Patterns in shapes

Numbers are not the only things that come in sequences. Shapes can grow too , a triangle that gains a side, a square that splits into smaller squares, a flower of petals. Watching shapes grow is how geometers spot rules.

Concept

Begin with the simplest shape sequence: line segments of growing length. The $1$st segment is $1$ unit long, the $2$nd is $2$ units, the $3$rd is $3$ units, and so on. Each segment has only $2$ endpoints, but its length follows the counting numbers.

Now look at growing triangles. Start with one small triangle. In the next stage, draw a triangle made of $3$ smaller copies. Then a triangle of $6$ smaller copies , wait, why $6$? Look carefully: an equilateral triangle made by stacking rows of unit triangles has $1$ triangle in row $1$, $3$ in row $2$, $5$ in row $3$, and so on. So the totals are $1$, $1+3 = 4$, $1+3+5 = 9$. The big triangles of side $n$ contain exactly $n^2$ small triangles. Squares hiding inside triangles!

Next, growing squares. The $n$th square in the family has side $n$. Its perimeter is $4n$ and its area is $n^2$. Sketch them side by side and the gap between consecutive areas is $2n + 1$ , the next odd number. Same pattern as the gnomon idea.

A famous shape sequence is the family of polygons: triangle ($3$ sides), quadrilateral ($4$), pentagon ($5$), hexagon ($6$), heptagon ($7$), and so on. The number of sides goes $3, 4, 5, 6, 7, \dots$ , just the counting numbers from $3$. The number of diagonals of a polygon with $n$ sides is $\frac{n(n-3)}{2}$ , a hidden formula that you can spot if you draw small examples and count.

Then come growing dot grids. A $2 \times 2$ grid has $4$ dots. A $3 \times 3$ has $9$ dots. A $4 \times 4$ has $16$ dots. The number of dots in an $n \times n$ grid is $n^2$. But how many line segments can you draw connecting horizontal or vertical neighbours? In an $n \times n$ grid that is $2n(n-1)$, another tidy formula coming from a picture.

The point is this: a growing shape carries hidden numbers. Counting sides, dots, regions, or diagonals at each stage reveals a sequence , and then you have a number puzzle and a shape puzzle in one. That double identity is the joy of geometry.

Worked examples

Example 1. A triangle of side $4$ is made by stacking unit triangles. How many unit triangles in total?

• For side $n$, total is $n^2$.
• Side $4 \Rightarrow 16$ unit triangles.

Example 2. What is the perimeter and area of a square of side $7$ units?

• Perimeter $= 4 \times 7 = 28$ units.
• Area $= 7^2 = 49$ square units.

Example 3. How many diagonals does a hexagon ($n = 6$) have?

• Formula: $\frac{n(n-3)}{2} = \frac{6 \times 3}{2} = 9$.
• So a hexagon has $9$ diagonals.

Example 4. A $5 \times 5$ grid of dots is drawn. How many dots in all? How many small horizontal segments connecting neighbouring dots?

• Dots: $5 \times 5 = 25$.
• Horizontal segments: in each row there are $4$ small segments, and there are $5$ rows, so $4 \times 5 = 20$ horizontal segments.

Try it yourself

1. Sketch the triangles of side $1, 2, 3, 4$. Write the number of unit triangles in each.
2. A square has side $9$. What is the area? What is the perimeter?
3. How many diagonals does a pentagon have?
4. In a $6 \times 6$ dot grid, how many dots are there?
5. The sides of the polygon family go $3, 4, 5, 6, \dots$. What polygon has $10$ sides called?
6. Draw the next shape in this sequence: a single line segment, then a triangle, then a square, then a pentagon, then a $\dots$?
7. Challenge: In an equilateral triangle of side $5$ made of unit triangles, some unit triangles point up and others point down. How many point up? How many point down?

Activity

Toothpick triangles. With matchsticks or toothpicks, build the triangle sequence: side $1$, side $2$, side $3$, side $4$. How many toothpicks do you need at each stage? Make a small table: side, total toothpicks. Can you spot the rule? (Hint: for side $n$, the total is $\frac{3n(n+1)}{2}$.)