Question

Each square has a side length of 12 units. Compare the areas of the shaded regions in the 3 figures. Which figure has the largest shaded region? Explain or show your reasoning.

Answer 1

Answer:They are equal

Step-by-step explanation:

they will add up they are just token apart or in pieces

Answer 2

Three different figures are given. The area of the shaded region of all the figures is the same.

We need to determine the highest shaded area among all.

Now, the area of the square is common to all.

Thus,

$\begin{aligned}\rm{Area\;of\;square}&=12^2\\&=144 \;\rm{units^2}\end{aligned}$

1). Calculate the shaded area for figure A.

The diameter of the circle is 12 units, hence its radius is 6 units.

Therefore,

$\begin{aligned}\rm{Area\;of\;circle}&=\pi \times 6^2\\&=36 \pi \;\rm{units^2}\end{aligned}$

$\begin{aligned} \rm{Shaded\; area} &= \rm{Area\; of\; the\; square - Area\; of\; the\; circle}\\&=144-36 \pi \end{aligned}$

Thus, the area of the shaded region for figure 1 is $114-36 \pi$.

2). Calculate the shaded area for the figure B.

The diameter of the circle is 6 units, hence its radius is 3 units.

Therefore,

$\begin{aligned}\rm{Area\;of\;circle}&=\pi \times 3^2\\&=9 \pi \;\rm{units^2}\end{aligned}$

$\begin{aligned} \rm{Shaded\; area} &= \rm{Area\; of\; the\; square - 4 \times Area\; of\; the\; circle}\\&=144-36 \pi \end{aligned}$

Thus, the area of the shaded region for figure 2 is $114-36 \pi$.

3). Calculate the shaded area for figure C.

The diameter of the circle is 4 units, hence its radius is 2 units.

Therefore,

$\begin{aligned}\rm{Area\;of\;circle}&=\pi \times 2^2\\&=4 \pi \;\rm{units^2}\end{aligned}$

$\begin{aligned} \rm{Shaded\; area} &= \rm{Area\; of\; the\; square - Area\; of\; the\; circle}\\&=144-36 \pi \end{aligned}$

Thus, the area of the shaded region for figure 3 is $114-36 \pi$.

Therefore, the area of the shaded region of all the figures is the same.

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