Find the blue shaded area | Geometry Problem | Important Geometry Skills explained

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Find the blue shaded area | Geometry Problem | Important Geometry Skills explained

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Radius=12, height P = 6 (BOP = 30deg)
Start with 1/6 circle AOP = 1/6*pi*12^2 = 24*pi
Add Triangle OPQ = 1/2*4*6 = 12
Remove triangle OAQ = 1/2*4*12 = 24
Answer then is 24*pi + 12 - 24 = 24*pi - 12

EddieDraaisma

r = 4+8 =12.
Draw OP = r, so we have equilateral AOP, so that
angle AOP = θ = π/3 radians, and angle POB = φ = π/6.
(Using the presenter's convention of "area of a figure" = [figure]...)
[Blue shaded region] = [sector AOP] - [triangle AOQ] + [triangle POQ]
= (1/2)θr² - (1/2)bh + (1/2)ab sin(φ)
= (1/2) [ θr² - bh + ab sin(φ) ]
= (1/2) [ (π/3)(12)² - (4)(12) + (4)(12)sin(π/6) ]
= (1/2) [ 48π - 48 + 24 ]
= 24π - 12 . Done!

timeonly

Please inform me, which program are you using for the presentation?

BabisKoutroulis

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شرح واضح مرتب. شكرا جزيلا لكم والله يحفظكم ويحميكم ويرعاكم جميعا . تحياتنا لكم من غزة فلسطين

ghhdcdvv

Radius(R)=OB=AO
AQ=16*sqrt(10)
Area quarter circle 36pi
【AOB】-【AOQ】=【QBP】
36pi-12pi-24=【QBP】=24pi-24

alinayfeh

Very interesting questions and solutions on Math Booster.

johnbrennan

Very good. I enjoy your videos! Thanks.

claudeabraham

I calculated [segment AP] = 24π - 36√3
Calculated [∆OQP] = 12
Then used properties of a 30-60-90 triangle and Pythagoras to calculate that QP = √(160 - 48√3)
Therefore ∆APQ has sides AP = 12, AQ = 4√10 and QP = √(160 - 48√3) so _could_ use Heron's formula to calculate [∆APQ] then add this to [segment AP].
However, I decided that this would be far too horrific, so your method is much better.

Grizzly

Radious of quarter circle is R = 8+4= 12
Angle OAP is θ=60°
Angle OAQ is atan 4/12 = 18, 435°
Angle QAP is α = 60 - 18, 435 = 41, 565°

Blue Area = Area circular segment + Area blue triangle

Area of circular segment:
A = ½ R² (θ - sin θ)
A = ½ 12² (60° - sin 60°)
A = 13, 044 cm²

Area blue triangle :
A = ½ . a. b. sin.α
A = ½ √(12²+4²).12. sin 41, 565°
A = 50, 354 cm²

Blue shaded area:
Area = A₁ + A₂
Area = 13, 044 + 50, 354
Area = 63, 34 cm² ( Solved √ )

marioalb

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