A circle is inscribed in an equilateral triangle of side 24 cm, as sho

Question

https://gmatclub.com/forum/download/file.php?id=54396 A circle is inscribed in an equilateral triangle of side 24 cm, as shown above. What is the area of the shaded region? A. 148√3 - 36π cm^2 B. 144√3 - 36π cm^2 C. 144√3 - 48π cm^2 D. 121√3 - 48π cm^2 E. 121√3 - 36π cm^2 Are You Up For the Challenge: 700 Level Questions Untitled.png

GMATinsight · Answer

STrategy : Area of Shaded Region =

Area of equilateral ∆  = (√3/4)*24^2 = 144√3

Area of Circle  = πr^2

CONCEPT : Radius of circle inscribed in an equilateral ∆ = (1/3) Height of equilateral = (1/3)*(√3/2)*Side

i.e. Radius of Circle  = (1/3)*(√3/2)*24 = 4√3

i.e. Area of Circle  = π(4√3)^2 = 48π

Now, Shaded Area  =

Answer: Option

GMATWhizTeam · Answer

Solution 
 • Let’s redraw the given diagram as below.
 https://www.dropbox.com/s/v8q6o9b414vnnia/circle_in_equi.png?dl=1 
• Area of the shaded region = area of triangle ABC – area of the circle.
• We know that when a circle is inscribed in an equilateral triangle, then centre of the circle coincide with the centre (or centroid) of the equilateral triangle. 
 o So, area of triangle AOB = area of triangle AOC = area of triangle BOC  = \frac{1}{2}*BC*r  
• Also, 3 * (area of triangle BOC) = area of triangle ABC
 o So,  3*\frac{1}{2}*24*r = \frac{\sqrt{3}}{4}*24*24 ⟹ r =4\sqrt{3}  
• Thus, area of the shaded region =  \frac{\sqrt{3}}{4}*24*24 - \pi*(4\sqrt{3})^2 = 144\sqrt{3} – \pi*48  
Thus, the correct answer is  Option C.

Archit3110 · Answer

radius of inscribed circle r = a* √3/6
a given = 24 so radius ; 4*√3 area of circle pi * 48
and area of equilateral ∆ = √3/4 * side ^2 side= 24 so area = 144√3 
shaded region area =  144√3 - 48π  cm^2
OPTION C

AnirudhaS · Answer

area of equilateral ∆ABC =  \frac{\sqrt{3}}{4}*side^2 = 144 \sqrt{3} 
And inradius =  \frac{1}{3}*12\sqrt{3} = 4 \sqrt{3} 
So required area = ∆ABC - area of incircle
= 144\sqrt{ 3} - 48 \pi 
Answer:  C

poojakhanduja3017 · Answer

IMO C  
 Concepts and Formulae used: 
Based on the question: Area of Shaded Region =  
Concept: Radius of circle inscribed in an equilateral triangle = (1/3)* Height of equilateral triangle = (1/3)*(√3/2)*Side
 
Now, the steps:
Step 1 : Area of equilateral ∆ =(√3/4)∗242=144√3=(√3/4)∗242=144√3
Step 2: Area of Circle =πr*2=πr*2
Step 3: Radius of Circle =(1/3)*(√3/2)*24=4√3=(1/3)*(√3/2)*24=4√3
Step 4: Area of Circle =π* (4√3)*2=π*48=π*(4√3)2=48π
Answer: Shaded Area =

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A circle is inscribed in an equilateral triangle of side 24 cm, as sho

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