Question #148925
3. A regular square pyramid has a height of 8m. If the slant height makes an angle of 45º with the base, find the lateral area of the pyramid.
1
Expert's answer
2020-12-08T19:18:07-0500

Consider a regular square pyramid

Let a=a= the base edge, h=h= the altitude, and s=s= the slant height.

The lateral area ALA_L of a regular square pyramid is


AL=12(4a)sA_L=\dfrac{1}{2}(4a)s


The slant height makes an angle α=45°\alpha=45\degree with the base.

From the right triangle


sinα=hs,tanα=ha/2\sin\alpha=\dfrac{h}{s}, \tan \alpha=\dfrac{h}{a/2}


s=hsinα=8sin45°=82(m)s=\dfrac{h}{\sin\alpha}=\dfrac{8}{\sin45\degree}=8\sqrt{2}(m)

a=2htanα=2(8)tan45°=16(m)a=\dfrac{2h}{\tan\alpha}=\dfrac{2(8)}{\tan45\degree }=16(m)


AL=2as=2(16)(82)=2562(m2)A_L=2as=2(16)(8\sqrt{2})=256\sqrt{2}(m^2)

AL=2562m2A_L=256\sqrt{2}m^2



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