面積與周界的關係(絶對不是功課)

l 在1.(1)-(5)及2.(1)-(5)各題中，設邊長為 y 厘米。
在1.(3)-(5)及2.(3)-(5)各題中，由正多邊形的中點與正多邊形任兩鄰點連線，形成一個等腰Δ。設該等腰Δ面積為 A 平方厘米，兩底角為θ°，兩腰長 a 厘米，Δ經多邊形中點的高為 h 厘米。
各題答案取 4 位有效數字。


1. (1)
設正Δ的高為 h cm。
tan60° = h/(y/2)
h = (ytan60°)/2

A = (1/2) x y x [(ytan60°)/2] = 100
y = √(400/tan60°)
3y = 3√(400/tan60°) = 45.59
三角形周界 = 45.59 厘米


(2)
A = y² = 100
y = 10
4y = 40
正方形周界 = 40 厘米


(3)
正五邊形內角 = [(2 x 5 - 4) x 90°] / 5 = 108°
θ = 108°/2 = 54°
tan54° = h/(y/2)
h = (ytan54°)/2

A = 5 x (1/2) x [(ytan54°)/2] x y = 100
y = √(80/tan54°)
5y = 5√(80/tan54°) = 38.12
正五邊形周界 = 38.12 厘米


(4)
正六邊形內角 = [(2 x 6 - 4) x 90°] / 6 = 120°
θ = 120°/2 = 60°
tan60° = h/(y/2)
h = (ytan60°)/2

A = 6 x (1/2) x [(ytan60°)/2] x y = 100
y = √(200/3tan60°)
6y = 6√(200/3tan60°) = 37.22
正六邊形周界 = 37.22 厘米


(5)
正七邊形內角 = [(2 x 7 - 4) x 90°] / 7
θ = {[(2 x 7 - 4) x 90°] / 7} / 2 = 64.29°
tan64.29° = h/(y/2)
h = (ytan64.29°)/2

A = 7 x (1/2) x [(ytan64.29°)/2] x y = 100
y = √(400/7tan64.29°)
7y = 7√(400/7tan64.29°) = 36.72
正七邊形周界 = 36.72 厘米


(6)
設圓半徑為 r 厘米。

πr² = 100
r = √(100/π)

2πr = 2π√(100/π) = 35.45
周界 = 圓周 = 35.45 厘米


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2.
(1)
3y = 100
y = (100/3)

設正Δ的高為 h cm。
由 1.(1) h = (ytan60°)/2
h = (50/3)tan60°

正三角形面積 = (1/2) x (100/3) x (50/3)tan60° = 481.1 平方厘米


(2)
4y = 100
y = 25
y² = 25² = 625

正方形面積 = 625 平方厘米


(3)
5y = 100
y = 20

由 1. (3): h = (ytan54°)/2
h = (20tan54°)/2
h = 10tan54°

正五邊形面積 = 5 x (1/2) x 20 x (10tan54°) = 688.2 平方厘米


(4)
6y = 100
y = 50/3

由 1. (4): h = (ytan60°)/2
h = (50/3)(tan60°)/2
h = (25tan60°)/3

正六邊形面積 = 6 x (1/2) x (50/3) x (25tan60°)/3 = 721.7 平方厘米


(5)
7y = 100
y = 100/7

由 1. (5): h = (ytan64.29°)/2
h = (100/7)(tan64.29°)/2
h = (50tan64.29°)/7

正七邊形面積 = 7 x (1/2) x (100/7) x (50tan64.29°)/7 = 741.8 平方厘米


(6) 2πr = 100
r = 50/π

圓面積 = π(50/π)² = 795.8 平方厘米

Im Grade 11 in Canada . Got 98% in Math. I hope I can help you :D

Let me show you !

1.1

Area = s^2 * root3 / 4
s = length of a side

100 = s^2 * root3 / 4
s = 15.2cm

1.2

Area = s^2
s = 10cm

1.3

Area = 5t^2 tan 54° / 4
t = length of a side

100 = 5t^2 tan 54° / 4
t = 7.6cm

1.4

Area = 3 * root3 * t^2 /2
t = 6.2cm

1.5

Area = 7/4 * a^2 cot π/7
100 = 7/4 * a^2 cot π/7
a = 5.2cm

1.6

Area = πr^2
Circumference = 2πr

C = 2π* root(100/π)
C = 35cm

* for the second part just use the formulae above and plug in the numbers &now I will just give you the answers
2.1

A = 25980.8 cm2

2.2

A = 100^2 = 10000cm2

2.3

A = 17204.8cm2

2.4

A = 25980.8cm2

2.5

A = 36339.1cm2

2.6

A = 80.6cm2

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周長是否指周界???