Fractals and Visualising Solids | A

Question 3

Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get?

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Solution
Understand the Question
  • An equilateral triangle has all 33 sides equal and all 33 interior angles equal to 60∘60^\circ.
  • When each side is divided into 33 equal parts and the 33 corners are cut off, 33 smaller equilateral triangles are removed from the vertices.
  • The resulting 66-sided polygon (hexagon) has all sides of equal length and all interior angles equal to 120∘120^\circ, forming a regular hexagon.

Step 1 · Divide Triangle Sides into Thirds

Consider an equilateral triangle ABC\text{ABC} with side length 3s3s.Diagram 1

Dividing each side into three equal parts of length ss: On side AB: AD1=D1D2=D2B=s\text{On side AB: } \text{AD}_1 = \text{D}_1\text{D}_2 = \text{D}_2\text{B} = s On side BC: BE1=E1E2=E2C=s\text{On side BC: } \text{BE}_1 = \text{E}_1\text{E}_2 = \text{E}_2\text{C} = s On side CA: CF1=F1F2=F2A=s\text{On side CA: } \text{CF}_1 = \text{F}_1\text{F}_2 = \text{F}_2\text{A} = s

Step 2 · Identify the Cut-off Corner Triangles

Cutting along the marks near each corner removes three corner triangles: ΔAD1F2\Delta\text{AD}_1\text{F}_2, ΔBE1D2\Delta\text{BE}_1\text{D}_2, and ΔCF1E2\Delta\text{CF}_1\text{E}_2.Diagram 2

For ΔAD1F2\Delta\text{AD}_1\text{F}_2:

  • AD1=AF2=s\text{AD}_1 = \text{AF}_2 = s
  • ∠A=60∘\angle\text{A} = 60^\circ

Since it is an isosceles triangle with an included angle of 60∘60^\circ, ΔAD1F2\Delta\text{AD}_1\text{F}_2 is equilateral, so: D1F2=s\text{D}_1\text{F}_2 = s

Similarly, ΔBE1D2\Delta\text{BE}_1\text{D}_2 and ΔCF1E2\Delta\text{CF}_1\text{E}_2 are equilateral triangles with side length ss: E1D2=s\text{E}_1\text{D}_2 = s F1E2=s\text{F}_1\text{E}_2 = s

Step 3 · Determine the Resulting Shape

The remaining polygon has 66 vertices: D1,F2,F1,E2,E1,D2D_1, F_2, F_1, E_2, E_1, D_2.

Calculating the lengths of all six sides: D1F2=s\text{D}_1\text{F}_2 = s

F2F1=AC−AF2−CF1=3s−s−s=s\begin{aligned} \text{F}_2\text{F}_1 &= \text{AC} - \text{AF}_2 - \text{CF}_1 \\[0.6em] &= 3s - s - s \\[0.6em] &= s \end{aligned}

F1E2=s\text{F}_1\text{E}_2 = s

E2E1=BC−CE2−BE1=3s−s−s=s\begin{aligned} \text{E}_2\text{E}_1 &= \text{BC} - \text{CE}_2 - \text{BE}_1 \\[0.6em] &= 3s - s - s \\[0.6em] &= s \end{aligned}

E1D2=s\text{E}_1\text{D}_2 = s

D2D1=AB−AD1−BD2=3s−s−s=s\begin{aligned} \text{D}_2\text{D}_1 &= \text{AB} - \text{AD}_1 - \text{BD}_2 \\[0.6em] &= 3s - s - s \\[0.6em] &= s \end{aligned}

Each interior angle of the resulting hexagon (e.g., at vertex D1D_1 on line segment AB\text{AB}) is:

∠F2D1D2=∠AD1D2−∠AD1F2=180∘−60∘=120∘\begin{aligned} \angle \text{F}_2\text{D}_1\text{D}_2 &= \angle \text{AD}_1\text{D}_2 - \angle \text{AD}_1\text{F}_2 \\[0.6em] &= 180^\circ - 60^\circ \\[0.6em] &= 120^\circ \end{aligned}

Since all 66 sides are equal (ss) and all 66 interior angles are equal (120∘120^\circ), the resulting shape is a regular hexagon.

Answer

A regular hexagon

Common Mistakes
  • Calling it simply a Hexagon: Stating only "hexagon" instead of regular hexagon; since all side lengths and interior angles are equal, the hexagon is specifically regular.
  • Incorrect Side Length of Cut: Assuming the cut line D1F2\text{D}_1\text{F}_2 has a different length from the remaining side F2F1\text{F}_2\text{F}_1, rather than showing that the cut corner is an equilateral triangle with side ss.

More questions in A

Q1

Build it in Your Imagination

We will start this section by practising visualisation. For each prompt, feel free to talk to your partner, gesture, draw it in the air — but do not actually draw on paper!

  1. Picture your name, then read off the letters backwards. Make sure to do this by sight, not by sound — really see your name! Now try with your friend's name.
Q2

Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What shape is left over? How can you reassemble the four corners to make another square?

Q3

Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get?

Q4

Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left?

Q5

Net of a sphere? Experiment and see if you can make a paper cutout that can perfectly wrap around a ball without leaving any wrinkles, gaps or overlaps.

Q6

Place an object in front of a plane, such as a wall of your room. Shine a torch light on the object in a direction perpendicular to the wall.

What do you see?

Q7

Observe what happens to the size of the shadow as you vary the distance between your torch and your object.

Q8

Context: Observe what happens to the size of the shadow as you vary the distance between your torch and your object.

Q. Why does this happen?

Q9

Construct a model of a cube and use your hands to keep it balanced on one corner vertex. Can you try to understand why all the projected edges have equal length?

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