G.GMD.1 STUDENT NOTES WS #1 1 Dimensions are important to understand in this unit. As we progress though the dimensions our units of measurement change. So for example, the items that we measure in one dimension are perimeters, distances, and circumferences. All of these are lengths and exist in one dimensional space. In two dimensions, we move into area and square units we begin to measure the number of square units within the shape. ONE DIMENSIONAL MEASUREMENT POLYGON PERIMETER Perimeter refers to the distance around the edge of a closed figured shape. Perimeter formulas are often quite simple because they sum of the sides of a polygon. In some figures we can create condensed forms of this relationship because of the side properties of the polygon. Square Rectangle Triangle Regular Pentagon C w d l P = 2l + 2w S b D c B P=b+c+d P = 4s S P = 5s CIRCLE PERIMETER – CIRCUMFERENCE The perimeter of a circle is called its circumference. The circumference of a circle is the distance around the edge of the circle or in other words, the arc length of the circle. To discover the circumference formula we need to look at the relationship between the diameter and the circumference. The ratio between these two values will reveal one of the greatest numbers of all time, π. d d d d d d d d ? ? Notice that each time we get just over three diameters. It was this pattern that led many to explore what the value was. The history of pi is a great historical topic to study and if time permitted I would strongly suggest you investigate it further. We learn that depending on the culture and the time period pi has received different approximate values for its irrational value. The ancient Babylonians use the value of 3, while the Egyptian Rhind Papyrus notated pi to be 3.1605. Archimedes of Syracuse (287-212 BC), one of the great Greek mathematicians using regular polygons to approximate a circle was able to narrow down the value between 3 1/7 and 3 10/71. As time continued many mathematicians were able to establish ways to determine more and more correct digits of pi, for instance by 1699 the first 71 digits had been calculated correctly. Determining of digits of pi was a mathematical pursuit by many great scholars. Again this is a fascinating history and it would be a nice exploration. Circumference = πd = 2π πr G.GMD.1 STUDENT NOTES WS #1 2 Determine the missing information. C = 9π cm r = 16 cm r = 3 cm C = π cm r = ____________ d = ____________ C = __________ (E) d = ____________ d=9 r = 4.5 cm d = 2r d = 32 cm C = 2πr C = 6π cm C = πd d = 1 cm Determine the perimeter of the given shapes. (Lines that appear to be perpendicular are perpendicular.) 8 cm 10 cm 2 cm 6 cm 11 cm 18 cm 8 cm P = 11 + 8 + 2 + 2 + 9 + 6 = 38 cm 62 + 82 = x2 x = 10 C = ½ πd = ½ π(8) = 4π P = 18 + 6 + 10 + 10 = 44 cm P = 8 + 4π cm G.GMD.7 WORKSHEET #7 Period that no matter what size of circle you will always have 6 and a bit radii that will fit along the circumference. Lenny disagrees he says that if the circle is really big then more than 6 complete radii will fit along the circumference. Who is right and why? k-l- } -( <-t ; o-"-*' u,r.- {t{4t--{.- l' 1. Jack says \ ..It ( ul't"- Jt"u\r. i', c. i 1-' i' ir-i- r, -_ -i,,\l (( {--, v r.l Pr. i'.r\','.1 (L'1,' 5Jl! 11, a f,/ ';\&.! - 2. Pi is an irrational number. What is an irrational number? Give another example of an irrational number. 3. Determine the circumference. (E) means leave as an exact value. a) c r=3cm = L. -tr t"yw' b) d=B.5cm (E) c - c) r=5..6 d) d = (E) (E) 5 -cm 4 f= (E) 4. Determine the missing information. a)d=3cm b) C = L6n cm c1, = g !.* d)C=5n = hf"r "''' tt'--' -L- cm 1E1 r = .L 5. Archimedes estimated the value of pi in 230 BC ,sine &rTai'folygons. While this is not the method that he used, we will also use a regular polygon to approximate the value of pi. The perimeter of the regular decagon is a close approximation for the circu mference. a) lf the side length is 6 cm, what is the perimeter of the regutar decagon? L (p\ Perimeter = 6c fri't b) Use trigonometry to determine the radius of the regular polygon. Radius = (4 decimal places) Diameter = (4 decimal places) c) Now divide the perimeter by the two times the radius to see how close our approximation is for pi. /"",c ['(.t'!tt"i Approximation for pi = J .Cq {*u(2 decimat ptaces) -72 4'L*l G.GMD.I WORKSHEET #7 2 6. Determine the perimeter of the following figures. (Lines that appear to be perpendicular are perpendicular.) b) a) d) c) I (rti 3 en: l ,6 (rY1 Scm Circumference Perimeter = X-L tvq Perimeter e) f) = iz't( t wr = Perimeter (E) = h) s) lO crn 2cm Circumference Perimeter = (E) = i) 1O m) ] U r {Arq i {-rlt1 cnl Perimeter = rrn n) ?i l-S cn" (E) r&1 rzrL Perimeter = 'vrrfv Perimeter = *u\ (E) o) '-"'" 10 {m L- 30 t* Perimeter = Perimeter :"fi" t/vv r) 4cm = Perimeter C tA {fi1 l0{ffi = {t; = 'L'8 k) i) Perimeter Perimeter (E) Perimeter = luF* &a cui,t g) loof 5a't9! G.GMD.I WORKSHEET #7 3 7. Determine the perimeter of the following figures. (Lines that appear to be perpendicular are perpendicular.) +(.1+t{}-ls+tfir) J + s.JL + K-.(r ) Ll b) 5cm dl"l a- ,5 rm Af f U (1 Y- I [/ = perimeter = bfl tT c,tt1g1 Perimeter n G d) a a t1 vZ I r-(r) 6r+'lvz Perimeter = (L dec.) = 5(m 6Xt q fZ t'\E\ Icm Perimeter (E) = e) *(rn rcrn *i j*, $tm Perimeter = 1 r-# rrF c'"fA, Perimeter = (E)

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