Revision and important questions: Exploring Some Geometric Themes

7.1 Formula box

• Sierpinski carpet: Rₙ = 8ⁿ, Hₙ = (8ⁿ - 1)/7, area = (8/9)ⁿ.
• Sierpinski triangle: Rₙ = 3ⁿ, Hₙ = (3ⁿ - 1)/2, area = (3/4)ⁿ.
• Koch snowflake: Sₙ = 3 × 4ⁿ, perimeter = 3 × (4/3)ⁿ.
• Prism with n-sided base: F = n + 2, E = 3n, V = 2n.
• Pyramid with n-sided base: F = n + 1, E = 2n, V = n + 1.
• Cube: 6 faces, 12 edges, 8 vertices, 11 nets.
• Regular tetrahedron: 4 faces, 6 edges, 4 vertices, 2 nets.
• Projection of a segment of length l: p = l cos θ; p² relations for the three views.
• Shortest path on a cuboid: unfold to a net and use the Baudhayana theorem.

7.2 Quick checks for drawing questions

Front view is the projection on the vertical plane, top view on the horizontal plane and side view on the side plane. Always give all three for an object, because one view rarely fixes the solid. For a net, check that every edge of the solid appears twice and folds without overlap. For a shortest-path problem, unfold so that both points lie in the net, join them with a straight line, and confirm the line does not leave the paper shape. For isometric drawings, the vertical grid direction is height, and the two diagonal directions are length and depth; keep parallel edges parallel. A cube stack is counted layer by layer, from the bottom up, so the total is the sum of the visible and hidden cubes in each layer.

7.3 Ten important questions

Question 1. A Sierpinski carpet starts with area 1. Find the area at steps 1, 2 and 3. Answer. (8/9), (8/9)² = 64/81, and (8/9)³ = 512/729.

Question 2. For the Sierpinski triangle, find R_5 and H_5. Answer. R_5 = 3⁵ = 243; H_5 = (243 - 1)/2 = 121.

Question 3. Find the number of sides and the perimeter of the Koch snowflake at step 3, starting with side 1. Answer. S_3 = 3 × 4³ = 192; P_3 = 3 × (4/3)³ = 3 × 64/27 = 64/9 units.

Question 4. A prism has 7-sided congruent faces. How many faces, edges and vertices? Answer. F = 7 + 2 = 9; E = 3 × 7 = 21; V = 2 × 7 = 14.

Question 5. A pyramid has a 6-sided base. How many faces, edges and vertices? Answer. F = 6 + 1 = 7; E = 2 × 6 = 12; V = 6 + 1 = 7.

Question 6. How many nets does a cube have? How many does a regular tetrahedron have? Answer. A cube has 11 nets up to rotation and reflection; a regular tetrahedron has 2.

Question 7. The projection of a line segment on a plane has length 6 cm while the segment is 10 cm. What is the angle between the line and the plane? Answer. p = l cos θ gives 6 = 10 cos θ, so cos θ = 0.6 and θ is the angle whose cosine is 0.6, about 53°.

Question 8. A cuboid unfolds so that two points form a right triangle with legs 9 cm and 12 cm. What is the shortest surface path between them? Answer. d = √(9² + 12²) = √225 = 15 cm.

Question 9. What are the front, top and side views of an upright cone? Answer. Front: a triangle. Top: a circle with a point at its centre. Side: a triangle congruent to the front view.

Question 10. How many different free shapes can be made from four cubes joined face to face? Answer. 8: the 5 flat arrangements and 3 three-dimensional arrangements.

7.4 Common mistakes to avoid

• Mixing the fractal factors: carpet multiplies the count by 8 and the area by 8/9; gasket multiplies the count by 3 and the area by 3/4.
• Forgetting that perimeter can grow while area stays bounded, as in the Koch snowflake.
• Counting a hole more than once; older holes remain at every step.
• Confusing a net with a drawing of the solid: the net must fold with no overlap.
• Using the number of sides of the base as the number of prism faces. The two end faces must be added.
• Forgetting that a pyramid has only one base face but a prism has two.
• Treating a single projection as if it determined the solid. Different solids can share a projection.
• Drawing isometric edges along grid directions that do not match the solid's three axes.

When an answer looks too small or too large, test it with a rough estimate before moving on; a missing square or a dropped zero usually shows up at once.