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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 (more specifically geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
), a semicircle is a two-dimensional geometric shape that forms half of a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
. Being half of a circle's 360°, the arc
Arc (geometry)

In geometry, an arc is a closed set segment of a differentiable curve in the two-dimensional manifold; for example, a circular arc is a segment of the circumference of a circle....
 of a semicircle always measures 180°. A triangle inscribed in a semicircle is always a right triangle.

micircle can be used to construct arithmetic
Arithmetic mean

In mathematics and statistics, the arithmetic mean of a list of numbers is the sum of all of the list divided by the number of items in the list....
 and geometric
Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the...
 means of two lengths using straight-edge and compass.






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Semicircle
In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 (more specifically geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
), a semicircle is a two-dimensional geometric shape that forms half of a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
. Being half of a circle's 360°, the arc
Arc (geometry)

In geometry, an arc is a closed set segment of a differentiable curve in the two-dimensional manifold; for example, a circular arc is a segment of the circumference of a circle....
 of a semicircle always measures 180°. A triangle inscribed in a semicircle is always a right triangle.

Uses

A semicircle can be used to construct arithmetic
Arithmetic mean

In mathematics and statistics, the arithmetic mean of a list of numbers is the sum of all of the list divided by the number of items in the list....
 and geometric
Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the...
 means of two lengths using straight-edge and compass. If we make a semicircle with a diameter of a+b, then the length its radius is the arithmetic mean (since it's half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common end and the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean, which can be proved using Pythagorean theorem
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
.

This method can be used to accomplish quadrature of a rectangle (since a square whose sides are equal to geometric mean of sides of a rectangle has the same area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 as the rectangle), and thus any figure for which we can construct a rectangle with equal area, such as any polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 (but not a circle).

See also

  • Thales' theorem
    Thales' theorem

    In geometry, Thales' theorem states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle....


External links



ca:Semicercle es:Semicírculo fr:Hémicycle pl:Pólokrag pt:Semicírculo tl:Kalahating bilog