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ICSE • कक्षा 7 • Mathematics • अध्याय 14
अनुमानित समय: 45 Mins
प्रगति: अध्ययनरत

त्रिविमीय आकारों का द्विविमीय निरूपण (Representing 3D in 2D)

In ICSE Class 7 Mathematics, "Representing 3-D in 2-D" provides an authoritative spatial visualization master study guide on polyhedra, nets, isometric sketching, and Euler's formula. This comprehensive chapter explores Solid Shapes (Two-dimensional [2-D] plane figures with length and breadth vs Three-dimensional [3-D] solid shapes with length, breadth, and height/depth), Anatomy of Polyhedrons (Faces $[F]$, Vertices $[V]$, and Edges $[E]$ of cubes, cuboids, triangular prisms, square pyramids, and triangular pyramids [tetrahedron]), Euler's Formula ($F + V - E = 2$ for any convex polyhedron), Types of Polyhedra (Convex vs Concave, Regular polyhedra / Platonic solids), Nets of 3-D Shapes (2-D folding templates that fold into 3-D solids; The 11 distinct nets of a cube, nets of cylinders, cones, and prisms), and 2-D Representation Techniques on Paper (1. Oblique Sketches [drawn on squared graph paper where front face shows true shape, depth edges drawn at an angle but not to scale], 2. Isometric Sketches [drawn on isometric dot paper where all three dimensions are drawn strictly to scale at $30^\circ$ isometric axes]), and Viewing Solids from Different Angles (Front view, Side view, Top view / Plan) aligned with the 2026–27 CISCE curriculum.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें
प्रस्तावना अध्याय अवधारणाएँ सूत्र एवं सर्वसमिकाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

How Did an 18th-Century Blind Swiss Genius Discover a Magic Formula That Connects Every Closed 3D Solid in the Universe?

In 1752, the blind mathematical titan Leonhard Euler sat in Saint Petersburg, running his fingers over wooden geometrical blocks—cubes, pyramids, prisms, and crystal octahedrons. Euler made a staggering discovery that seemed like pure sorcery: no matter how many flat faces a solid shape has, no matter how many pointed corners or straight edges you carve into it, if you count its Faces ($F$), count its Vertices ($V$), and subtract its Edges ($E$), the result is ALWAYS the magic number TWO!

$$F + V - E = 2$$

A cube has 6 faces, 8 vertices, and 12 edges: $6 + 8 - 12 = 2$! A triangular pyramid has 4 faces, 4 vertices, and 6 edges: $4 + 4 - 6 = 2$! A soccer ball polyhedron with 32 faces: $32 + 60 - 90 = 2$! This is Euler's Formula! How do architects and video game 3D modelers draw 3D blocks on flat 2D computer screens using Isometric Sketches? How can a flat sheet of cardboard fold into a 3D box using Nets? Let's master spatial geometry.

यह अध्याय क्यों महत्वपूर्ण है

Visualizing three-dimensional space on two-dimensional surfaces is the foundational core of engineering blueprints, CAD drafting, architectural modeling, manufacturing packaging, video game animation engines, and robotics navigation. Euler's formula is a cornerstone of topology and ICSE geometry.

अध्ययन से पूर्व (आवश्यक ज्ञान)

  • Basic 2D shapes: Squares, rectangles, triangles, circles.
  • Basic 3D terminology: Cube, cuboid, cylinder, cone, sphere.
  • Elementary integer addition and subtraction.

इस अध्याय के लक्ष्य

  • Differentiate 2D plane figures from 3D solid shapes and identify their faces, vertices, and edges.
  • Verify and apply Euler's Formula ($F + V - E = 2$) to determine unknown faces, vertices, or edges.
  • Differentiate between Prisms and Pyramids based on base geometry and lateral faces.
  • Identify and draw valid 2D folding Nets for cubes, cuboids, cylinders, cones, and pyramids.
  • Differentiate between Oblique Sketches and Isometric Sketches.
  • Draw Front, Side, and Top (Plan) views of solid arrangements.

अध्याय रूपरेखा एवं प्रगति

1 1. Anatomy of Polyhedrons: Faces, V...
2 2. Euler's Formula for Convex Polyh...
3 3. 2-D Nets of 3-D Solids
4 4. 2-D Sketching: Oblique vs Isomet...

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Anatomy of Polyhedrons: Faces, Vertices & Edges

Understand
A. Polyhedron Definition:

A Polyhedron (plural: polyhedra) is a three-dimensional solid figure whose entire boundary is composed of flat polygonal faces:

  • Face ($F$): A flat polygonal surface bounding the solid.
  • Edge ($E$): The straight line segment where two adjacent faces meet.
  • Vertex ($V$): The point (corner) where three or more edges intersect.
  • Non-Polyhedrons: Solids with curved surfaces (Cylinder, Cone, Sphere) are NOT polyhedrons.
B. Prisms vs Pyramids:
  • Prism: A polyhedron whose two end faces (bases) are congruent, parallel polygons, and whose lateral faces are parallelograms / rectangles (e.g., Triangular Prism, Cuboid).
  • Pyramid: A polyhedron whose base is any polygon, and whose lateral faces are triangles meeting at a single common point called the Apex (e.g., Square Pyramid, Tetrahedron).

2. Euler's Formula for Convex Polyhedrons

Euler's Formula

For any Convex Polyhedron, the number of Faces ($F$), Vertices ($V$), and Edges ($E$) satisfy the invariant topological theorem:

$$\mathbf{F + V - E = 2} \quad \text{or} \quad \mathbf{F + V = E + 2}$$
PolyhedronFaces ($F$)Vertices ($V$)Edges ($E$)Euler Verification ($F + V - E$)
Cube / Cuboid6812$6 + 8 - 12 = 14 - 12 = \mathbf{2}$
Triangular Prism5 (2 $\Delta$, 3 rect)69$5 + 6 - 9 = 11 - 9 = \mathbf{2}$
Square Pyramid5 (1 sq, 4 $\Delta$)58$5 + 5 - 8 = 10 - 8 = \mathbf{2}$
Tetrahedron (Triangular Pyramid)4 (all $\Delta$)46$4 + 4 - 6 = 8 - 6 = \mathbf{2}$

3. 2-D Nets of 3-D Solids

Nets of Solids

A Net is a two-dimensional flat skeleton outline that can be cut and folded along its edges to form the three-dimensional solid shape:

  • Cube: Has exactly 11 distinct valid nets composed of 6 squares joined edge-to-edge.
  • Cylinder: Formed by a central rectangle (lateral surface) with two congruent circles attached to opposite edges (bases).
  • Cone: Formed by a sector of a circle (curved surface) and a circular base.
  • Square Pyramid: Formed by a central square surrounded by four triangles attached to each edge.

4. 2-D Sketching: Oblique vs Isometric Sketches

Sketching Techniques
A. Oblique Sketch:
  • Drawn on plain squared graph paper.
  • The front face is drawn with its exact true shape and dimensions.
  • The receding depth edges are drawn parallel at an angle (usually $45^\circ$), but are foreshortened and NOT drawn to true scale.
B. Isometric Sketch:
  • Drawn on Isometric Dot Paper (featuring dots arranged in an equilateral triangular grid).
  • The three spatial axes make equal angles of $120^\circ$ with each other (receding axes at $30^\circ$ to the horizontal).
  • True Scale: All edge lengths are drawn strictly proportional to their actual measurements.

महत्वपूर्ण सूत्र, सर्वसमिकाएँ एवं प्रमेय

Euler's Polyhedron Formula
F + V - E = 2
Valid for any simple convex polyhedron.
Prism and Pyramid Vertex-Edge Rules
$$\text{n-gonal Prism}: F = n+2, V = 2n, E = 3n; \quad \text{n-gonal Pyramid}: F = n+1, V = n+1, E = 2n$$
Quick formulas for n-sided polygonal bases.

Polyhedrons: Faces, Vertices, Edges & Nets

Representing 3-D in 2-D: Euler's Formula, Solids & Nets CUBE / CUBOID • Faces (F) = 6 • Vertices (V) = 8 • Edges (E) = 12 F + V - E = 6+8-12 = 2 EULER'S FORMULA F + V - E = 2 • Triangular Prism: F=5, V=6, E=9 ⇒ 5+6-9 = 2 • Square Pyramid: F=5, V=5, E=8 ⇒ 5+5-8 = 2 • Tetrahedron (Tri. Pyramid): F=4, V=4, E=6 ⇒ 4+4-6 = 2 • Curves (cylinder/sphere) NOT polyhedra NETS & SKETCHES • 2-D Folding Nets:   • Cube has 11 distinct nets   • Cylinder = 1 rectangle + 2 circles   • Cone = 1 sector + 1 circle • Oblique Sketch:   Front face true size; depth not to scale • Isometric Sketch:   Drawn on 30° isometric dot paper;   ALL dimensions drawn to TRUE scale! • Views: Front, Side, and Top (Plan) EULER'S TOPOLOGY: F + V - E = 2 FOR ALL CONVEX POLYHEDRONS • ISOMETRIC SKETCHES ARE TO SCALE

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1
A polyhedron is a 3D solid bounded entirely by flat polygonal faces.
मुख्य बिंदु 2
Faces are flat surfaces; Edges are line segments where faces meet; Vertices are corner intersection points.
मुख्य बिंदु 3
Solids with curved surfaces (cylinders, cones, spheres) are NOT polyhedra.
मुख्य बिंदु 4
Euler's Formula: For any convex polyhedron, $F + V - E = 2$ (or $F + V = E + 2$).
मुख्य बिंदु 5
A cube has 6 faces, 8 vertices, and 12 edges ($6 + 8 - 12 = 2$).
मुख्य बिंदु 6
A triangular prism has 5 faces, 6 vertices, and 9 edges ($5 + 6 - 9 = 2$).
मुख्य बिंदु 7
A square pyramid has 5 faces, 5 vertices, and 8 edges ($5 + 5 - 8 = 2$).
मुख्य बिंदु 8
A net is a 2D flat template that folds along edges into a 3D solid; a cube has 11 distinct nets.
मुख्य बिंदु 9
Oblique sketches show front faces in true size, but depth is distorted.
मुख्य बिंदु 10
Isometric sketches are drawn on triangular dot paper at 30-degree axes with all dimensions drawn strictly to scale.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

1
A polyhedron has 20 faces and 12 vertices. Find the number of edges it has using Euler's formula.
उत्तर एवं व्याख्या देखें
उत्तर:

Euler's Formula states: $F + V - E = 2$.
Given: $F = 20$, $V = 12$.

$$20 + 12 - E = 2$$


$$32 - E = 2$$


$$E = 32 - 2 = \mathbf{30}$$

.
The polyhedron (an icosahedron) has $30\text{ edges}$.


Apply $F + V - E = 2$: $20 + 12 - E = 2 \implies 32 - E = 2 \implies E = 30$.
2
Can a polyhedron have 10 faces, 20 edges, and 15 vertices? Give reasons.
उत्तर एवं व्याख्या देखें
उत्तर:

Check using Euler's Formula $F + V - E = 2$:
Given: $F = 10, V = 15, E = 20$.

$$\text{LHS} = F + V - E = 10 + 15 - 20 = 25 - 20 = 5$$


Since $\text{LHS} = 5 \neq 2$, it violates Euler's Formula.
Therefore, NO such polyhedron can exist!


Substitute into $F + V - E$: $10 + 15 - 20 = 5 \neq 2$. Since it does not equal 2, it is impossible.
3
Find the number of faces, vertices, and edges of a "Hexagonal Prism". Verify Euler's formula.
उत्तर एवं व्याख्या देखें
उत्तर:

A hexagonal prism has two regular hexagons as top and bottom bases, and 6 rectangular lateral faces:
• Faces ($F$): 2 hexagonal bases + 6 rectangular faces $= \mathbf{8\text{ faces}}$.
• Vertices ($V$): 6 on the top base + 6 on the bottom base $= \mathbf{12\text{ vertices}}$.
• Edges ($E$): 6 on top + 6 on bottom + 6 vertical $= \mathbf{18\text{ edges}}$.
• Euler Verification:

$$F + V - E = 8 + 12 - 18 = 20 - 18 = \mathbf{2}$$

.
Euler's formula is verified!


For an $n$-gonal prism: $F = n+2$, $V = 2n$, $E = 3n$. For $n=6$: $F=8, V=12, E=18$. $8+12-18=2$.
4
Differentiate between a "Prism" and a "Pyramid" with examples.
उत्तर एवं व्याख्या देखें
उत्तर:

• Prism: A polyhedron that has two congruent, parallel polygonal bases (top and bottom), and all its lateral side faces are parallelograms or rectangles.
Examples: Triangular prism, Cuboid (rectangular prism), Hexagonal prism.
• Pyramid: A polyhedron that has only ONE polygonal base, and all its lateral side faces are triangles that meet at a single common vertex (Apex).
Examples: Square pyramid (Great Pyramid of Giza), Tetrahedron (triangular pyramid).


Prisms have 2 congruent parallel bases and rectangular sides; Pyramids have 1 base and triangular sides meeting at an apex.
5
What is a "Net" of a 3D solid? Describe the net of a closed right circular cylinder.
उत्तर एवं व्याख्या देखें
उत्तर:

• Net: A flat two-dimensional geometrical pattern of connected plane faces that can be folded along edges to construct the 3-dimensional solid shape.
• Net of a Closed Cylinder: Consists of three components:
1. One Rectangle representing the unrolled curved lateral surface (where length $= 2\pi r$ and breadth $= h$).
2. Two identical Circles (of radius $r$) attached to opposite edges of the rectangle, representing the top and bottom circular bases.


A 2D folding pattern; the net of a cylinder consists of a rectangle and two circular base lids.
6
Differentiate between an "Oblique Sketch" and an "Isometric Sketch".
उत्तर एवं व्याख्या देखें
उत्तर:

• Oblique Sketch:
- Drawn on standard squared graph paper.
- The front face is drawn with its true geometrical shape and exact dimensions.
- The receding depth edges are drawn at a slanting angle (usually $45^\circ$), but are NOT drawn to true scale.
• Isometric Sketch:
- Drawn on Isometric Dot Paper with triangular grid dots.
- The three axes are inclined at equal angles of $120^\circ$ ($30^\circ$ to the horizontal).
- True Scale: All measurements (length, breadth, height) are drawn strictly proportional to their actual scale.


Oblique: front face true size, depth not to scale; Isometric: drawn on 30° dot paper, ALL dimensions strictly to true scale.
7
What are the three standard projections used to view a 3D solid? Explain with reference to a brick.
उत्तर एवं व्याख्या देखें
उत्तर:

The three orthogonal projection views are:
1. Front View (Elevation): What the object looks like when viewed directly from the front (for a brick, a rectangle of length $\times$ height).
2. Side View (End Elevation): What the object looks like when viewed directly from the side (for a brick, a rectangle of breadth $\times$ height).
3. Top View (Plan): What the object looks like when viewed directly from above (for a brick, a rectangle of length $\times$ breadth).


Front view (elevation), Side view, and Top view (plan).
8
Why is a cylinder NOT considered a polyhedron?
उत्तर एवं व्याख्या देखें
उत्तर:

• By definition, a polyhedron is a 3D solid bounded entirely by flat polygonal faces (rectangles, triangles, squares, etc.) that meet at straight line edges.
• A cylinder has a smooth, continuous curved lateral surface, and its two bases are circles (which are curved shapes, not polygons).
• Because it contains curved surfaces and no straight polygon edges, a cylinder is NOT a polyhedron.


Polyhedra must be bounded entirely by flat polygonal faces; cylinders contain curved surfaces and circular non-polygonal bases.
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त्रिविमीय आकारों का द्विविमीय निरूपण (Representing 3D in 2D) में कोई संदेह या प्रश्न है? हमारे AI अध्ययन मित्र से तुरंत समझें।

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