NCERT Solutions for Class 7 Maths Chapter 13: Visualising Solid Shapes — Free PDF

This chapter helps you see the world in three dimensions. You will learn to identify and describe 3D shapes (solids) and understand how they relate to 2D representations.

Key topics:
- Faces, edges, and vertices of 3D shapes
- Nets of 3D shapes
- Drawing oblique sketches and isometric sketches
- Viewing 3D shapes from different positions (top view, front view, side view)
- Euler's formula: $F + V - E = 2$

The chapter has 4 exercises. Below are solved examples from each.

Understanding solid shapes is essential not just for mathematics but also for subjects like science (molecular geometry), art (perspective drawing), and engineering (design and manufacturing). The ability to mentally rotate objects, interpret 2D diagrams of 3D shapes, and count faces, edges, and vertices builds spatial reasoning skills that are valuable throughout your academic career. This chapter also introduces you to Euler's formula, one of the most elegant results in geometry.

Solid (3D shape): An object that occupies space and has three dimensions — length, breadth, and height.

Polyhedron: A solid whose faces are all flat polygons. Examples: cube, cuboid, prism, pyramid. Non-examples: sphere, cylinder, cone (they have curved surfaces).

Face: A flat surface of a solid. A cube has $6$ faces.

Edge: A line segment where two faces meet. A cube has $12$ edges.

Vertex: A point where three or more edges meet. A cube has $8$ vertices.

Euler's formula: For any convex polyhedron:

Net: A 2D pattern that can be cut out and folded to form a 3D shape. Each solid has one or more valid nets.

Oblique sketch: A freehand 3D drawing on plain paper where the front face is drawn in its true shape, and the depth is shown at an angle.

Isometric sketch: A more accurate 3D drawing made on isometric dot paper (dots arranged in equilateral triangular patterns at $60^\circ$ angles).

Cross-section: The 2D shape you get when you slice through a 3D solid with a flat plane.

Views of a solid: A solid can be viewed from different directions:
- Front view (elevation): what you see from the front
- Top view (plan): what you see looking down from above
- Side view (profile): what you see from the side

Every 3D shape (polyhedron) has faces (flat surfaces), edges (line segments where faces meet), and vertices (corner points).

Q1. Count the faces, edges, and vertices of a cuboid.

• Faces: $6$ (all rectangles)
  - Edges: $12$
  - Vertices: $8$

Verify with Euler's formula: $F + V - E = 6 + 8 - 12 = 2$. Correct.

Q2. Count for a triangular prism.

• Faces: $5$ ($2$ triangular $+$ $3$ rectangular)
  - Edges: $9$
  - Vertices: $6$

Check: $5 + 6 - 9 = 2$. Correct.

Q3. Count for a square pyramid.

• Faces: $5$ ($1$ square base $+$ $4$ triangular faces)
  - Edges: $8$
  - Vertices: $5$

Check: $5 + 5 - 8 = 2$. Correct.

Q4. Complete the table:

**Q5. A polyhedron has $8$ faces and $6$ vertices. How many edges does it have?**

Using Euler's formula: $F + V - E = 2$
$8 + 6 - E = 2$
$E = 12$

**Q6. Can a polyhedron have $10$ faces, $20$ edges, and $15$ vertices?**

Check: $F + V - E = 10 + 15 - 20 = 5 \neq 2$.

No, this is not possible because it violates Euler's formula.

A net is a 2D pattern that can be folded to form a 3D shape.

Q1. Which of the following is a net for a cube?

A valid cube net has exactly $6$ squares arranged so that when folded, each pair of opposite faces doesn't overlap. There are $11$ different nets for a cube.

Key rule: In a cube net, no more than $4$ squares can be in a row, and there must be exactly $6$ squares total.

How to check if a net is valid: Imagine folding it mentally. Each face must have a unique position when folded. Two squares that would overlap when folded make the net invalid.

Q2. Describe the net of a cylinder.

The net of a cylinder consists of:
- $2$ circles (top and bottom)
- $1$ rectangle (the curved surface unrolled)

The rectangle's width equals the height of the cylinder, and its length equals the circumference of the circle ($2\pi r$). When you roll up the rectangle and attach the two circles, you get back the cylinder.

Q3. Describe the net of a cone.

The net of a cone consists of:
- $1$ circle (the base)
- $1$ sector of a circle (the curved surface unrolled)

The radius of the sector equals the slant height of the cone, and the arc length of the sector equals the circumference of the base circle.

Q4. Can this net form a valid tetrahedron?

A tetrahedron net has $4$ equilateral triangles. A valid net arranges them so they fold into $4$ faces without overlap. The most common tetrahedron net is a row of $3$ triangles with the $4$th attached to one side of the middle triangle.

Q5. Draw the net of a square pyramid.

A square pyramid net consists of:
- $1$ square (the base)
- $4$ isosceles triangles (the lateral faces), each attached to one side of the square

When folded up, the four triangles meet at the apex.

Oblique sketch: A rough 3D drawing on plain paper where the front face is drawn in its true shape.

Isometric sketch: A more precise 3D drawing on isometric dot paper (dots at $60°$ angles).

**Q1. Draw an oblique sketch of a cuboid $4 \times 3 \times 2$.**

Steps:
1. Draw the front face as a $4 \times 3$ rectangle.
2. Draw the back face as a smaller, shifted rectangle (shifted diagonally upward and to the right).
3. Connect the corresponding corners with slanted lines.
4. Use dashed lines for the edges that would be hidden behind the solid.

The front face retains its true shape and proportions, while the depth is shown at an angle (typically $45^\circ$) and is often drawn at half the actual length for a more realistic appearance.

Q2. How do you draw an isometric sketch of a cube on dot paper?

Steps:
1. Draw a rhombus for the top face using the isometric dots.
2. Draw vertical lines downward from the three visible corners.
3. Complete the bottom edges using the dot grid.
4. Each edge of the cube appears as an equal length segment along the isometric axes.

On isometric dot paper, three axes meet at $120^\circ$ angles, which gives the sketch a realistic 3D appearance.

Q3. What are the three standard views of a 3D object?

• Front view (elevation)
  - Top view (plan)
  - Side view (profile)

Examples of views for common solids:

Q1. What cross-section do you get when you cut a cube with a plane parallel to its base?

A square (same size as the base).

Q2. What cross-section do you get when you vertically cut a cylinder?

A rectangle (height $=$ cylinder height, width $=$ diameter).

Q3. What cross-section do you get when you cut a cone horizontally (parallel to base)?

A circle (smaller than the base). The closer the cut is to the apex, the smaller the circle.

Q4. What shape do you see from the top of a cone?

A circle (with a point at the centre representing the apex).

Q5. Match the solid with its top view:

Q6. What cross-section do you get when you cut a cube diagonally from one edge to the opposite edge?

A rectangle (specifically, a rectangle whose length is the diagonal of a face). If the diagonal cut passes through $4$ edges symmetrically, you can even get a regular hexagon.

Q7. A solid shape has the same top view, front view, and side view — all circles. What is the solid?

A sphere. It is the only common solid that looks the same from every direction.

Example 1: Using Euler's formula to find missing values

A polyhedron has $12$ edges and $8$ vertices. Find the number of faces. Identify the solid.

Solution:
$F + V - E = 2$
$F + 8 - 12 = 2$
$F = 6$

A solid with $6$ faces, $8$ vertices, and $12$ edges is a cuboid (or cube if all faces are squares).

Example 2: Verifying a prism

A hexagonal prism has two hexagonal bases. Find $F$, $V$, and $E$ and verify Euler's formula.

Solution:
- Faces: $2$ hexagonal bases $+$ $6$ rectangular lateral faces $= 8$
- Vertices: Each hexagon has $6$ vertices $\times 2 = 12$
- Edges: $6$ edges on top $+$ $6$ edges on bottom $+$ $6$ vertical edges $= 18$

Check: $8 + 12 - 18 = 2$ ✓

Example 3: Identifying a solid from its net

A net consists of $2$ equilateral triangles and $3$ rectangles. What solid does it form?

Solution: A triangular prism. The two triangles form the end faces, and the three rectangles form the lateral faces.

Example 4: Cross-section of a sphere

What shape do you get when you cut a sphere with any flat plane?

Solution: Always a circle. If the plane passes through the centre of the sphere, you get the largest possible circle (called a great circle). Otherwise you get a smaller circle.

Mistake 1: Confusing faces and surfaces.
A cylinder has $3$ surfaces ($2$ flat circles and $1$ curved surface), but Euler's formula applies only to polyhedra with flat faces. Do not apply Euler's formula to cylinders, cones, or spheres.

Mistake 2: Miscounting edges at the base of a pyramid.
A square pyramid has $4$ base edges $+$ $4$ slant edges $= 8$ edges total. Students often forget the base edges or the slant edges, getting $4$ instead of $8$.

Mistake 3: Thinking all arrangements of 6 squares form a cube net.
There are only $11$ valid cube nets out of $35$ possible hexominoes (arrangements of $6$ connected squares). A common invalid net is a $2 \times 3$ rectangle — this cannot fold into a cube because opposite faces overlap.

Mistake 4: Drawing hidden edges as solid lines.
In oblique and isometric sketches, edges that are hidden behind the solid should be drawn as dashed lines, not solid lines. This is important for clarity and marks in exams.

Mistake 5: Confusing prism and pyramid.
A prism has two identical parallel bases connected by rectangles. A pyramid has one base with all lateral faces meeting at a single point (apex). A triangular prism has $5$ faces; a triangular pyramid (tetrahedron) has $4$ faces.

Q1. A polyhedron has $5$ faces and $6$ vertices. How many edges does it have? Name the solid.

Q2. How many faces, edges, and vertices does a pentagonal pyramid have?

Q3. Can a net made of $5$ equilateral triangles form any solid? If so, which one?

Q4. What cross-section do you get when you cut a triangular prism parallel to its triangular face?

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Answers:

A1. $F + V - E = 2 \implies 5 + 6 - E = 2 \implies E = 9$. A solid with $5$ faces, $6$ vertices, and $9$ edges is a triangular prism.

A2. Faces: $1$ pentagonal base $+ 5$ triangular faces $= 6$. Vertices: $5$ base vertices $+ 1$ apex $= 6$. Edges: $5$ base edges $+ 5$ slant edges $= 10$. Check: $6 + 6 - 10 = 2$ ✓.

A3. No standard convex polyhedron is formed by exactly $5$ equilateral triangles. A tetrahedron uses $4$ and an octahedron uses $8$. Five equilateral triangles cannot fold into a closed solid.

A4. A triangle (congruent to the base triangle). Cutting parallel to a base of a prism always produces a cross-section identical to that base.

Use this table to quickly check your face/edge/vertex counts. Every entry satisfies Euler's formula $F + V - E = 2$.

Patterns for prisms: An $n$-sided prism has $n + 2$ faces, $2n$ vertices, and $3n$ edges.

Patterns for pyramids: An $n$-sided pyramid has $n + 1$ faces, $n + 1$ vertices, and $2n$ edges.

These formulas are extremely useful for quickly finding any missing value without counting individually. Simply plug in $n$ (the number of sides of the base polygon) to get all three counts.

Solids that do NOT satisfy Euler's formula (non-polyhedra):
- Cylinder: $3$ surfaces ($2$ flat + $1$ curved) — Euler's formula does not apply
- Cone: $2$ surfaces ($1$ flat + $1$ curved) — Euler's formula does not apply
- Sphere: $1$ surface (curved) — Euler's formula does not apply