science

Gyroelongated pentagonal cupola

Gyroelongated pentagonal cupola is a science 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 Gyroelongated pentagonal cupola rather than just read about it. In short: In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J24). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J5) by attaching a decagonal antiprism to its base.

Key takeaways

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

Reference excerpt

In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J24). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J5) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J46) with one pentagonal cupola removed. A Johnson solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.

Area and volume With edge length a, the surface area is

A = 1 4 ( 20 + 25 3 + ( 10 + 5 ) 5 + 2 5 ) a 2 ≈ 25.240003791... a 2 , {\displaystyle A={\frac {1}{4}}\left(20+25{\sqrt {3}}+\left(10+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\approx 25.240003791...a^{2},}

and the volume is

V = ( 5 6 + 2 3 5 + 5 6 2 650 + 290 5 − 2 5 − 2 ) a 3 ≈ 9.073333194... a 3 . {\displaystyle V=\left({\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\approx 9.073333194...a^{3}.}

Dual polyhedron The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.

References

External links Weisstein, Eric W., "Gyroelongated pentagonal cupola" ("Johnson solid") at MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Gyroelongated pentagonal cupola

Start with the simplest possible case. Write down what Gyroelongated pentagonal cupola claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Gyroelongated pentagonal cupola 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 Gyroelongated pentagonal cupola 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 Gyroelongated pentagonal cupola

In research: Gyroelongated pentagonal cupola appears in science 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 Gyroelongated pentagonal cupola 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: Gyroelongated pentagonal cupola is common in secondary-school and first-year university syllabi. It links to neighbouring topics Johnson solids, Polyhedron stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Gyroelongated pentagonal cupola 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 Gyroelongated pentagonal cupola in 20 minutes

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

Frequently asked questions

What is Gyroelongated pentagonal cupola in simple terms?

Why does Gyroelongated pentagonal cupola matter?

Because it connects several science 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 Gyroelongated pentagonal cupola?

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 Gyroelongated pentagonal cupola.