Question

Estimate the area of the irregular shape. Explain your method and show your work?

Answer 1

Hint: As the figure is a little difficult to figure out so, we will line them up with straight lines. After this we will apply the formulas $\dfrac{1}{2}\left( height \right)\left( side1+side2 \right),\dfrac{1}{2}\left( base\times height \right)$ to find the area of trapezium and triangle respectively. Also, we will use the height of trapezium between the parallel lines only. Finally, we will add up all areas and get the area of the given diagram.

Complete step by step solution:
If we look at the given diagram carefully, then we come to know that it looks similar to the shape of a trapezium. To clear this point, we will understand about some properties of trapezium. Every trapezium will have one thing common which is its parallel side. This means that any trapezium will have one side parallel to the other side. This is the only pair of parallel sides a trapezium will have.
If we look at the diagram carefully, then it is difficult to imagine its shape. But if we somehow create straight lines approximately, then we can assume it as a trapezium. This can be understood from the following figure.

So, we can find the area of trapezium 1. For this we will take the help of the formula of trapezium, $\dfrac{1}{2}\left( height \right)\left( side1+side2 \right)\,...(i)$. According to the diagram, the height of trapezium 1 is 5 units along with sides 3 units and 2 units respectively. Substituting these values in (i) we get
$\dfrac{1}{2}\left( 5 \right)\left( 3+2 \right)=\dfrac{1}{2}\times 5\times 5=\dfrac{25}{2}=12.5\,units\,...(ii)$
Similarly, we can apply this formula to the second trapezium as below.

So, we can find the area of trapezium 2. For this we will take the help of the formula of trapezium, $\dfrac{1}{2}\left( height \right)\left( side1+side2 \right)\,...(i)$. According to the diagram, the height of trapezium 1 is 5 units along with sides 4 units and 2 units respectively. Substituting these values in (i) we get
$\dfrac{1}{2}\left( 5 \right)\left( 4+2 \right)=\dfrac{1}{{2}}\times 5\times {{{6}}^{3}}=15\,units\,...(iii)$
We can see in this above figure that there is some region left on the left side of the figure. This must be covered too. After approximately lining up this space, we come to know that it is a triangle as shown below.

To find the area of the triangle as marked in the above diagram, we will apply the formula of it as $\dfrac{1}{2}\left( base\times height \right)$. After substituting these values, we get $\dfrac{1}{2}\left( 1\times 4 \right)=\dfrac{{{{4}}^{2}}}{{2}}=2\,units\,...(iv)$.
Now, we will add up all the areas as in, equation (ii) + equation (iii) + equation (iv). Thus, we get
$12.5units+15units+2units=29.5units.$
Hence, the total area of the given region is 29.5 units.

Note: Sometimes, it becomes difficult for us to figure out the shape of such diagrams which are not even defined to us. For example in this diagram we have a drawing which cannot be figured out at first. So, we will apply approximately straight lines and get to the result. Here we come to know that the diagram that is made after using straight lines is trapezium and a little triangle. After this we will use the formulas of trapezium and triangle. To get the area of the whole figure we need to add up all the small areas and get to the desired result. One should focus on the question while solving otherwise, there might be a chance to get a wrong solution.