Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

This hard level explores advanced concepts including detailed comparisons, multiple nets, and complex solid structures.

Master these concepts to excel in spatial reasoning and higher-level geometry.

1. Write Euler's formula.

Euler's formula: F + V - E =

Perfect! This fundamental formula relates faces, vertices, and edges.

2. Which solid has only one vertex and one edge?

Answer:

Correct! A cone has a unique structure with just 1 vertex at the apex.

3. Name the solid which has one curved face and two circular faces.

Answer:

Excellent! A cylinder has 2 circular faces (top and bottom) and 1 curved surface.

4. What do we call the flat surfaces of a solid?

Answer:

Perfect! Faces are the flat or curved surfaces that bound a solid.

5. How many faces does a pentagonal pyramid have?

Answer: faces

Great! 1 pentagonal base + 5 triangular faces = 6 faces total.

Drag each property to its correct solid:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Write the number of faces, vertices, and edges of a triangular prism.

Faces = (2 triangular + 3 rectangular)

Vertices = (3 on each triangular face)

Edges = (3 + 3 + 3)

Excellent! F + V - E = 5 + 6 - 9 = 2 ✓

2. Verify Euler's formula for a triangular pyramid.

Faces (F) = (all triangles)

Vertices (V) = (3 base + 1 apex)

Edges (E) = (3 base + 3 lateral)

Verification: F + V - E = 4 + 4 - 6 = ✓

Perfect! Euler's formula is verified.

3. Compare the features of a cone and a cylinder.

Similarities:

Both have circular basescircular faces and .

Differences:

Cone has circular face, Cylinder has

Cone has vertex, Cylinder has vertices

Great comparison! Both are important curved solids.

4. Draw rough sketches of: (a) Cone (b) Triangular pyramid.

Note: Draw these on your answer sheet:

(a) Cone: Circular base with curved surface meeting at apex

(b) Triangular pyramid: Triangular base with 3 triangular faces meeting at apex

Did you complete both sketches? YesDone

Excellent! Remember to label all parts clearly.

5. Identify the shape formed when a cube is sliced parallel to its base.

Answer:

Perfect! A cross-section parallel to the base of a cube is always a square.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Verify Euler's formula F + V - E = 2 for: (a) Square pyramid (b) Triangular prism.

(a) Square pyramid:

Faces (F) = (1 square + triangles)

Vertices (V) = (4 base + 1 apex)

Edges (E) = (4 base + 4 lateral)

F + V - E = 5 + 5 - 8 = ✓

(b) Triangular prism:

Faces (F) = (2 triangles + 3 rectangles)

Vertices (V) = (3 + 3)

Edges (E) = (3 + 3 + 3)

F + V - E = 5 + 6 - 9 = ✓

Excellent! Euler's formula is verified for both solids.

2. Draw two different nets of a cube. Label all faces and explain how they fold into a cube.

Note: Draw on your answer sheet:

A cube has different possible nets

Each net must have exactly squares

Common net patterns:

Net 1: Cross/T-shape (4 in a row, 1 above, 1 below)

Net 2: L-shape or zigzag pattern

When folded, opposite faces of the net come together to form the .

Did you complete both nets with labels? YesDone

Great! Understanding nets helps in visualizing 3D structures.

3. Write a detailed comparison between prisms and pyramids with examples and diagrams.

PRISMS:

Have identical parallel bases

All lateral faces are rectanglesparallelograms

Example: Triangular prism has faces

PYRAMIDS:

Have base (polygon)

All lateral faces are meeting at apex

Example: Square pyramid has faces

Key Difference: Prisms have uniform cross-sectionsparallel bases, pyramids taper to a pointhave an apex.

Did you draw diagrams on your sheet? YesDone

Excellent comparison! Both are important polyhedra families.

4. A cylinder is considered as a combination of two circles and one rectangle. Justify this statement using its net and surface structure.

Net of Cylinder:

Contains circles (top and bottom bases)

Contains rectangle (when curved surface is unrolled)

Rectangle dimensions:

Width = height of cylinderh

Length = circumference of base circle2πr

Assembly:

When rectangle is rolled, its edges form a

Circles are attached to top and bottomthe ends of this curved surface

This creates the complete .

Perfect! Understanding nets helps us see how 3D shapes are constructed.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. The 3D shape that has 1 curved and 1 flat circular surface is:

(a) Cone (b) Cylinder (c) Sphere (d) Cube

Correct! A cone has 1 flat circular base and 1 curved surface.

7. A solid having 6 faces, 8 vertices, and 12 edges is:

(a) Cube (b) Cuboid (c) Both a and b (d) Cylinder

Perfect! Both cube and cuboid have the same F, V, E values.

8. The 3D figure that rolls easily is:

(a) Cube (b) Sphere (c) Cuboid (d) Pyramid

Excellent! A sphere rolls perfectly in all directions due to its curved surface.

9. The solid with only one vertex is:

(a) Cube (b) Cone (c) Cylinder (d) Sphere

Perfect! A cone has exactly one vertex at its apex (top point).

10. The number of faces in a hexagonal prism is:

(a) 5 (b) 6 (c) 8 (d) 12

Correct! 2 hexagonal bases + 6 rectangular lateral faces = 8 faces total.

🎉 Exceptional Achievement! You've Mastered Advanced 3D Geometry!

Here's what you learned:

• Euler's Formula (F + V - E = 2):

  • Universal formula for all polyhedra
  • Verification examples:
    • Cube: 6 + 8 - 12 = 2
    • Square pyramid: 5 + 5 - 8 = 2
    • Triangular prism: 5 + 6 - 9 = 2

• Advanced Solid Properties:

  Solid	Faces	Vertices	Edges
  Triangular Prism	5	6	9
  Square Pyramid	5	5	8
  Pentagonal Pyramid	6	6	10
  Hexagonal Prism	8	12	18

• Multiple Nets:

  • A cube has 11 different possible nets
  • All nets must have 6 connected squares
  • Different arrangements fold into the same cube

• Prisms vs Pyramids:

  • Prisms: 2 parallel identical bases, rectangular lateral faces
  • Pyramids: 1 base, triangular lateral faces meeting at apex
  • Both follow Euler's formula

• Cylinder Structure:

  • Net consists of: 2 circles + 1 rectangle
  • Rectangle dimensions: height × circumference (2πr)
  • When rolled and assembled, forms complete cylinder

• Special Properties:

  • Cone: Only solid with 1 vertex and 1 edge
  • Sphere: Only solid with 0 vertices, 0 edges
  • Cylinder: 2 circular faces, 1 curved surface
  • Hexagonal prism: 8 faces total

• Cross-Sections:

  • Slicing parallel to base preserves base shape
  • Cube sliced parallel → Square
  • Cylinder sliced parallel → Circle

• Spatial Reasoning:

  • Understanding how 2D nets fold into 3D shapes
  • Visualizing internal structure of complex solids
  • Comparing and contrasting similar shapes

These advanced concepts are essential for architecture, engineering, and higher mathematics!