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Semicircle

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Semicircle

Area	⁠πr²/2⁠
Perimeter	$(π+2)r$

In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, $π$ radians, or a half-turn). It only has one line of symmetry (reflection symmetry).

In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points.

By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex.

All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle.

Arithmetic and geometric means

[edit]

A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. For a semicircle with a diameter of a + b, the length of its radius is the arithmetic mean of a and b (since the radius is 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 endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean. This can be proven by applying the Pythagorean theorem to three similar right triangles, each having as vertices the point where the perpendicular touches the semicircle and two of the three endpoints of the segments of lengths a and b.^[1]

The construction of the geometric mean can be used to transform any rectangle into a square of the same area, a problem called the quadrature of a rectangle. The side length of the square is the geometric mean of the side lengths of the rectangle. More generally, it is used as a lemma in a general method for transforming any polygonal shape into a similar copy of itself with the area of any other given polygonal shape.^[2]

Farey diagram

[edit]

The Farey sequence of order n is the sequence of completely reduced fractions which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. With a restricted definition, each Farey sequence starts with the value 0, denoted by the fraction ⁠0/1⁠, and ends with the fraction ⁠1/1⁠. Ford circles can be constructed tangent to their neighbours, and to the x-axis at these points. Semicircles joining adjacent points on the x-axis pass through the points of contact at right angles.^[3]

Equation

[edit]

The equation of a semicircle with midpoint $(x_{0},y_{0})$ on the diameter between its endpoints and which is entirely concave from below is

y=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}}

If it is entirely concave from above, the equation is

y=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}}

Arbelos

[edit]

An arbelos is a region in the plane bounded by three semicircles connected at their endpoints, all on the same side of a straight line (the baseline) that contains their diameters.

References

[edit]

External links

[edit]

Weisstein, Eric W. "Semicircle". MathWorld.

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from Grokipedia

A semicircle is a two-dimensional geometric figure consisting of a diameter of a circle and the arc of the circle that connects the endpoints of the diameter, forming half of the full circle.^[1] It is bounded by this straight diameter and the curved semicircular arc, with the center of the semicircle coinciding with the center of the original circle.^[1] The arc subtends a central angle of 180 degrees, covering exactly half the circumference of the circle.^[2] In geometry, the semicircle holds significant properties, including the theorem that any angle inscribed in a semicircle—formed by two chords from the endpoints of the diameter to a point on the arc—is a right angle measuring 90 degrees.^[3] This inscribed angle theorem, rooted in classical Euclidean geometry, underscores the semicircle's role in demonstrating perpendicularity and has applications in constructions and proofs involving circles.^[4] The area of a semicircle with radius

r

is half the area of the full circle, given by the formula

A = \frac{1}{2} \pi r^2

.^[5] Its perimeter, or boundary length, comprises the semicircular arc length

\pi r

plus the diameter

2r

, totaling

\pi r + 2r

.^[6] Semicircles appear in various mathematical and real-world contexts, such as in the derivation of circle properties through integration or in modeling arches, lenses, and segments in engineering and physics.^[7] They also feature in neutral geometry classifications, where a semicircular arc is defined as the intersection of a circle with a closed half-plane bounded by the diameter.^[8]

Definition and Basic Properties

Definition

A semicircle is a fundamental geometric figure consisting of half of a circle, defined by a straight diameter and the curved arc that connects its two endpoints, spanning an angle of 180 degrees or π radians at the center.^[9] This arc represents exactly one-half of the full circumference of the circle, with the diameter serving as the base that bisects the circle into two congruent parts.^[10] The term "semicircle" can refer to different aspects of the shape depending on context: the semicircular arc, which is the curved boundary alone; the closed semicircular curve, comprising the arc and the adjoining diameter; or the semicircular disk, which includes the interior region bounded by the arc and diameter.^[11] In all cases, the straight diameter acts as the foundational line, while the semicircular arc provides the curved boundary.^[12] The concept of the semicircle originates in ancient Greek geometry, notably in Euclid's Elements (circa 300 BCE), where it is described as "the figure contained by the diameter and the circumference cut off by it," used in discussions of basic circle divisions and properties.^[1] Etymologically, "semicircle" derives from the Latin semicirculus, combining the prefix semi- (meaning "half") with circulus (meaning "circle" or "small ring").^[13] The shape exhibits a line of symmetry along the perpendicular bisector of the diameter.^[12]

Geometric Characteristics

A semicircle is a two-dimensional figure formed by a diameter of a circle and the arc connecting its endpoints, where the diameter serves as the chord subtending a central angle of 180 degrees.^[9]^[1] This configuration positions the semicircle as half of the full circle, sharing the same center at the midpoint of the diameter, and inheriting the constant radius property of the parent circle.^[9] The semicircle exhibits reflection symmetry along a single line, which is the perpendicular bisector of the diameter passing through its midpoint and the apex of the arc.^[9] This line of symmetry divides the figure into two congruent halves, highlighting its bilateral balance despite lacking the infinite rotational symmetries of a full circle. Along the curved arc, the semicircle maintains the geometric property that any radius drawn from the center to a point on the arc is perpendicular to the tangent line at that point of contact.^[9]^[3] As a limiting case of a circular segment, the semicircle occurs when the segment's height equals the radius of the originating circle, resulting in the chord becoming the full diameter and the arc spanning exactly half the circumference.^[14]^[9] This bounded structure, enclosed by the straight diameter and the semicircular arc, contrasts with the fully enclosed nature of a complete circle, enabling the semicircle to symmetrically divide the circle into two equal regions while providing a distinct boundary for geometric constructions.^[9]

Formulas and Equations

Area and Perimeter

The area of a semicircle with radius

r

is given by the formula

A = \frac{1}{2} \pi r^2

.^[9] This derives from the area of a full circle,

\pi r^2

, which is halved to account for the semicircle comprising exactly half the disk.^[9] The derivation can be shown using integration: the area is

A = 2 \int_0^r \sqrt{r^2 - x^2} \, dx

A=2∫0rr2−x2

dx, which evaluates to $\frac{1}{2} \pi r^2$ .^[9] The perimeter of a semicircle, which includes the curved arc and the straight diameter, is $P = \pi r + 2r = r(\pi + 2)$ .^[9] The curved portion, or semicircumference, is half the full circumference of the circle, $\frac{1}{2} \times 2\pi r = \pi r$ , derived via the arc length integral $s = \int_{-r}^r \sqrt{1 + (x')^2} \, dy$

dy where $x = \sqrt{r^2 - y^2}$

, simplifying to $\pi r$ .^[9] The straight portion is the diameter, $2r$ , added to complete the boundary.^[9] Here, $r$ denotes the radius, defined as half the diameter of the full circle from which the semicircle is formed; units for area are typically square units (e.g., square meters), while perimeter uses linear units (e.g., meters).^[9] For example, if $r = 5$ cm, the area is $A = \frac{1}{2} \pi (5)^2 = 12.5 \pi$ cm² ≈ 39.27 cm², and the perimeter is $P = 5(\pi + 2)$ cm ≈ 21.57 cm.^[9] These formulas enable practical measurements in geometry problems, such as calculating material for semicircular arches or fields, where the radius is measured from the center to the arc.^[9]

Cartesian Equation

The Cartesian equation of a semicircle is derived from the equation of a full circle by solving for one variable and imposing a restriction to half the plane. The standard equation for a circle centered at

(h, k)

with radius

r

(x - h)^2 + (y - k)^2 = r^2

.^[15] To obtain the explicit equation for a semicircle, solve for

y

(y - k)^2 = r^2 - (x - h)^2

y - k = \pm \sqrt{r^2 - (x - h)^2}

y−k=±r2−(x−h)2

$y = k \pm \sqrt{r^2 - (x - h)^2}$

The upper semicircle corresponds to the positive root, $y = k + \sqrt{r^2 - (x - h)^2}$

, while the lower semicircle uses the negative root, $y = k - \sqrt{r^2 - (x - h)^2}$

.^[16] These equations are defined over the domain $x \in [h - r, h + r]$ , where the expression under the square root is non-negative, ensuring the semicircle traces the arc between the endpoints of the diameter. In standard positioning, the diameter lies along the x-axis with the center at the origin ( $h = 0$ , $k = 0$ ), so the upper semicircle simplifies to $y = \sqrt{r^2 - x^2}$

for $x \in [-r, r]$ .^[17] For the unit semicircle centered at the origin ( $r = 1$ ), the upper half is given by $y = \sqrt{1 - x^2}$

with $x \in [-1, 1]$ .^[18] This form highlights the semicircle's role as a function, unlike the full circle, and facilitates applications such as integration over the curve. An alternative implicit representation of the upper semicircle retains the circle's equation with an inequality constraint: $(x - h)^2 + (y - k)^2 = r^2$ where $y \geq k$ . This form is useful for describing the semicircle without isolating $y$ , particularly in contexts involving implicit differentiation or boundary conditions.^[19] The domain restriction on $x$ remains essential to avoid extraneous points outside the arc.

Applications in Geometry

Thales' Theorem

Thales' theorem asserts that if A and B are the endpoints of the diameter of a semicircle, and C is any point on the remaining arc, then ∠ACB measures exactly 90 degrees.^[20] This property highlights a fundamental relationship between the semicircle and right angles, positioning the theorem as a cornerstone of classical geometry.^[21] The theorem is attributed to Thales of Miletus, an ancient Greek philosopher and mathematician active around 624–546 BCE, and is considered one of the earliest recorded geometric theorems in Western tradition.^[22] Ancient historian Eudemus of Rhodes, in his lost History of Geometry (as referenced by later commentators like Proclus), credits Thales with this discovery.^[23] A standard proof relies on the inscribed angle theorem for a full circle, which states that an angle at the circumference subtended by a given arc equals half the central angle subtended by the same arc.^[21] Here, the diameter AB subtends a central angle of 180 degrees, so the inscribed angle ∠ACB at point C on the circumference is half of 180 degrees, or 90 degrees.^[20] This approach leverages the semicircle's symmetry in completing the full circle, ensuring the arc ACB aligns precisely with the theorem's conditions.^[22] As an implication, every triangle ABC formed this way is right-angled at C, with the diameter AB serving as the hypotenuse, generalizing the configuration to all such inscribed triangles.^[21] Visually, this manifests as a right-angled triangle inscribed within the semicircle, where the right angle lies on the curved arc opposite the diameter, emphasizing the theorem's elegant geometric inscription.^[20] In applications, Thales' theorem enables the construction of right angles using only a compass and straightedge: draw segment AB as the intended hypotenuse, construct the semicircle with AB as diameter, and select any point C on the arc to form ∠ACB at 90 degrees.^[21] This method underscores the theorem's practical value in ancient surveying and modern geometric constructions.^[22]

Arithmetic and Geometric Means

One classical compass-and-straightedge construction employs a semicircle to determine both the arithmetic mean and geometric mean of two given line segments of lengths

a

and

b

. To perform the construction, first place the segments end-to-end along a straight line to form diameter

AB

of length

a + b

, with the junction point denoted as

C

such that

AC = a

and

CB = b

. Next, draw the semicircle with diameter

AB

. Then, erect a perpendicular line at

C

that intersects the semicircle at point

D

. The length

CD

equals the geometric mean

\sqrt{ab}

, while the arithmetic mean $(a + b)/2$ is the distance from $A$ (or $B$ ) to the midpoint of $AB$ along the diameter.^[24] This result follows from the geometric mean theorem in right triangles, as established in Euclidean geometry. The semicircle ensures that $\angle ADB = 90^\circ$ by Thales' theorem, making $AB$ the hypotenuse of right triangle $ADB$ . Here, $CD$ serves as the altitude from the right angle at $D$ to the hypotenuse $AB$ , landing at $C$ . By the properties of similar triangles—specifically, $\triangle ACD \sim \triangle ADB$ and $\triangle BCD \sim \triangle ADB$ —it holds that $CD^2 = AC \cdot CB = ab$ , so $CD = \sqrt{ab}$

. Alternatively, by the power of point $C$ with respect to the circle, the intersecting chords theorem yields $AC \cdot CB = CD \cdot CD'$ , where $D'$ is the symmetric point below the diameter; since the construction is symmetric, $CD = CD'$ , confirming $CD^2 = ab$ .^[24]^[25] The construction provides a visual demonstration of the arithmetic mean-geometric mean (AM-GM) inequality, which states that $(a + b)/2 \geq \sqrt{ab}$

for positive $a$ and $b$ , with equality if and only if $a = b$ . In the semicircle of radius $r = (a + b)/2$ , the maximum height from the diameter to the arc occurs at the midpoint and equals $r$ , representing the arithmetic mean. However, unless $C$ coincides with the midpoint (i.e., $a = b$ ), the height $CD$ at $C$ is strictly less than $r$ , illustrating $\sqrt{ab} < (a + b)/2$

<(a+b)/2. When $a = b$ , $C$ is the midpoint, $CD = r$ , and equality holds./04:_Picture_Proofs/4.02:_Arithmetic_and_Geometric_Means) This method originates in ancient Greek geometry, as described by Euclid in the Elements (Book VI, Proposition 13), where it serves to construct a mean proportional between two magnitudes without requiring numerical computation or advanced tools.^[24] For example, with $a = 4$ and $b = 9$ , the diameter $AB = 13$ , the geometric mean $CD = \sqrt{36} = 6$

=6, and the arithmetic mean is $13/2 = 6.5$ , satisfying $6 < 6.5$ .^[24]

Arbelos

The arbelos, derived from the Greek word for "shoemaker's knife," is a plane region bounded by three semicircles sharing a common baseline diameter AB, where the largest semicircle has diameter AB and the two smaller semicircles have diameters AC and CB, with C a point between A and B.^[26] The figure is formed on the same side of the baseline, creating a shaded region resembling a knife blade between the arc of the large semicircle and the combined arcs of the two smaller ones, with a cusp at point C where the smaller arcs meet.^[27] This geometric construction was first systematically studied by the ancient Greek mathematician Archimedes (c. 287–212 BCE) in his work known as the Book of Lemmas, a collection of propositions on circles and related figures.^[27] Archimedes explored the arbelos through several lemmas, highlighting its intriguing properties that connect areas of curved regions in unexpected ways.^[28] Geometrically, the arbelos exhibits the property that the sum of the areas of the two smaller semicircles equals the area of the large semicircle minus the area of the arbelos itself, emphasizing the region's role as a difference of circular segments.^[26] A key result from Archimedes' Book of Lemmas (Proposition 4) states that the area of the arbelos equals the area of a circle whose diameter is the perpendicular distance from the cusp at C to the point where it intersects the arc of the large semicircle.^[28] Further propositions (5 and 6) demonstrate equal areas for certain circles inscribed in the arbelos that are tangent to the three semicircular arcs or to specific lines and arcs within the figure.^[28] The arbelos holds significance in classical geometry for illustrating relationships between circular areas and straight-line segments, contributing to early efforts in quadrature problems where curved areas are equated to polygonal or circular ones.^[26] Its name evokes the analogy of a shoemaker's tool in ancient texts, underscoring its practical yet elegant form in demonstrating non-trivial equalities without advanced tools.^[27]

Farey Diagrams

In a Farey diagram of order

n

, semicircles are drawn in the upper half-plane to connect adjacent fractions

p/q

and

r/s

from the Farey sequence

F_n

of reduced fractions between 0 and 1, where these fractions satisfy

|ps - qr| = 1

.^[29] Each such semicircle has its diameter along the baseline (the real axis) from

p/q

r/s

and is tangent to this baseline, with radius

1/(2qs)

.^[29] These semicircles exhibit key geometric properties when superimposed on Ford circles, which are full circles centered at

(p/q, 1/(2q^2))

with radius

1/(2q^2)

, tangent to the baseline at each rational

p/q

.^[29] Specifically, the semicircles intersect the corresponding Ford circles orthogonally, meaning at right angles, due to the curvature relations in the upper half-plane model.^[29] This orthogonality highlights the non-intersecting nature of the diagram's edges except at vertices, forming a tessellation of curvilinear triangles.^[29] The construction of the Farey diagram begins with the Farey sequence

F_n

, which lists all reduced fractions

a/b

with

0 \leq a \leq b \leq n

in increasing order, starting from

0/1

and

1/1

.^[29] Adjacent pairs in

F_n

determine the semicircles, and the diagram approximates the action of the modular group

SL(2,\mathbb{Z})

on the upper half-plane, generating infinite strips or triangles for irrational limits.^[29] Intersections between semicircles or with Ford circles occur at points corresponding to the mediant

(p + r)/(q + s)

of the endpoints, preserving the neighbor relation under Farey addition.^[29] Mathematically, Farey diagrams provide a visualization of rational approximations and continued fractions, relating to the Poincaré disk model of hyperbolic geometry through the upper half-plane tessellation.^[29] They facilitate proofs in number theory, such as Lagrange's theorem on periodic continued fractions for quadratic irrationals, by encoding Diophantine properties and class groups via lattice norms.^[29] The Farey sequences underlying these diagrams were introduced in the early 19th century by John Farey, but the geometric diagram itself was developed in the late 19th century by Adolf Hurwitz in 1894, with Ford circles added later in 1938 by Lester S. Ford.^[29]^[30] For example, in the Farey diagram of order 2, the sequence is

0/1, 1/2, 1/1

, yielding semicircles between

0/1

and

1/2

(radius

1/4

) and between

1/2

and

1/1

(radius

1/4

), each intersecting the respective Ford circles orthogonally.^[29]

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References

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