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 Geometry:10.5PartIIWorksheet
“Unofficial”Worked‐OutSolutionsbyEarlWhitney
Note:someoftheproblemsstatethattheanswershouldbewrittenasasimplifiedradical.
1 Areaofaregularhexagonwithaside
. useradical Step 1: How many sides? 6 6 ∙ 8
Step 2: Find the perimeter: 48 Step 3: Find the apothem: Createthe“littleguy”triangle:
Ourgoalistofind .
Thelengthofthebaseofthe“littleguy”triangleis:8
2
Thesumoftheanglesinthefigure upperright is: 6
Eachangleofthefiguremeasures:720˚
∠
120˚
2
6
2 ∙ 180˚
60˚
√ .
Then,the“littleguy”triangleisa30˚‐60˚‐90˚triangle.So,
∙ √ ∙
√ Step 5: (Optional) Compare result to the area of a square with side 2 . 720˚
120˚
Step 4: Calculate the area: The area in Step 4 is 96 ∙ √3~166.3 This should be a little less than a square with side 2
Rectangle area is: 8√3 ∙ 8√3 64 ∙ 3
8√3 192  Page 2 of 8 2 Areaofaregulardecagonwithaperimeter
. Step 1: How many sides? 10 50
Step 2: Find the length of a side: 10
5 Step 3: Find the apothem: Createthe“littleguy”triangle:
Ourgoalistofind .
.
Thelengthofthebaseofthe“littleguy”triangleis:5
2
Thesumoftheanglesinthefigure upperright is: 10
Eachangleofthefiguremeasures:1,440˚
∠
144˚
2
Then,tan 72˚
So,
10
. 2 ∙ 180˚
1,440˚
144˚
72˚
.
. ∙ tan 72˚
. ∙ 3.0777
.
. Step 4: Calculate the area: ∙ .
∙
Step 5: (Optional) Compare result to the area of a square with side 2 . The area in Step 4 is 192.4 This should be a little less than a square with side 2
Rectangle area is: 15.388 ∙ 15.388
2 ∙ 7.694
236.8  15.388 Page 3 of 8 3 Areaofaregularhexagonwithaside
. useradical Step 1: How many sides? 6 6 ∙ 10
Step 2: Find the perimeter: 60
Step 3: Find the apothem: Createthe“littleguy”triangle:
Ourgoalistofind .
Thelengthofthebaseofthe“littleguy”triangleis:10
2
Thesumoftheanglesinthefigure upperright is: 6
2 ∙ 180˚
Eachangleofthefiguremeasures:720˚
∠
120˚
2
6
120˚
60˚
Then,the“littleguy”triangleisa30˚‐60˚‐90˚triangle.So,
√ Step 4: Calculate the area: ∙ √ ∙
√ Step 5: (Optional) Compare result to the area of a square with side 2 . The area in Step 4 is 150∙ √3~259.8 This should be a little less than a square with side 2
Rectangle area is: 10√3 ∙ 10√3 720˚
100 ∙ 3
10√3 300  .
Page 4 of 8 4 Areaofasquarewithapothem
√ .
Step 1: How many sides? 4 √
Step 2: Find the length of a side: 2 ∙ 5√2
√ 10√2 MethodA:UsingtheApothemFormula
4 ∙ 10√2
Step 3: Find the perimeter: 40√2 Step 4: Calculate the area: ∙ √ ∙
√ ∙
∙
MethodB:UsingtheSquareAreaFormula
Step 3: Calculate the area: √
∙
Page 5 of 8 5 Areaofaequilateral i.e.,regular trianglewithradius
.
Step 1: How many sides? 3 Step 2: Find the measure of ∠ : Thesumoftheanglesinatriangle
180˚
Eachangleofthefiguremeasures:180
∠
60˚
2
3
60˚
30˚
Step 3: Find the apothem: Createthe“littleguy”triangle:
Ourgoalsaretofind andb. Notethatthe“littleguy”triangleisa30˚‐60˚‐90˚triangle.
So,
8 2
.
Step 4: Find the perimeter: Thelengthofthebaseofthe“littleguy”triangleis:
Thelengthofasideofthemaintriangleis:2 ∙ √ Perimeterthenis
3 ∙ 8√3
√ ∙ √3
√ √ MethodA:UsingtheApothemFormula
Step 5: Calculate the area: ∙ ∙
√
√ ~
. MethodB:UsingtheTriangleAreaFormula
Step 5: Calculate the area: ∙ √ ∙
√ ~
. Page 6 of 8 6 Areaofaregularhexagonwithaperimeter
∙ √ ∙
andapothem √ . useradical √ 7 Areaofaregularpentagonwithapothem
. 1decimal Step 1: How many sides? 5 Step 2: Find the measure of ∠ : Thesumoftheanglesinthefigure lowerright is: 5
Eachangleofthefiguremeasures:540
∠
108˚
2
5
2 ∙ 180˚
540˚
108˚
54˚
Step 3: Find the length of the base of the pentagon: Createthe“littleguy”triangle:
Ourgoalistofindb. Then,tan 54˚
So,
˚
.
Thelengthofasideofthemainfigureis:2 ∙ .
.
.
Step 4: Find the perimeter of the pentagon: Perimeterthenis
5 ∙ 10.1716
.
Step 5: Calculate the area: ∙ ∙
.
Step 6: (Optional) Compare result to the area of a square with side 2 . The area in Step 5 is 178 This should be a little less than a square with side 2
2∙7
Rectangle area is: 14 ∙ 14 196  14 Page 7 of 8 8 Areaofaregularoctagonwithaside
. 1decimal Step 1: How many sides? 8 8 ∙ 20
Step 2: Find the perimeter: 160
Step 3: Find the apothem: Createthe“littleguy”triangle:
Ourgoalistofind . Thelengthofthebaseofthe“littleguy”triangleis:20 2
Thesumoftheanglesinthefigure upperright is: 8
Eachangleofthefiguremeasures:1,080˚
∠
135˚
2
Then,tan 67.5˚
So,
8
2 ∙ 180˚
1,080˚
135˚
67.5˚
∙ tan 67.5˚
∙ 2.4142
.
Step 4: Calculate the area: ∙ .
∙
,
. Step 5: (Optional) Compare result to the area of a square with side 2 . The area in Step 4 is 1,931.4 This should be a little less than a square with side 2
Rectangle area is: 48.284 ∙ 48.284
2 ∙ 24.142
2,331  48.284 Page 8 of 8 9 Areaofaregularpentagonwithaperimeter
decimal ∙ .
∙
. andapothem .
. 1
Note: a regular pentagon with perimeter 30
should have an apothem of 4.13
, which would generate an area of 61.9
10 Areaofaregularhexagonwhen ofitis
∙
49 7 .
11 Circumferenceofacirclewhenareais
. nounitsgiven 49 2
2 ∙ 7