Question

Diagram of the adjacent picture frame has outer dimensions $28{\text{cm}} \times 24{\text{cm}}$ and inner dimensions ${\text{20cm}} \times 16{\text{cm}}{\text{.}}$ Find the area of the shaded part of the frame, if the width of each section is the same.

Answer 1

Hint: The given shaded area is seem to be a trapezium, and we know the formula to find the area of a trapezium with length of its parallel sides equal to $m\;{\text{and}}\;n\;{\text{units}}$ and height equals to $h$ which is given as $A = \dfrac{1}{2}(m + n)h$. Width of each section of frame is equal, with this information you can find the height of the trapezium.

Complete step-by-step solution:
The shaded area in the given frame is a trapezium whose area is given as the product of half of the sum its parallel sides and its height, in the given figure we can see that the length of the parallel sides are given that are equal to $28{\text{cm}}\;{\text{and}}\;20{\text{cm}}$, and we have to find the height for the trapezium in order to calculate its area.
Let us take the width of each section to be $x$

Now from the figure, we can write that
$
   \Rightarrow x + 16 + x = 24 \\
   \Rightarrow 2x + 16 = 24 \\
   \Rightarrow 2x = 24 - 16 \\
   \Rightarrow x = \dfrac{8}{2} \\
   \Rightarrow x = 4{\text{cm}} \\
 $
We can see that the width of each section and the height of trapezium are equal,
$ \Rightarrow h = x = 4{\text{cm}}$
Now coming to the formula to find the area of the shaded region or the trapezium
$A = \dfrac{1}{2}(m + n)h,\;{\text{where}}\;m\;{\text{and}}\;n$ are length parallel sides and $h$ is the height.
Substituting the values,
$
  A = \dfrac{1}{2} \times (20 + 28) \times 4 \\
   = 2 \times 48 \\
   = 96{\text{c}}{{\text{m}}^2} \\
 $
$\therefore 96{\text{c}}{{\text{m}}^2}$ is the area of the shaded region.

Note: In this types of question, always first try to figure out the shape of the shaded region, if you recognized the shape then directly apply the formulas available for that shape or if you don’t recognize the shape then try to split the shaded area into some known shapes.