Right-Triangle Formulas. sin! = Opp Hyp = y z. Right-Triangle Area. Rt. Triangle Area = 1! base! height
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1 MATH 7 Area of Triangle Dr. Neal, WKU We now will ue te rigt-triangle trig formula to find te area of rigt triangle, equilateral triangle, and iocele triangle. We ten will ue te iocele triangle to find te area of regular n -ided polygon. Finally, we will ue Heron Formula to find te area of oter calene triangle. Rigt-Triangle Formula x + y = z z = x + y x = z y y = z x co = Adj Hyp = x z in = Opp Hyp = y z tan = Opp Adj = y x x = zco and y = zin Rigt-Triangle Area Given a rigt triangle, we can find te area uing Rt. Triangle Area = ae eigt Te ae and eigt are te adjacent and oppoite ide of te two acute angle, o we alo can ay Area = opp adj or Area = x y. Example. Find te area of te following rigt triangle: (i) (ii)
2 8 55º 0 0º 30º (iii) (iv) (v) Solution. (i) We ave te ae and eigt, o te area i (30)(50) = 750 q. unit. (ii) Te lat ide i x = = 33, and te area i (33)(56) = 9 q. unit. (iii) Te two lateral ide are given y x = co0º and y = in 0º. So te area i given y Area = co0ºin 0º = co 0º in0º q. unit. (iv) To find te eigt y, we ue tan(55º) = y 8 8 8tan(55º) 5.7 q. unit. and y = 8tan(55º). So te area i given y (v) To find te ae x, we ue tan(30º) = 0 x and x = 0. So te area i given y tan(30º) q. unit. tan(30º)
3 z Anoter general form of rigt-triangle area can e given wen we ave te ypotenue z and one angle. Firt, recall tat in() = in co o tat in co = in( ). Becaue x = zco and y = zin, we otain Area = zco" zin " = z co" in " = z # % $ in(" ) & ( = z in(" ). Te form to ue are ' x y = Rt. Triangle Area = z co in or Rt. Triangle Area = z in( ) In Example (iii) aove, we ave Area = in( 0º) q. unit. Example (iv) and (v) alo demontrate oter formula tat can e ued. If we ave an angle in a rigt triangle wit x eing adjacent and y eing oppoite, ten Rt. Triangle Area = x tan and Rt. Triangle Area = y tan In mot cae toug, it i eaiet to ue Area = ae eigt, and imply find te ae and eigt uing rigt-triangle trig. Equilateral Triangle Area Given an equilateral triangle wit tree ide of lengt and tree 60º angle, we can till find te area uing ae eigt. We note tat in60º= wic give = in60º= 3.
4 60º 60º Te ae of te entire triangle i and te eigt i equilateral triangle i (/) 3 = 3. 3, o te area of an Equilateral Triangle Area = 3 Example. Find te area of an equilateral triangle wit ide of lengt 0 ince. Solution. Te area i quare ince. Iocele Triangle Area Given an iocele triangle, we can find te area uing ae eigt provided we know te ae angle and eiter te ae lengt or vertical ide. Generally, we are given only one of or.
5 If we ave, ten = co" and = in", wic give Iocele Triangle Area = = co" in " If we ave, ten tan = / ( / ) wic give = tan. Te area i ten Iocele Triangle Area = = tan " Example 3. Find te area of te following iocele triangle: (i) Vertical ide of lengt 0 ince and a vertical angle of 30º (ii) A ae of feet and ae angle of 0º 80 " 30 Solution. (i) If te vertical angle i 30º, ten eac ae angle i = = 75º. So te eigt i = 0 in 75º and alf te ae i / = 0 co75º. So te area i 30º = 0 co 75º in 75º = 5 in. 0 (i) 75º / (ii) 6 0º (ii) Wit = and = 0º, ten / 6 = tan 0º or = 6 tan 0º. So te area i " % = 6 6tan 0º = tan 0º $ # ' & = 30. ft.
6 Regular n-sided Polygon A regular n -ided polygon make n congruent iocele triangle were te ae of a triangle equal one ide of te polygon. If tere are n ide, ten te vertical angle of eac interior triangle i V = 360º n. In order to find te area of te polygon, we firt mut find te area of eac interior iocele triangle. But now we will do o in term of te vertical angle V. We note tat tan V $ # & = / " %, o tat = / tan V. $ # & " % V V V V / / Te area of one interior iocele triangle i ten = and V = 80º n, we otain te area of a regular n -ided polygon: " tan V % $ ' # &. Uing V = 360º n Regular n-gon area = n tan 80º $ # & " n % Example. Find te area of a regular octagon (n = 8) wit ide of lengt 0 ince. Solution. Eac vertical angle i V = 360º/8 = 5º. Biecting an interior triangle, we ave tan(.5º ) = 5 ; o = 5 tan(.5º). Te overall area i ten 8 " % $ # ae t' = & " % 8 0 $ ', or # tan(. 5º )& tan 80º, wic give aout 8.87 quare ince. " % $ ' # 8 &
7 Scalene Triangle and Heron Formula Suppoe a triangle a ide of lengt a,, and c. Ten Heron Formula give te area a Area = ( a)( )( c) were = (a + + c) Example 5. Find te area of te following triangle: 0 Solution. Let = ( + + 0) = 3. Ten te area i 3(3 0)(3 )(3 ) = = q. unit.
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