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Sub 505 (Easy)|   Geometry|      
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Bunuel
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The figure shows an equilateral triangle where each vertex is the cent 16 Nov 2020, 01:15
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91% (02:32) correct 9% (02:51) wrong based on 67 sessions

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The figure shows an equilateral triangle where each vertex is the center of a circle. Each circle has radius 20. What is the area of the shaded region?


A. \(400\sqrt{3}- 100\pi\)

B. \(400\sqrt{3}- 150\pi\)

C. \(400\sqrt{3}- 175\pi\)

D. \(400\sqrt{3}- 200\pi\)

E. \(400\sqrt{3}- 210\pi\)


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D
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AjitJangale
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The figure shows an equilateral triangle where each vertex is the cent Updated on: 16 Nov 2020, 05:13
Originally posted by AjitJangale on 16 Nov 2020, 02:58.
Last edited by AjitJangale on 16 Nov 2020, 05:13, edited 1 time in total.
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Area of shaded region = Area of equilateral triangle - Area of 3 segments.

Radius of Circle = 20

Side of equilateral triangle = 40

Area of equilateral triangle = (√3/4) * \((side)^2\)

= 400√3

Area of 1 segment = (60/360) * π * \((20)^2\)

Area of 3 segments = 3 * (60/360) * π * \((20)^2\)

= 200π

Area of shaded region = 400√3 - 200π.

IMO Answer is D.
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The figure shows an equilateral triangle where each vertex is the cent 16 Nov 2020, 04:09
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AjitJangale
Area of shaded region = Area of equilateral triangle - Area of 3 segments.

Radius of Circle = 20

Side of equilateral triangle = 40

Area of equilateral triangle = (√3/4) * \((side)^2\)

= 400√3

Area of 1 segment = (60/360) * π * \((20)^2\)

Area of 3 segments = 3 * (60/360) * π * \((20)^2\)

= 200π

Area of shaded region = 400√3 - 200π.

IMO Answer is E.
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I am afraid that you have mistyped your final answer,
400√3 - 200π would be (D).
And thank you for your explanation :thumbsup:
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The figure shows an equilateral triangle where each vertex is the cent 16 Nov 2020, 05:14
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DongminShin
AjitJangale
Area of shaded region = Area of equilateral triangle - Area of 3 segments.

Radius of Circle = 20

Side of equilateral triangle = 40

Area of equilateral triangle = (√3/4) * \((side)^2\)

= 400√3

Area of 1 segment = (60/360) * π * \((20)^2\)

Area of 3 segments = 3 * (60/360) * π * \((20)^2\)

= 200π

Area of shaded region = 400√3 - 200π.

IMO Answer is E.

I am afraid that you have mistyped your final answer,
400√3 - 200π would be (D).
And thank you for your explanation :thumbsup:
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You are correct. Thanks for finding error.

Now corrected my error.
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The figure shows an equilateral triangle where each vertex is the cent 16 Nov 2020, 08:36
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Are of Triangle
=(side)*(Side)*\sqrt{3}/4 = 40*40 * \sqrt{3} / 4 = 400\sqrt{3}....(1)

Area of sectors
Each segment subtends an angle of 60 degrees. Three circles are subtending 180 degrees (60x3 degrees)
Total area of 3 sector combined (inside the triangle)= Area of 1 circle * (180 degrees / 360 degrees) = 400\pi * (1/2) = 200\pi ....(2)

Area of shaded region = 400\sqrt{3} - 200\pi ......(1)-(2)

CORRECT ANSWER: D

Bunuel

The figure shows an equilateral triangle where each vertex is the center of a circle. Each circle has radius 20. What is the area of the shaded region?


A. \(400\sqrt{3}- 100\pi\)

B. \(400\sqrt{3}- 150\pi\)

C. \(400\sqrt{3}- 175\pi\)

D. \(400\sqrt{3}- 200\pi\)

E. \(400\sqrt{3}- 210\pi\)


Project PS Butler


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The figure shows an equilateral triangle where each vertex is the cent 16 Nov 2020, 18:55
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The figure shows an equilateral triangle where each vertex is the center of a circle. Each circle has radius 20. What is the area of the shaded region?

Radius = 20
Area = \(\pi r^2\) = 400 \(\pi\)

Equilateral triangle = \(180^{\circ} \)

Area of shaded region = Area of triangle - Sum of Area of 3 non shaded part of 3 circles

Area of triangle:
Area of equilateral triangle = √3/4 * \(side^2 \)

Side length of the triangle = 40

√3/4 * \(40^2 \) = 400√3

Area of non shaded region:
Sum of area of 3 parts of the circle enclosed in triangle = 3* \(\pi 20^2\) * \(1/6\) = 200\(\pi\)

Area of shaded region = 400√3 - 200\(\pi\)

Ans: D
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