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Type: Open-Ended
Category: Two Dimensional Shapes
Level: Grade 10
Standards: HSG-MG.A.3
Author: nsharp1
Created: 4 years ago

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Two Dimensional Shapes Question

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Tessellation, or tiling, refers to covering a surface with shapes so that there are no gaps or overlaps. These can be different shapes, or the same shape. In the questions that follow, a regular tessellation will be considered- that is, a tessellation that uses only one type of regular polygon.

Grade 10 Two Dimensional Shapes CCSS: HSG-MG.A.3

For a tessellation, one can look at the sum of all the edge lengths. For a regular tessellation, this is equal to the total number of edges multiplied by the length of one edge (or side) of the polygon being used. Edges common to two polygons are only counted once. For example, a tessellation with two squares (each with side length of x) would have 7 edges in total and the sum of the edge lengths would be 7x.

Find the sum of the edge lengths for each of the first 10 tiles of a regular tessellation for all possible shapes, each with an area of 1 square unit. To clarify, if a square can be part of a regular tessellation, find the sum of the edge lengths for a 1 square tessellation, then a 2 square tessellation, ... and finally for a 10 square tessellation. Repeat this for all other regular polygons which can form a tessellation.

Note that the MINIMUM sum of the edge lengths should be found. For example, a tessellation of 4 squares should have a total of 12 edges, not 13 (the squares should form a larger square, not a rectangle), and the total edge length will then be this number multiplied by the side length of the square.