Semicircle

Key takeaways

Semicircle belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Semicircle to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Semicircle from memory before moving on to harder problems.

Reference excerpt

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

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.

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.

Farey diagram

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.

Equation The equation of a semicircle with radius r {\displaystyle r} and midpoint ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} on the diameter between its endpoints and which is entirely concave from below is

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

If it is entirely concave from above, the equation is

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

Arbelos

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.

See also Amphitheater Archimedes' twin circles Archimedes' quadruplets Great semicircle Salinon Wigner semicircle distribution

References

External links Weisstein, Eric W. "Semicircle". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Semicircle

Start with the simplest possible case. Write down what Semicircle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Semicircle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Semicircle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Semicircle

In research: Semicircle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Semicircle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Semicircle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Semicircle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Semicircle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Semicircle means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Semicircle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Semicircle in simple terms?

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).

Why does Semicircle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Semicircle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Semicircle.