The surface area of a mathematical object denotes area of its total surface.
The given pyramid is composed of a square and 4 congruent triangles, thus surface area of pyramid is sum of area of square and of those 4 triangles.
The expression which doesn't give surface area A of given pyramid is:
Option D: [tex]\dfrac{1}{2}.A_{Base}.h[/tex]
Given that:
- Slant height of pyramid = l units
- Pyramid is regular and of square base.
- Base's edge's length = b units.
- The height of pyramid from base to top = h units.
The surface area of a mathematical object denotes area of its total surface.
The given pyramid is composed of a square and 4 congruent triangles, thus surface area of pyramid is sum of area of square and of those 4 triangles.
Finding surface area of pyramid:
Surface area of pyramid = Area of square at the bottom of pyramid + sum of area of all 4 triangles of that pyramid.
or
since all triangles are congruent(since base of pyramid is square and that pyramid is regular), thus area of those triangles is same.
Since count of those triangles is 4, thus we have:
Surface area of pyramid = [tex]A_{QPRS} + 4 \times A_{QPR}[/tex]
Now since area of triangle is half of product of its base and height, thus we have:
[tex]A_{QPR} = \dfrac{1}{2}. b.l \text{\: \: (since the height of the triangles is slant height l of pyramid)}[/tex]
and
[tex]A_{PQRS} = b^2[/tex]
Thus, surface area of pyramid = [tex]b^2 + \dfrac{1}{2}.b.l[/tex]
The last option is wrong since surface area of pyramid from base and height is derived by formula:
[tex]A = A_{PQRS} +2b\sqrt{\dfrac{A_{PQRS}}{4} + h^2}[/tex]
Thus the expression which doesn't give surface area A of given pyramid is:
Option D: [tex]\dfrac{1}{2}.A_{Base}.h[/tex]
Learn more about surface area of regular pyramid here:
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