Question

Top surface of a raised platform is in the
shape of a regular octagon as shown in the figure.
Find the area of the octagonal figure.

​

Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;:-}}}$

Here the Concept of Areas of Trapezium and Area of Rectangle has been used. We know that all the sides of a regular octagon are equal. Thus the length of all sides of octagon is equal to 10 cm. The given octagon is divided into three figures that is Trapezium GFED, Rectangle GHCD and Trapezium ABCH. Firstly we can find separately the areas of each and thus add them to get final answer.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\pink{Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{2}\:\times\:(Sum\;of\;Parallel\;Sides)\:\times\:Height}}}}$

$\\\;\boxed{\sf{\pink{Area\;of\;Rectangle\;=\;\bf{Length\;\times\;Breadth}}}}$

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★ Solution :-

» Length of Each Side of Octagon = 10 m

» Length of FQ = 8 m

» Length of GD = 22 m

» Length of PB = 8 m

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~ For the Area of Trapezium GFED ::

We know that,

$\\\;\sf{:\rightarrow\;\;Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{2}\:\times\:(Sum\;of\;Parallel\;Sides)\:\times\:Height}}$

• Here First Parallel Side = FE = 10 m

• Second Parallel Side = GD = 22 m

• Height = FQ = 8 m

By applying values in formula, we get

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;GFED\;=\;\bf{\dfrac{1}{2}\:\times\:(FE\;+\;GD)\:\times\:FQ}}$

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;GFED\;=\;\bf{\dfrac{1}{2}\:\times\:(10\;+\;22)\:\times\:8}}$

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;GFED\;=\;\bf{\dfrac{1}{2}\:\times\:32\:\times\:8}}$

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;GFED\;=\;\bf{32\:\times\:4}}$

$\\\;\bf{:\Longrightarrow\;\;Area\;of\;Trapezium\;GFED\;=\;\bf{\orange{128\;\;m^{2}}}}$

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~ For the Area of Rectangle GHCD ::

We know that,

$\\\;\sf{:\rightarrow\;\;Area\;of\;Rectangle\;=\;\bf{Length\;\times\;Breadth}}$

• Length = GD = 22 m

• Breadth = DC = 10 m

By applying values, we get

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Rectangle\;GHCD\;=\;\bf{GD\;\times\;DC}}$

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Rectangle\;GHCD\;=\;\bf{22\;\times\;10}}$

$\\\;\bf{:\Longrightarrow\;\;Area\;of\;Rectangle\;GHCD\;=\;\bf{\orange{220\;\;m^{2}}}}$

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~ For the Area of Trapezium ABCH ::

We know that,

$\\\;\sf{:\rightarrow\;\;Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{2}\:\times\:(Sum\;of\;Parallel\;Sides)\:\times\:Height}}$

• Here First Parallel Side = AB = 10 m

• Second Parallel Side = CH = 22 m

• Height = PB = 8 m

By applying values, we get

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;ABCH\;=\;\bf{\dfrac{1}{2}\:\times\:(AB\;+\;CH)\:\times\:PB}}$

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;ABCH\;=\;\bf{\dfrac{1}{2}\:\times\:(10\;+\;22)\:\times\:8}}$

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;ABCH\;=\;\bf{32\:\times\:4}}$

$\\\;\bf{:\Longrightarrow\;\;Area\;of\;Trapezium\;ABCH\;=\;\bf{\orange{128\;\;m^{2}}}}$

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~ For the Area of Octagon ::

$\\\;\sf{\green{\mapsto\;\;Area\;of\;Octangon\;=\;\bf{Ar.(GFED)\;+\;Ar.(GHCD)\;+\;Ar.(ABCH)}}}$

On applying values, we get

$\\\;\sf{\mapsto\;\;Area\;of\;Octangon\;=\;\bf{128\;+\;220\;+\;128}}$

$\\\;\sf{\mapsto\;\;Area\;of\;Octangon\;=\;\bf{\blue{476\;\;m^{2}}}}$

$\\\;\underline{\boxed{\tt{Area\;\;of\;\;Octagon\;=\;\bf{\purple{476\;\;m^{2}}}}}}$

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★ More to know :-

$\;\qquad\underline{\underline{\rm{\red{Properties\;of\;Rectangle}}}}$

• Opposite sides are equal and parallel.

• The adjacent sides form angle of 90° with each other.

• Diagonals bisect each other at 90°

$\;\qquad\underline{\underline{\rm{\red{Properties\;of\;Trapezium}}}}$

• There is a pair of opposite parallel sides.

• Also there is a pair of opposite non - parallel sides.

• Diagonals bisect each other.

Answer 2

Answer:

Here the Concept of Areas of Trapezium and Area of Rectangle has been used. We know that all the sides of a regular octagon are equal. Thus the length of all sides of octagon is equal to 10 cm. The given octagon is divided into three figures that is Trapezium GFED, Rectangle GHCD and Trapezium ABCH. Firstly we can find separately the areas of each and thus add them to get final answer.

Let's do it !!

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