science

Pentagonal orthobirotunda

Pentagonal orthobirotunda 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 Pentagonal orthobirotunda rather than just read about it. In short: In geometry, the pentagonal orthobirotunda is a polyhedron constructed by attaching two pentagonal rotundae along their decagonal faces, matching like faces. It is a Johnson solid.

Key takeaways

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

Reference excerpt

In geometry, the pentagonal orthobirotunda is a polyhedron constructed by attaching two pentagonal rotundae along their decagonal faces, matching like faces. It is a Johnson solid.

Construction The pentagonal orthobirotunda is constructed by attaching two pentagonal rotundas to their base, covering decagon faces. The resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to the icosidodecahedron (or pentagonal gyrobirotunda), an Archimedean solid: the difference is that one of its rotundas is twisted around 36°, making the pentagonal faces connect to the triangular one, a process known as gyration. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The pentagonal orthobirotunda is one of them, enumerated as the 34th Johnson solid J 34 {\displaystyle J_{34}} .

Properties The surface area of an icosidodecahedron A {\displaystyle A} can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron V {\displaystyle V} can be determined by slicing it into two pentagonal rotundae, after which summing up their volumes. Therefore, its surface area and volume can be formulated as:

A = ( 5 3 + 3 25 + 10 5 ) a 2 ≈ 29.306 a 2 V = 45 + 17 5 6 a 3 ≈ 13.836 a 3 . {\displaystyle {\begin{aligned}A&=\left(5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}&\approx 29.306a^{2}\\V&={\frac {45+17{\sqrt {5}}}{6}}a^{3}&\approx 13.836a^{3}.\end{aligned}}}

References

External links Weisstein, Eric W., "Pentagonal orthobirotunda" ("Johnson solid") at MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Pentagonal orthobirotunda

Start with the simplest possible case. Write down what Pentagonal orthobirotunda 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 Pentagonal orthobirotunda 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 Pentagonal orthobirotunda 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 Pentagonal orthobirotunda

In research: Pentagonal orthobirotunda 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 Pentagonal orthobirotunda 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: Pentagonal orthobirotunda 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 Pentagonal orthobirotunda 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 Pentagonal orthobirotunda in 20 minutes

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

Frequently asked questions

What is Pentagonal orthobirotunda in simple terms?

In geometry, the pentagonal orthobirotunda is a polyhedron constructed by attaching two pentagonal rotundae along their decagonal faces, matching like faces. It is a Johnson solid.

Why does Pentagonal orthobirotunda 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 Pentagonal orthobirotunda?

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 Pentagonal orthobirotunda.