Maths-
General
Easy
Question
Thomas is making a sign in the shape of a regular hexagon with 4 -inch sides, which he will cut out from a rectangular sheet of metal, as shown in the figure above. What is the sum of the areas of the four triangles that will be removed from the rectangle?
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The correct answer is:
HINT: Find the measure of the angles and sides. Then find the areas of removed triangles.
Complete step by step solution:
Given that it is a regular hexagon, so PQ = QD = DS = SR = RC = CP = 4 inches
Each angle of regular hexagon can be found out by the formula 
, where n is the number of sides.
Here, n = 6, we get 120° as each angle of the regular hexagon.
So, we get ∠CPQ = ∠PQD = ∠QDS = ∠DSR = ∠SRC = ∠RCP = 120°
We have ∠CPA = 180 - 120 = 60° (supplementary angles)
From this, ∠PCA = (180 - (90 + 60)) = 30°
So we get a 30 - 60 - 90 triangle, ΔPAC
Similarly, we have ΔQBD, ΔDFS, ΔREC to be 30 - 60 - 90 triangles.
In ΔPAC, we have hypotenuse PC = 4
Then, the shorter side PA = 2 (In 30 - 60 - 90 triangle, the shorter side is half the hypotenuse)
And the longer side AC = 
(In 30 - 60 - 90 triangle, the longer side is 
times the shorter side)
Likewise, QB = FS = RE = 2 and BD = DF = EC =
So, area of one right angles triangle, ΔPAC = 
We have 4 such similar triangles, so their total area will be 
Hence the total area of four triangles that will be removed from the rectangle =
Hence option A is the correct answer.
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