Hard Level Worksheet Questions

Interactive Exploring Geometrical Figures Worksheet

Advanced geometrical figures involve complex polyhedra, polygon angle calculations, Euler's formula applications, and 3D visualization. Understanding these concepts is crucial for higher geometry, engineering design, and spatial reasoning in mathematics.

Let's explore advanced concepts of 2D and 3D geometrical figures.

1. Define a polyhedron and give one example.

Definition: 3D figure with flat faces2D polygonCurved surface figureCircle

Example: CubeSphereCylinderCircle

Perfect! A polyhedron is a 3D figure with flat polygonal faces, like a cube.

2. How many faces does a triangular prism have?

Step 1: Count bases: 2134 triangular faces

Step 2: Count lateral faces: 3245 rectangular faces

Step 3: Total faces = 2 + 3 = 5467

Excellent! Triangular prism has 2 triangular bases + 3 rectangular faces = 5 total.

3. Name the polygon with all sides equal and all angles equal. Regular polygonIrregular polygonScalene polygonRight polygon

Correct! A regular polygon has all sides and all angles equal.

4. Write the number of vertices of a hexagonal pyramid.

Step 1: Base vertices: 6578 (hexagon has 6 vertices)

Step 2: Apex vertices: 1203

Step 3: Total vertices = 6 + 1 = 7689

Perfect! Hexagonal pyramid: 6 base vertices + 1 apex = 7 vertices.

5. What is the sum of the interior angles of a pentagon?

Step 1: Apply formula: (n-2) × 180° where n = 5463

Step 2: Sum = (5-2) × 180° = 3 × 180° = 540°480°600°360°

Excellent! Pentagon interior angles sum: (5-2) × 180° = 540°.

6. Write Euler's formula for a polyhedron. F + V - E = 2F - V + E = 2F + V + E = 2F - V - E = 2

Perfect! Euler's formula: Faces + Vertices - Edges = 2.

7. Name a 3D figure that has only one curved surface. SphereConeCylinderCube

Correct! A sphere has only one continuous curved surface.

8. In a quadrilateral, if three angles are 75°, 95°, and 110°, find the fourth angle.

Step 1: Sum of angles in quadrilateral = 360°180°540°720°

Step 2: Fourth angle = 360° - (75° + 95° + 110°) = 360° - 280° = 80°70°90°85°

Great! Quadrilateral angles must sum to 360°, so fourth angle = 80°.

9. How many edges are there in a cube?

Method: Count systematically: 128610 edges

Perfect! Cube has 12 edges (4 on top + 4 on bottom + 4 vertical).

10. Write the number of diagonals in an octagon.

Step 1: Apply formula: n(n-3)/2 where n = 8796

Step 2: Diagonals = 8(8-3)/2 = 8×5/2 = 20162418

Excellent! Octagon has 20 diagonals using formula n(n-3)/2.

Drag each concept to its correct category:

Part B: Short Answer Questions (2 Marks Each)

1. Find the measure of each exterior angle of a regular decagon.

Step 1: Apply exterior angle formula

Each exterior angle = 360°÷numberofsides360° × number of sides

Each exterior angle = 360° ÷ 10 = 36°30°40°45°

Perfect! Regular decagon: each exterior angle = 360°/10 = 36°.

2. The sum of interior angles of a polygon is 1260°. Find the number of sides.

Step 1: Apply interior angle sum formula

(n-2) × 180° = 1260°

Step 2: Solve for n

n-2 = 1260° ÷ 180° = 7689

n = 7 + 2 = 98107

Excellent! The polygon has 9 sides (nonagon).

3. A polyhedron has 12 faces and 20 vertices. Find the number of edges using Euler's formula.

Step 1: Apply Euler's formula

We know that Euler's formula: F + V - E = 2F + V + E = 2

+ - E = 2

Step 2: Solve for E

E = 30283226

Perfect! This polyhedron (icosahedron) has 30 edges.

4. A quadrilateral has angles in the ratio 3 : 4 : 5 : 6. Find all angles.

Step 1: Set up equation

Let angles be 3x, 4x, 5x, 6x

Sum: 3x + 4x + 5x + 6x = x = °

Step 2: Solve for x

x = 360° ÷ 18 = 20°18°22°24°

Step 3: Find all angles

Angles: 3×20° = 60°54°66°72°, 4×20° = 80°72°88°96°, 5×20° = 100°90°110°120°, 6×20° = 120°108°132°144°

Outstanding! The angles are 60°, 80°, 100°, and 120°.

Part C: Long Answer Questions (4 Marks Each)

1. The sum of interior angles of a polygon is 1620°. Find: (i) Number of sides (ii) Each interior angle if regular.

Step 1: Find number of sides

Using formula: (n-2) × 180°(n+2) × 180° = 1620°

n-2 = 1620° ÷ 180° = 98107

Therefore, n = 9 + 2 = 1110129

Step 2: Find each interior angle if regular

Each interior angle = Total sum ÷ Number of sidesTotal sum × Number of sides

Each interior angle = 1620° ÷ 11 = 147.27°145°150°140°

Excellent! The polygon has 11 sides, each interior angle = 147.27°.

2. A polyhedron has 8 triangular faces and 6 vertices. Find edges and verify Euler's formula.

Step 1: Given information

Faces (F) = , Vertices (V) =

Step 2: Apply Euler's formula

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

E = 8 + 6 - 2 = 12101416

Step 3: Verify Euler's formula

F + V - E = 8 + 6 - 12 = 2014

Perfect! This octahedron has 12 edges, and Euler's formula is verified.

3. The exterior angle of a regular polygon is 40°. Find: (i) Number of sides (ii) Each interior angle.

Step 1: Find number of sides

For regular polygon: Each exterior angle = 360° ÷ n180° ÷ n

n = 98107

Step 2: Find each interior angle

Each interior angle = 180° - exterior angle180° + exterior angle

Each interior angle = 180° - 40° = 140°135°145°150°

Outstanding! The polygon has 9 sides (nonagon), each interior angle = 140°.

4. Draw net for square pyramid. Write its faces, vertices, and edges.

Step 1: Analyze square pyramid structure

Net consists of: 1 square base + 4 triangular faces

Step 2: Count faces

Number of faces = 1 + 4 = 5467

Step 3: Count vertices

Number of vertices = 4 (base) + 1 (apex) = 5463

Step 4: Count edges

Number of edges = 4 (base) + 4 (lateral) = 86107

Step 5: Verify using Euler's formula

F + V - E = 5 + 5 - 8 = 2013

Excellent! Square pyramid: F=5, V=5, E=8, satisfying Euler's formula.

5. A polygon has twice as many diagonals as its number of sides. Find the number of sides.

Step 1: Set up equation

Let number of sides = n

Number of diagonals = n(n-3)/2n(n+3)/2

Given: Number of diagonals = 2n

Step 2: Solve equation

n(n-3)/2 = 2n

n(n-3) =

n22n - 3n3 = 4n

n2 - =

n(n-7n+7) = 0

Since n > 0, n = 7685

Step 3: Verify

Diagonals in heptagon = 7(7-3)/2 =

Twice the number of sides = 2×7 =

Perfect! The polygon is a heptagon (7 sides).

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. A polyhedron that has only triangular faces is called:

(a) Cube (b) Prism (c) Tetrahedron (d) Cylinder

Correct! A tetrahedron has 4 triangular faces - the simplest polyhedron.

2. The sum of exterior angles of any polygon is always:

(a) 180° (b) 360° (c) 540° (d) Depends on sides

Correct! The sum of exterior angles of any polygon is always 360°.

3. A solid having one curved surface and no edges is a:

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

Correct! A sphere has only one continuous curved surface and no edges or vertices.

4. The number of diagonals in a hexagon is:

(a) 6 (b) 9 (c) 12 (d) 15

Correct! Using formula n(n-3)/2: hexagon has 6(6-3)/2 = 9 diagonals.

5. The measure of each interior angle of a regular pentagon is:

(a) 100° (b) 108° (c) 120° (d) 90°

Correct! Regular pentagon: (5-2)×180°/5 = 540°/5 = 108°.

🎉 Outstanding! You've Mastered Hard Level Geometrical Figures! Here's what you accomplished:

✓ Advanced Polygon Properties: Interior/exterior angle calculations, diagonal formulas

✓ Complex Polyhedra Analysis: Euler's formula applications, faces/edges/vertices counting

✓ Regular Polygon Mastery: Understanding angle relationships and classifications

✓ 3D Visualization Skills: Nets, orthographic views, and spatial reasoning

✓ Problem-Solving Strategies: Ratio problems, equation setup, and systematic analysis

✓ Geometric Classifications: Distinguishing between different types of 3D figures

✓ Formula Applications: Using mathematical formulas for geometric calculations

✓ Real-World Connections: Understanding how geometry applies to engineering and design

Your expertise in advanced geometrical concepts prepares you for trigonometry, coordinate geometry, solid geometry, and engineering applications!