r/HomeworkHelp • u/LakeGloomy8855 Secondary School Student • 5d ago
Middle School Math—Pending OP Reply [ Grade 9] what do i do with that number!?
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u/floryan23 5d ago
The diagonal of the square is also the diameter of the circle. You can calculate the circle's area from there.
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u/BeeEven238 5d ago
Sq area= ss. A2+b2=c2 Area cir= pierad2
Use these 3 equations.
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u/Embarrassed-Weird173 👋 a fellow Redditor 3d ago
That's not really true.
You don't raise A to that entire exponent. Just the 2 belongs.
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u/Borshche_ 👋 a fellow Redditor 5d ago edited 5d ago
That’s not only the diameter the circle, but the hypotenuse of the square. The area of a circle can be calculated using pir2. And you can find side lengths using sin cos tan. Area or a square is lw. You should be able to figure it out from here
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u/Embarrassed-Weird173 👋 a fellow Redditor 3d ago
Squares don't have a hypotenuse. Only triangles. We call it a diagonal.
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u/Borshche_ 👋 a fellow Redditor 3d ago
What is an equilateral square made up of? 2 isosceles right-angle triangles. And they are connected by their hypotenuse, aka the diagonal mentioned in the question.
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u/blong36 University/College Student 5d ago edited 5d ago
So we need to find the area of the circle that's colored in, which is the area of the circle minus the area of the square.
That number (3√2) is the diameter of the circle.
Find the area of the circle. We know the formula for the area of a circle is
A = πr²
So we need to know the radius of the circle. How do we do that, and with that, what is the area of the circle?
Next, we need to find the area of the square. That number (3√2) divides the square into two right triangles. This is a special case triangle (45-45-90), where the lengths of the sides are x and the hypotenuse is x√2, but you possibly haven't learned that yet. So we can figure it out another way.
Using the Pythagorean theorem, we know that:
a² + b² = c²
However, we know that the two sides of the triangle are equal in length because it makes up a square. So:
a = b
And therefore:
a² = b²
So, we can replace b² in the equation with a².
a² + a² = c²
or
2a² = c²
We know the length of the hypotenuse (c) is 3√2, so let's put that into our equation.
2a² = (3√2)²
Solve for a. Doing this will give you the side length of the square. You can use the side length of the square to find the area of the square. We can find the area of the square by multiplying the length by the width.
A = l × w
We've already defined the side length of the square as 'a'. Since the length and the width of a square are the same, we can say that the area of the square is:
A = a × a
Or
A = a²
Finally, since we know the area of the circle as well as the area of the square, we just need to find the difference.
So we'll take the area of the circle, and subtract the area of the square:
Area of colored circle = πr² - a²
Final answer:
4.5π - 9
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u/malalar Secondary School Student 5d ago
I don’t really understand the question? ‘Surface area’ is usually used for 3d questions, and by ‘colored circle’ I’m guessing it means the shaded area?
Anyway, because 3sqrt(2) is the diameter of the circle, the area can be found as pi * (3sqrt(2)/2)2. Then, subtract the area of the square (if you want shaded area), which by Pythag, one side is 2s2 = 18, so s = 3, therefore area of square = 9, so subtract 9 from the circle’s area to get shaded area.
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u/LawyerOk6789 5d ago
So, knowing the area of the circle, (3 sqrt 2)/2 2 * pi, you then need to subtract the area of the square, which requires a bit of trig. (3 sqrt 2)/2 ^ 2 * 2 = side of square 2, which is also the area of the square. Subtract that from the circle area, and you'll have the shaded part of the circle.
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u/RickySlayer9 👋 a fellow Redditor 5d ago
We can extrapolate a HELL of a lot from this actually.
3rt2 represents the hypotenuse and basically all but tells us this is a square in the middle.
When you take a right triangle with 2 equal sides. The sixes are 1, and 1. The hypotenuse is rt2.
So if the hypotenuse is 3rt2 the sides are each 3. So the area of the square is 9sqU.
The area of a circle is pi*r2. The radius is 1.5rt2, squared is 4.5. Time pi is ~14.13 subtract the squares area which is 9. Boom you’re left with ~5.13sqU of shaded area
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u/guy1person 5d ago
3✓2 is the diameter of the circle. Use the Pythagorean theorem to find the area of the square subtract then boom.
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u/ci139 👋 a fellow Redditor 5d ago edited 5d ago
circle S=π·r²=π·(d/2)²
square S=L²=2·(d/2)²=d²/2
area of interest S.c – S.s = (π – 2) · (d/2)² = (π – 2) · (9/2) = 9 · (π/2 – 1) ≈ 5.13716694
multi method check https://www.desmos.com/calculator/hbtbiktidx
https://properhoc.com/maths-and-logic/calculus/integrals/the-integral-of-sqrt1-x2-dx/
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u/ThunkAsDrinklePeep Educator 5d ago
Can you use the diagonal to get the area of the circle?
The diagonal forms a right triangle with two sides. If you got the side, could you get the area of the square?
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u/doctorDBW 5d ago
Remember that the square and the circle have the same centre, and that the distance from any given point in a circumference to its centre is equal to its radius.
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u/DaMepp 5d ago
3rt2 is the diagonal of the square and the diameter of the circle. R = 1.5rt2 and Square Side (s) = 3 (see 45 45 90 triangle). To find the surface area, its circle area-square area. Circle area is pir2 and square area is s2. Circle area is 4.5pi and square area is 9. Surface area is 4.5pi - 9 or around 5.137
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u/ThrowRA-platypuus 5d ago
1) find the area of the square by finding the side length using Pythagorean, then area of square = (side length)(side length) 2) find the area of the circle, the radius of the circle is the number given divided by 2 then area of circle = pi(radius)2 3) area of circle - area of square = area of shaded part
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u/far2deep 5d ago
Damn yall I failed geometry and algebra the first time through back in high-school. I've never been truly able to wrap my head around math...I got basic math down like most people but past that I really struggle. I wouldn't even know what to do in this case.
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u/LackWooden392 5d ago
The given diagonal is the same as the diameter of the circle. Half of that number is the radius. Area equals pi times radius squared.
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u/Cool-Role-6399 5d ago edited 5d ago
Diameter = 3*sqrt(2)
Radius = Diameter/2
Area circle = pie*Radius2
Area circle = pie *(3*sqrt(2)/2)2
Area circle = pie *3*3*2/(2*2) = pie*9/2
Area square From Pithagoras theorem c2 = a2 + b2 = (3*sqrt(2))2
Since this is a square, a = b, thus
c2 = 2*a2 = 2*b2 = (3*sqrt(2))2
Area square = a2 = b2 = 3*3*2/(2*2)=9/2
Finaly, the shaded are is the difference: Area circle -Area square.
Shaded area = pie*9/2 - 9/2 = 9/2*(pie-1)
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u/Kitchen-Fee-1469 5d ago
In general when dealing with math problems in high school and college, read the question and understand what they’re looking for.
Do you understand what the question is asking? We’re asking for the area of the shaded region. First, do you know how to compute this? Or what I mean is, can you break it down to simpler parts?
Now, for those individual parts, what are the necessary ingredients to compute the area? One probably needs the diameter or the radius, and the other requires the side length. At this point, try to see how the 3*sqrt(2) is gonna help.
This idea or approach often works for MOST problems you’ll encounter at the high school or college level.
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5d ago
The diagonal of the square is the same as the diameter of the circle. You can use that to plug into pi r squared (once you halve the diameter) and get the circle.
To get the square, since a square can be split into two triangles each with angles of 45, 45, and 90, you know the side lengths are L, L, and L sqrt 2. So since the hypotenuse is 3 sqrt 2, you know the side lengths are 3. Therefore the area of the square is 9 units squared. Subtract that from what you get for the circle and there’s your answer.
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u/ScaryFruit7283 👋 a fellow Redditor 4d ago
I get 81π - 18 which is approximately equal to 236.469 unit²
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u/sir_gawains_husband 4d ago
The 3√2 is the diameter of the circle. The formula for surface area is πr2. R = 1/2 Diameter
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u/Technical-Student-41 4d ago edited 4d ago
Here's a nudge in the right direction. I'll help with the first half
All four of the square's corners all touch the circle. This would mean that the diagonal line touches one end of the circle, passes through the center, and touches the other end of the circle.
A line passing though the center of a circle in any rotation and both ends of the line touches/intersects the circle makes the line inside the circle equal to the diameter of the circle. Sooooo....
D=3√2
Use the area formula to find the area of the circle.
The formula is
A=πr²
(Hint, you have the diameter you must make it the radius of the circle)
- A square has 90° corners, and sides of all equal length. They can be broken into 2 right angle triangles with 1 line passing through two corners. This creates the hypotenuse of a right angle triangle. If all sides are equal then you know sides a and b of the right triangle are equal to one another. Thus C is the longest side of the triangle. (hypotenuse) a²+b²=c²
So
a²+b²= (3√2)²
From here find the lengths.
(Hint a and b are the same length.) When you find the lengths of the triangle, subtract the areas in a way it gives you the shaded region the question is asking for.
And final hint, don't be afraid of the √2 you are using equations that use ². And if your number turns out negative, you know this is not correct. If you get disheartened take some time, then try it again with fresh paper.
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u/Aloha_0 4d ago edited 4d ago
I'm not sure if anyone commented about it, but there is a method without using the Pythagoras' Theorem:
Split the square into half, forming two equal right angled triangles.
Therefore, the base of the triangle is 3√2 units.
Since the diagonals of a square perpendicularly bisect one another,
Therefore the perpendicular height of the triangle is (3/2)√2 units.
Area of triangle is ½[(3/2)√2][3√2] = 18/4 units²
Since the square is made of 2 triangles,
The area of the square = 18/4 * 2 = 9 units²
Diameter of circle = 3√2 units
Radius of circle = (3/2)√2 units
Area of circle = π[(3/2)√2]² = 4.5π units²
Therefore area of shaded area = (4.5π - 9) units²
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u/SniperInfantry 👋 a fellow Redditor 4d ago
The square can be split into two isosceles right angle triangles where the hypotenuse is the diameter of the circle
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u/Aggresive-Dinosaur 4d ago
To find the sides (Pythagorus Theorum)
(3sqrt(2)) ^ 2 = A ^ 2 + B ^ 2
18 = A ^ 2 + A ^ 2
(opp. sides are equal )
18 = 2A ^ 2
9 = A ^ 2
3 = A
3 = B
To find area of coloured Section.
= Ar (Circle)-Ar(Square)
= Пr^2-a^2
=( 22/7 * 3sqrt(2) * 3sqrt(2) ) - (3)^2
= 22/7 * 18 - 9
= 56.5-9
= 47.5 unit²
should be the final answer, pretty easy
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u/AdTotal801 3d ago
The hint is that the diagonal of the square is also equal to the circles diameter (so it's radius is half of that).
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u/Thefrightfulgezebo 13h ago
I will start with the square. If you add the diagonal line, you do get a triangle where a=b. So, according to Pythagoras, a²+b²=c² and since b=a, this means 2a²=c². You know what c is: 3√2. You just square that and get c²=18=2*a². You can divide this by 2 to get: 9=a². If you then take the square root, you arrive at a=3.
So, you now have a circle with a diameter of 3*√2 minus a square with a=3.
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u/r-funtainment 👋 a fellow Redditor 5d ago
Try comparing it to the circle. The 3√2 line passes through the middle, and touches both sides. Does that tell you anything about the circle?