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Exploring Some Geometric Themes - Visualising Solids

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Difference between 2D and 3D shapes: Two-dimensional shapes like triangles and squares have length and breadth, whereas three-dimensional solids like cubes and spheres have length, breadth, and height (or depth).

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Polyhedrons: These are 3D solids made up of polygonal faces. Faces (FF) are flat surfaces, Edges (EE) are line segments where faces meet, and Vertices (VV) are points where edges meet.

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Convex Polyhedrons: A polyhedron is convex if the line segment joining any two points on its surface lies entirely inside or on the polyhedron.

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Regular Polyhedrons: A polyhedron is regular if its faces are made up of regular polygons and the same number of faces meet at each vertex (e.g., a Cube).

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Prisms and Pyramids: A prism is a polyhedron whose base and top are congruent polygons and whose other faces (lateral faces) are parallelograms. A pyramid is a polyhedron whose base is a polygon and whose lateral faces are triangles with a common vertex.

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Visualising Views: 3D objects can be viewed from different positions, typically referred to as the Front view, Side view, and Top view.

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Mapping Space: Maps use symbols and scales to represent the location of objects and the distance between them. Unlike a picture, a map does not show perspective.

📐Formulae

F+V−E=2F + V - E = 2

F+V=E+2F + V = E + 2

💡Examples

Problem 1:

A polyhedron has 3030 edges and 1212 vertices. Find the number of faces using Euler's formula.

Solution:

Given E=30E = 30 and V=12V = 12. According to Euler's formula: F+V−E=2F + V - E = 2 Substituting the values: F+12−30=2F + 12 - 30 = 2 F−18=2F - 18 = 2 F=2+18=20F = 2 + 18 = 20

Explanation:

By applying Euler's formula for convex polyhedrons, we can solve for the unknown number of faces when vertices and edges are provided.

Problem 2:

Can a polyhedron have 1010 faces, 2020 edges, and 1515 vertices?

Solution:

Check using Euler's formula F+V−E=2F + V - E = 2. Given F=10F = 10, V=15V = 15, and E=20E = 20. Calculate F+VF + V: 10+1525\begin{array}{r} 10 \\ +15 \\ \hline 25 \end{array} Now check F+V−EF + V - E: 25−20=525 - 20 = 5 Since 5≠25 \neq 2, the given dimensions do not satisfy Euler's formula.

Explanation:

For any polyhedron to exist, it must satisfy the condition F+V−E=2F + V - E = 2. Since the result here is 55, such a polyhedron is not possible.

Problem 3:

Identify the number of faces, vertices, and edges for a Pentagonal Pyramid.

Solution:

A pentagonal pyramid has a pentagonal base and 55 triangular lateral faces. Therefore: F=1F = 1 (base) +5+ 5 (lateral) =6= 6. V=5V = 5 (base vertices) +1+ 1 (apex) =6= 6. E=5E = 5 (base edges) +5+ 5 (lateral edges) =10= 10. Check: 6+6−10=12−10=26 + 6 - 10 = 12 - 10 = 2.

Explanation:

A pyramid with an nn-sided base always has n+1n+1 faces, n+1n+1 vertices, and 2n2n edges.

Visualising Solids Class 8 Notes & Examples | CBSE Maths