Question

The top surface of a raised platform, as shown in the figure , is in the shape of a regular octagon of side 5m . What is the area of the octagonal surface?

Answer 1

Hint: It is given that the regular shape of an octagon has sides 5 m. We have to find the area of the octagonal surface. We will divide the octagon intro trapezium and rectangle after that with available information we will find the area.

Complete step-by-step answer:
Area of octagonal surface = Area of trapezium ABCH+ Area of rectangle HCDG+ Area of trapezium GDEF
We can say that by symmetry Area of trapezium ABCH will be same to Area of GDEF
Let's find area of trapezium ABCH
AB and HC are two parallel side
Area of trapezium ABCH =1/2(Sum of two parallel sides)(height)
All sides of octagon are equal so
\[AB = BC = CD = DE = EF = FG = GH = 5m\]
It is given that HC=11m
Area of Trapezium ABCH = $\dfrac{1}{2}(AB + CH)(height)$
 $
   = \dfrac{1}{2}(5 + 11)(4) \\
   = \dfrac{1}{2}(16)(4) \\
   = 32{m^2} \;
 $
Area of rectangle HCDG = Length × breadth
$
   = 11 \times 5 \\
   = 55{m^2} \;
 $
Area of trapezium GDEF will be the same as the Area of trapezium ABCH.
Area of octagonal surface
 $
   = 32 + 55 + 32 \\
   = 119{m^2} \;
 $
So the octagonal surface area is 119 metre square.

Note: Octagon , pentagon such type of question will be solved by breaking them into trapezium, square, rectangle whichever is possible. Mensuration is based upon formulas whether it is cone cylinder anything. In this question every information was given to us. We broke the octagon into trapezium and rectangles and solved it further.