Intermediate Math Circles November 5, 2008 Geometry II
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1 1 Univerity of Waterloo Faculty of Matematic Centre for Education in Matematic and Computing Intermediate Mat Circle November 5, 2008 Geometry II Geometry 2-D Figure Two-dimenional ape ave a perimeter and an area. Perimeter i te lengt of te outline of a ape. rea i te urface te ape cover. Te table below give formula for te perimeter P and area of ome common 2-D figure. w P = 4 P = 2(l + w) = 2 = lw l a a b c b P = 2(a + b) = b d P = a + b + c + d = b+d 2
2 2 a c r b P = a + b + c P = 2πr = b 2 = πr 2 Example: rectangle a dimenion 24cm 15cm. If te longer ide i decreaed by 6cm, by ow many centimetre mut te oter ide be increaed for te new rectangle to ave te ame area a te original? Solution: Te original rectangle a area = = 360cm 2. Te new rectangle a lengt 24 6 = 18cm, and widt (15 + x)cm. Terefore, it area i = 18(15 + x) = x. However, ti area i te ame a te original o 360 = x or 90 = 18x and o x = 5cm. Terefore, te orter ide mut be increaed by 5cm to keep te area te ame.
3 3 3-D Figure Tree-dimenional figure ave a urface area and a volume. Surface area i te two-dimenional area te urface of a 3-D figure take up. Volume i te amount of pace a figure take up. Te table below give formula for te urface area S and volume V of ome 3-D figure. S = 6 2 V = 3 w l S = 2(lw + l + w) V = lw r Let be te area of te bae. S = 4πr 2 V = 1 3 V = 4 3 πr3 Note tat te triangle-baed pyramid, quare-baed pyramid, and cone are all pecial cae of te formula in te bottom left of te table. Te bae of tee olid would be a triangle, a quare, and a circle, repectively.
4 4 Example: cube wit ide lengt 10 ret inide a pere, o tat eac of te 8 vertice of te cube touc te urface of te pere. Wat i te volume of ti pere? Solution: Since all vertice of te cube are toucing te pere, te diagonal of te cube will be equal to te diameter of te pere. Te diameter of te pere i terefore d = d = 3 2 d = 3 d = 10 3 Terefore, ince d = 2r, r = 5 3. So, te volume of te pere i V = 4 3 πr3 V = 4 3 π(5 3) 3 V = 4 3 π(375 3) V = 500 3π
5 5 Problem Set 1. piece of tring, 40cm long, i formed into a circle wit te end of te tring toucing eac oter. Wat i te radiu of te circle? 2. quare of ide lengt 2n + 1 a anoter quare of ide lengt 2n inide it. Wat i te area between te two quare in term of n? 3. Te lengt, in ince, of te tree edge meeting at eac corner of a rectangular prim are 1, 2 and 3. Wat i te lengt of te diagonal of ti prim? 4. In te diagram, B i te diameter of te emicircle. If C = 8, CB = 6, and CB = 90, wat i te area of te aded portion? C B cube i painted red and i ten cut into 27 unit cube. How many of tee cube ave paint on exactly two of teir face? 6. wire 60cm in lengt i cut into two part in te ratio 2 : 1. Eac part i bent to form a quare. Wat i te total area of te two quare? 7. In te diagram, circular arc P Q, QR and ST ave centre T, S and Q repectively. If P T = QT = QS = SR = 1, ten wat i te perimeter of figure P QRST? P Q 60 R 60 T 60 S 8. n open box i made from a quare piece of tin by removing a 6 inc quare from eac corner and folding te ide upward. If te volume of te box i 864 cubic ince, find te area of te original quare. 9. Te tip of a traigt reed growing in te centre of a pond 8 feet in diameter reace one foot above te water. Wen te reed i pulled over to te edge, wit it bottom fixed, te tip jut touce te edge of te pond. Wat i te dept of te pond in feet?
6 6 10. In te diagram, BC i an equilateral triangle aving ide lengt 2, and LB, BMC and KC are arc of circle aving centre C, and B, repectively. Find te total area of te aded region in te diagram. L K B M C 11. BCD i a parallelogram in wic P = 12, DQ = 16, CS = 10, P Q = 5, and QR = 2. Find te area of figure BRSC. D C B P Q R S 12. rectangular oue wic meaure 20m 10m a an outide electrical outlet at a corner of te oue. n electric mower, connected by a cord to te outlet, can reac a maximum ditance of 15m. Wat i te larget area of lawn wic can be cut? 13. circle of radiu r i rolled around te outide of a rectangle of perimeter p, alway maintaining contact wit te rectangle. Wat i te ditance travelled by te centre of te circle, wen te circle a travelled once around te rectangle? 14. paper cone, wen cut along it lant eigt, and flattened out, form a emi-circle of radiu 10cm. Find te eigt of te cone. r
7 7 15. manufacturer ell clear platic tape on a pool wit radiu 1cm. Te tape i 0.02cm tick and 1.5cm wide. Te combined radiu of te pool and te tape i 3cm. Wat i te bet approximation of te lengt of te tape on ti pool, in metre, to one decimal place? 1cm 3cm
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