mathematics

Pentadecagon

Pentadecagon is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pentadecagon rather than just read about it. In short: In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. Regular pentadecagon A regular pentadecagon is represented by Schläfli symbol {15}.

Key takeaways

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

Reference excerpt

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

Regular pentadecagon A regular pentadecagon is represented by Schläfli symbol {15}. A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by

A = 15 4 a 2 cot ⁡ π 15 = 15 4 7 + 2 5 + 2 15 + 6 5 a 2 = 15 a 2 8 ( 3 + 15 + 2 5 + 5 ) ≃ 17.6424 a 2 . {\displaystyle {\begin{aligned}A={\frac {15}{4}}a^{2}\cot {\frac {\pi }{15}}&={\frac {15}{4}}{\sqrt {7+2{\sqrt {5}}+2{\sqrt {15+6{\sqrt {5}}}}}}a^{2}\\&={\frac {15a^{2}}{8}}\left({\sqrt {3}}+{\sqrt {15}}+{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}\right)\\&\simeq 17.6424\,a^{2}.\end{aligned}}}

Construction As 15 = 3 × 5, a product of distinct Fermat primes, a regular pentadecagon is constructible using compass and straightedge: The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's Elements.

Compare the construction according to Euclid in this image: Pentadecagon In the construction for given circumcircle: F G ¯ = C F ¯ , A H ¯ = G M ¯ , | E 1 E 6 | {\displaystyle {\overline {FG}}={\overline {CF}}{\text{,}}\;{\overline {AH}}={\overline {GM}}{\text{,}}\;|E_{1}E_{6}|} is a side of equilateral triangle and | E 2 E 5 | {\displaystyle |E_{2}E_{5}|} is a side of a regular pentagon. The point H {\displaystyle H} divides the radius A M ¯ {\displaystyle {\overline {AM}}} in golden ratio: A H ¯ H M ¯ = A M ¯ A H ¯ = 1 + 5 2 = Φ ≈ 1.618 . {\displaystyle {\frac {\overline {AH}}{\overline {HM}}}={\frac {\overline {AM}}{\overline {AH}}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Pentadecagon

Start with the simplest possible case. Write down what Pentadecagon 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 Pentadecagon 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 Pentadecagon 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 Pentadecagon

In research: Pentadecagon 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 Pentadecagon 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: Pentadecagon is common in secondary-school and first-year university syllabi. It links to neighbouring topics Constructible polygons, Polygons by the number of sides, so understanding it makes those chapters shorter.
In everyday life: Look for Pentadecagon 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.

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How to study Pentadecagon in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pentadecagon 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 Pentadecagon out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pentadecagon in simple terms?

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. Regular pentadecagon A regular pentadecagon is represented by Schläfli symbol {15}.

Why does Pentadecagon 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 Pentadecagon?

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 Pentadecagon.