Consider a regular square pyramid
Let a = a= a = the base edge, h = h= h = the altitude, and s = s= s = the slant height.
The lateral area A L A_L A L of a regular square pyramid is
A L = 1 2 ( 4 a ) s A_L=\dfrac{1}{2}(4a)s A L = 2 1 ( 4 a ) s
The slant height makes an angle α = 45 ° \alpha=45\degree α = 45° with the base.
From the right triangle
sin α = h s , tan α = h a / 2 \sin\alpha=\dfrac{h}{s}, \tan \alpha=\dfrac{h}{a/2} sin α = s h , tan α = a /2 h
s = h sin α = 8 sin 45 ° = 8 2 ( m ) s=\dfrac{h}{\sin\alpha}=\dfrac{8}{\sin45\degree}=8\sqrt{2}(m) s = sin α h = sin 45° 8 = 8 2 ( m )
a = 2 h tan α = 2 ( 8 ) tan 45 ° = 16 ( m ) a=\dfrac{2h}{\tan\alpha}=\dfrac{2(8)}{\tan45\degree }=16(m) a = tan α 2 h = tan 45° 2 ( 8 ) = 16 ( m )
A L = 2 a s = 2 ( 16 ) ( 8 2 ) = 256 2 ( m 2 ) A_L=2as=2(16)(8\sqrt{2})=256\sqrt{2}(m^2) A L = 2 a s = 2 ( 16 ) ( 8 2 ) = 256 2 ( m 2 )
A L = 256 2 m 2 A_L=256\sqrt{2}m^2 A L = 256 2 m 2
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