What it does
A fractal is a shape where each small part looks like the whole. You make one by repeating a simple rule again and again. This page builds two famous ones from an equilateral triangle:
- Koch snowflake: cut every side into three equal parts and build a small triangle outwards on the middle part. Every side becomes four sides, each a third as long.
- Sierpinski triangle: join the midpoints of every filled triangle and take out the middle one. Every triangle becomes three, each half as wide.
Move the step slider and watch the picture change. A table lists every step from 0 to 6, with the number of pieces, their side, the perimeter and the area, and the formulas are shown with your numbers. The surprise: the snowflake’s perimeter grows by a third at every step, without end, yet its area never passes 8/5 of the first triangle. The Sierpinski triangle keeps three quarters of its area each step, so its area shrinks towards zero.
It fits Class 8 Mathematics, exploring some geometric themes (patterns, self-similar shapes and repeated growth), and gives practice with powers and fractions. Patterns that repeat by turning and tiling are in the tessellation studio; for the Mandelbrot set and fractals you can zoom into, go to the fractal explorer.
How to use it
- Pick Koch snowflake or Sierpinski triangle.
- Move n from 0 to 6 steps. The dashed outline is the first triangle.
- Type the side of the first triangle in cm to change all the lengths and areas in the table.
- The highlighted row of the table is the step you are looking at.
A worked example
Koch snowflake from a triangle with 9 cm sides:
- Step 0: 3 sides, perimeter 27 cm, area √3/4 × 81 ≈ 35.074 cm².
- Step 1: 12 sides of 3 cm, perimeter 36 cm (27 × 4/3). Three small triangles of side 3 cm are added, so the area is 35.074 × 4/3 ≈ 46.765 cm².
- Step 2: 48 sides of 1 cm, perimeter 48 cm. Area 35.074 × (1 + 3/5 × (1 − 16/81)) ≈ 51.962 cm².
- The area can never pass 8/5 × 35.074 ≈ 56.118 cm², but after 6 steps the perimeter is already 27 × (4/3)⁶ ≈ 151.7 cm, and it keeps growing.
Sierpinski triangle from 8 cm sides: after 3 steps there are 27 triangles with 1 cm sides, and the area left is (3/4)³ = 27/64 of the start.
Check yourself
Four fresh questions each time: sides of a snowflake, a snowflake’s perimeter, triangles in a Sierpinski triangle, and the fraction of area left. Type the answer and press Check. Nothing is saved.
FAQ
How can a shape have an endless edge but a limited area? Each step adds less and less area (4/9 as much as the step before), so the total settles down. The edge is multiplied by 4/3 each time, and that never settles.
Why stop at 6 steps? After 6 steps the snowflake already has 12 288 sides and the pieces are too small to see on a phone. A true fractal is what you would get after endless steps.
Is this from the NCERT book? It covers the same topic with fibub’s own pictures and examples. fibub is not part of NCERT.
Where the numbers come from
- The Koch snowflake (Helge von Koch, 1904) and the Sierpinski triangle (Wacław Sierpiński, 1915), with their perimeter and area formulas, are standard mathematics. Wording, pictures and examples are fibub's own.
- Checked by measuring the drawn shapes directly (shoelace area and summed side lengths, steps 0 to 6) against the formulas, and against the limit of 8/5 for the snowflake's area.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.