Can you find area of the Purple shaded region? | (Rectangle inscribed in a circle) |
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- Опубліковано 8 лют 2025
- Learn how to find the area of the Purple shaded region in the circle. Yellow rectangle is inscribed in a circle. Yellow rectangle area and perimeter are given as 264 and 68 respectively. Important Geometry skills are also explained: area of the rectangle formula; Pythagorean theorem; area of the circle formula. Step-by-step tutorial by PreMath.com
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Hello from Brazil, 👋 Love ur content!!
Glad you like them!
Thanks ❤️
Two Premath videos. What a glorious morning!
Glad you like them!
Thanks dear❤️
Very nice sharing sir❤❤❤
Thanks for visiting❤️
This one is pretty easy compared to your other Premath videos. Thanks for the brain teaser.👍
Let a = rectangle width and b = rectangle length. So 2a+2b = 68 and a*b= 264. That's 2 equations with 2 unknowns. Solve and get a = 12 and b = 22. Diagonal of rectangle is diameter. Use Pyth. theorem to get diameter (2*radius), i.e 12^2 + 22^2 = (2R) ^ 2. so 628=4R^2. R^2 = 157. Purple = 157*PI - 264
Yes, I did it in a similar way. Not identical but mostly the same components. Thank you.
@ 3:22 , I absolutely love it! 🙂
Glad to hear that!
Thanks ❤️
Purple Region Area = Circle O Area - Rectangle ABCD Area
The rectangle area was already given as 264 square units. To find the circle area, we need to find its radius.
Label the length of the rectangle l and the width w. Therefore, lw = 264 and 2(l + w) = 68.
Draw a diagonal of rectangle ABCD. (I chose segment BD) This is the diameter of circle O. So, l² + w² = d² by definition of a diagonal.
Divide both sides of the second equation by 2. Thus, l + w = 34.
(l + w)² = 34²
l² + 2lw + w² = 1156
d² + 2(264) = 1156
d² + 528 = 1156
d² = 628
d = √628
= √(2 * 2 * 157)
= 2√157
Since d is the diameter of circle O, the radius will be half its length. Thus, the radius BO is √157 units.
Now, we are ready to find the circle area.
A = πr²
= π(√157)²
= 157π
Purple Region Area = 157π - 264
So, the area of the purple region is 157π - 264 square units (exact), or about 229.23 square units (approximation).
Thanks ❤️
Always think your channel give me great time
At 6:14, I found it easier to first simplify the measure of the diameter, then divide by 2 for the radius. Otherwise, the method used here is essentially how I solved it.
Also, had this figure been to scale, a=22, b=12.
Great short cut! and neat lesson
Thanks for sharing❤
Thank you!
Area = x•y = 264
Perimeter = 2(x+y) = 68
Form two numbers x and y from all the factors of 264 such that x+y = 68/2 = 34
Prime factors of 264: 2•2•2•3•11
24+11 = 35 -- X
8 + 33 = 41 -- X
12 + 22 = 34 -- ✓
Therefore AB and CD are 12 and BC and DA are 22. Drop a perpendicular to point P from O to bisect AB. AP and BP are 6 and OP is 11.
Triangle ∆APO:
a² + b² = c²
r² = 6² + 11² = 36 + 121 = 157
A = πr² - 264 = 157π - 264 ≈ 229.23
Thanks ❤️
Very Very useful video sir 🎉🎉🎉🎉🎉
Thanks and welcome❤️
That's very easy.
l and L beeing the dimensions of the rectangle, we have lL= 264 and l + L= 34, so l and L are the solutions of the equation x^2 - 34x + 264 = 0
The reduct delta is 17^2 - 264 = 25, so the solutions are l = 17 - 5 = 12 and L = 17 + 5 = 22.
Now if R is the radius of the circle, the Pythagore gives that R^2 = (12 / 2)^2 + (22 / 2)^2 = 121 + 36 = 157
Finally the purple area is 157. Pi - 264.
Thanks ❤️
The area of a circle is also πd²/4 where d is the diameter of the circle. Thus, we can get directly 628/4 😉
2a+2b=68→ a+b=34 ; a*b=264→ b=264/a→ a+(264/a)=34→ a²-34a+264=0→ a=12 ; b=22 → 12²+22²=AC=4r²→ r²=157 → Área púrpura =157π -264 =229.23
Gracias y saludos.
Purple shaded area=157π-264=229.23 square units. ❤❤❤ Thanks.
Thanks ❤️
Solution:
Area = Base . Height
A = b . h
264 = b . h
b . h = 264. Eq. 1
Perimeter = 2 Base + 2 Height
68 = 2b + 2h
2 (b + h) = 68
b + h = 34. Eq. 2
b . h = 264
b + h = 34
(34 - h) . h = 264
34h - h² - 264 = 0 (.-1)
h² - 34h + 264 = 0
h = 34 ± √[(-34)² - 4.1.264] / 2
h = 34 ± √100 / 2
h = 34 + 10 / 2
h = 22 rejected
h = 34 - 10 / 2
*h = 12 accepted*
*b = 22*
r² = (11)² + (6)²
r² = 121 + 36
*r² = 157*
Area Purple Region = Area Circle - Area Rectangle
APR = πr² - b.h
APR = 157π - (22.12)
*APR = 157π - 264 Square Units*
*==============*
*APR = 229,23 Square Units*
*=============*
Thanks Sir
That’s very nice
With glades
❤❤❤❤❤❤
I did it the long way by chord intersect theorem
Thanks dear ❤️
Thank you
Yes I can!!
Solving this System of Equations:
x*y = 264
2x + 2y = 68 2(x + y) = 68 x + y = 34
So:
x = 22 and y = 12 or,
x = 12 and y = 22
Finding the value of the Diagonal (D):
12^2 + 22^2 = D^2
144 + 484 = D^2
628 = D^2
D = sqrt(628)
D ~ 25,06 lu
So, the Radius (R) is equal to: sqrt(628) / 2
Area of the Circle (A):
A = Pi * R^2
A = Pi * 628 / 4
A = Pi * 157
A ~ 493,23 su
Finding the Purple Shaded Area (P):
P = 493,23 - 264 ~ 229,23 su
Answer: The Purple Shaded Area is equal to app. 229,23 su
Hello from Rezende RJ - Brazil
I used the quadratic formula to figure out the width and height of the rectangle. I knew the sum of the squares of the width and height would be 4 times the radius squared.
157π-264≈228,98≈229😊
Thanks ❤️
228
Thanks ❤️
Nice video but today's problem was a little too easy
Thanks ❤️
Yea, I did it in like 30 seconds
asnwer=150
It is not 157*PI ..it is 132**PI...
Let's do some math:
.
..
...
....
.....
First of all we calculate the side lengths x and y of the rectangle from the given values for the area A and the perimeter P:
A = x*y = 264
P = 2*(x + y) = 68
x + y = 34
x + 264/x = 34
x² + 264 = 34x
x² − 34x + 264 = 0
x = 17 ± √(17² − 264) = 17 ± √(289 − 264) = 17 ± √25 = 17 ± 5
Finally we have the side lengths x=22 and y=12 or x=12 and y=22. The length of the diagonals of the rectangle corresponds to the length of the circles diameter. So it follows:
d² = x² + y² = 22² + 12² = 484 + 144 = 628
r² = d²/4 = 628/4 = 157
A(purple) = A(circle) − A(rectangle) = πr² − A = 157π − 264 ≈ 229.23
Best regards from Germany
Thanks ❤️
I thought to myself … what are the factors of
264 = (1 2 3 4 6 8 11 12 22 24 33 44 66 88 132 264) ... or in multiplicative pairs
1, 264
2, 132
3, 88
4, 66
6, 44
8, 33
11, 24
12, 22
Then knowing that the perimeter is 68, which is (2𝒂 ⊕ 2𝒃), it means the (𝒂 + 𝒃) = 34. Well, what in the pairs above equals 34? ... 12 and 22.
Sure enough (12 × 22) = 264 square units, so the rectangle is satisfied. And
2 × (12 + 22) = 68, so the perimeter is equally satisfied.
The radius is √((𝒂 ÷ 2)² + (𝒃 ÷ 2)²), which evaluates to 12.529964
OK, Then the area of the whole circle is A = π𝒓² = 3.141593 × 12.529964² = 493.2300
… then just subtract out the 264 area of the rectangle, leaving
Purple = 493.2300 - 264.0000 = 229.2300 u²
Which is the same as what your solution provided!
By a rather different route.
Yay.
⋅-⋅-⋅ Just saying, ⋅-⋅-⋅
⋅-=≡ GoatGuy ✓ ≡=-⋅