Find the area of the shaded region in the figure below:

enter image description here

I am completely stuck on how to start off this question. Please help on some guidance on how to start it off.

  • $\begingroup$ Unless I am very mistaken, Mind Your Decisions made a video on this exact problem. $\endgroup$ – Mohammad Zuhair Khan Oct 27 '18 at 8:08
  • $\begingroup$ @Raptor, I've checked his YouTube channel, you're right thanks. An alternative approach will also be appreciated. $\endgroup$ – Meghan C Oct 27 '18 at 8:16
  • $\begingroup$ Hint: the sum of areas of two quadrilaterals on upper-left and lower-right equal to the sum of areas of the two quadrilaterals on upper-right and lower-left. $\endgroup$ – achille hui Oct 27 '18 at 8:16
  • 2
    $\begingroup$ @achillehui, thanks, I will attempt that approach. May I ask why this is true? $\endgroup$ – Meghan C Oct 27 '18 at 8:20

A square into 4 areas

Split the square into $8$ triangles, convince yourself you can group them into 4 pairs and each pair has same area. Let the area of the triangles be $a, b, c, d$ as illustrated above.

You are given $c + d = 20$, $b + c = 32$ and $a + d = 16$. The area of the quadrilateral (in cyan) is $$a + b = (a + d) + ( b + c) - ( c + d) = 16 + 32 -20 = 28$$

  • $\begingroup$ +1 Nice Answer!! $\endgroup$ – the_candyman Oct 27 '18 at 8:57
  • $\begingroup$ @achillehui, I really appreciate your help. Thank you. $\endgroup$ – Meghan C Oct 27 '18 at 9:18

Alternatively, refer to the figure:

$\hspace{4cm}$![enter image description here

Let $x$ be the half of the side of the large square. Then the side of the smaller oblique square is $x\sqrt{2}$, how:

Pythagorean theorem.

The total green area is $x^2$, how:

$$\frac12 \cdot x\sqrt{2}\cdot h_1+\frac12 \cdot x\sqrt{2} \cdot h_2=\frac12\cdot x\sqrt{2}\cdot (h_1+h_2)=\frac12\cdot x\sqrt{2}\cdot x\sqrt{2}=x^2.$$

The total area of grey and green regions is $2x^2=16+32=48$, how:

Green area is $x^2$ and grey area is $2\cdot \frac{x^2}{2}=x^2$.

Hence, the required area is $96-(16+20+32)=28$, how:

the area of the large square $(4x^2)$ minus the total area of grey, green and white regions $16+20+32$.


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