Six circles of radius 1 unit are drawn tangent to the sides of a regular hexagon. Each circle is also tangent to two other circles as shown in the drawing. What is the perimeter of the hexagon? Express your answer in simplest radical form.

Step-by-step explanation:

if we connect center of circles which tangent each other, then we get bigger regular hexagon with side length 2 unit but it differ 60° from original (smaller) hexagon. hence smaller hexagon has height 2•2–2•1 = 2. from center of smaller hexagon draw line to any two neigbouring vertexes so we get an equilateral triangle with height 1. by using pythagorean or trigonometry we get side length of smaller hexagon

so the perimeter is

unit

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.