can see that the perimeter is the curved line, which is half the circumference of a circle, along with the diameter line across the bottom.

The Perimeter which is also called as the Circumference of the circle is given by 2 $\pi$ r.

The curved part of this perimeter will be half of it, which is p r, and the diameter line is twice that of radius or 2 r.

Thus, the formula for Perimeter of Semicircle is given by :
Perimeter = $\pi$ r + 2r
                   = r ($\pi$ + 2) where r is the radius of the circle from which it is made and p is pi which is approximately 3.142.
Volume is defined for any three dimensional figures. A semicircle is a two-dimensional figure and so volume cannot be determined but area can be defined. The equation of a semicircle is same as that of a full circle but the values of the dependent variable are restricted to be either only positive or only negative.
For example: y = $( r^2 - x^2 )^{\frac{1}{2}}$  with only y $\geq$ 0 allowed and x² can never be greater than r² in either a full circle or semi circle.