A square is inscribed in a circle with radius = 5, what is the area of the square?

[@coolcutsgal2]

For someone who is 68 years old, who took all of this math in high school 40 + years ago I'm very grateful that you made this video so that I could refresh how to do a Square inscribed in a circle I am taking my accuplacer test in a few weeks and then after that I have to take a nursing test and a pre-nursing admissions test and as you know nursing is a lot of Science and a lot of math of which I excelled in in high school however based on the fact that I've been out of school for 45 + years I'm grateful for your content and I'm also grateful for other math tutors content so that I can glean the information from various sites various teachers that I feel are good teachers so thank you so much from a future nurse

[@jazzcats1000]

I simply divided the square into 4 triangles, which I recombined as 2 squares 5x5 each. 2x (5x5) = 50!

[@vijayakumaryalla1825]

The quickest way

[@hisham56hamilton50]

Can U elaborate ?

[@davidbroadfoot1864]

​@@hisham56hamilton50put in a bit of effort yourself. It's not rocket surgery.

[@hisham56hamilton50]

Thanx Big Foot@@davidbroadfoot1864 I will .

[@hisham56hamilton50]

Hi this is what I mustered .. Divid the square into 4 wright triagles in the center . Tringle Area * 4 = square Area. Square area = ( (5*5) / 2) *4 = 50 sq Un@@davidbroadfoot1864

[@CM-er8tv]

I just made 2 triangles with the hypotenuse as the base of 10 (2r), with a height of 5 (1r). Then used 1/2 x B x H to find the areal of the triangle and added the 2 triangle areas together to get 50 units^2.

[@berry1235]

thats how i did it too

[@CarolSperoni]

@@berry1235 Me too, why make hard work out of such a simple problem?

[@adrianm.2043]

I think the explanation is aimed at the struggling student, who would be a school child, equally suitable for someone older who missed out at school or has children and doesn't remember so well what they learned at school. The explanation was thorough and easily understood it first revised what the young student would have already learned and showed how to apply that learning to solving this question. indeed I had done it in my head but couldn't have explained it so well. I have grand children and they will soon be at this level. Having left school more than fifty years ago and since studied to honours degree level in engineering I did a lot of math. I would never say I was completely confident in it particularly at the higher level, but I got though it with high marks. (in the degree) but a lifetime later a lot of the things learned at school and university at a higher level I do not remember so well having not used them in my work since. I do remember the basics but perhaps not well enough to teach them confidently to a young child, this channel is just the job for that I can see the logical steps of explaining it to a child which I will adopt. Watching these video's a few minutes a day is also a way of occupying myself when the weather is bad. I am looking forward to the higher stuff like calculus, which I always struggled with, perhaps with this form of explanation all will become clear @@CarolSperoni

[@robinpinnock2678]

A little Pythagoras goes a long way!

[@CarolSperoni]

Fair enough, good point@@adrianm.2043

[@MrLeftshot]

c) 50. Simply draw a line through two opposing points of the square creating two right-angle triangles. Since you now know the length of the hypotenuse of the right triangle (radius 5 x 2 = 10 inches) you can now use Pythagorean Theorem to calculate the area of each triangle to be 25. As there are two of them the answer is c) 50.

[@ag-om6nr]

For all of the naysayers ! If you are so smart why are you wasting your time watching this channel ? This gentleman is taking the time to explain basic mathematics to people who are not strong mathematically ! And he is doing an excellent job !

[@thomasharding1838]

100% Correct.

[@tomtke7351]

we recently worked on a square where we found a diagonal creates two triangles inside the square each having angles: 45°/45°/90° The ratios of sides in such a triangle is 1/1/sqrt(2) in this case the diagonal is, thus, two radius lengths. define: D : diagonal of square S : side of square R : radius of circle A : area of square where: R = 5 units D = 2R A = S^2 for 1/1/sqrt(2) triangle abbreviated : a/a/b where : a = b/sqrt(2) in our problem b = 2R = 2(5) =10 a = b/sqrt(2) = 10/sqrt(2) herein: S = a = 10/sqrt(2) A = S^2 = (10/sqrt(2))^2 = 100/2 = 50 units^2

[@Big_Mike724]

His solution is actually much easier, but my mind went automatically to the isosceles right triangle as well. Using the relationships of the sides of such a triangle (x/x/x(sqrt(2))) leads to the same result, but is maybe a little more advanced than he wanted to show. I actually like his method better, but my mind always jumps to the hardest solution because that's the way I learned it lol.

[@njlauren]

50. The diagonal of the state is 10 ( 2xradius= 2x5= 10). That diagonal is the hypotenuse of a right triangle, where the 2 sides are the sides of the square,call them A Pythagorean theorem is C²= A²+B². But in a square, A=B, so we have 10²= A²+A²= 2A². Divide both sides by 2, get A²=50. By definition, the area of a square is S², in our case A², so the area is 50.

[@wonderchads1223]

I got 50 in a much quicker way. Area of a triangle ! Base is 5+5 Height is 5 Area of one triangle is 1/2bxh 5x5=25 x2 triangles =50

[@Siris35]

He loves to talk

[@idjbrown]

Did the same, thought the described answer was very convoluted. In general i enjoy these videos though

[@briansignorelli7090]

That's exactly the way I did it took about 5 Seconds

[@TrevorSachko]

Even quicker, the base x the height of one triangle...

[@quakers200]

Or you can use both diaginals making four triangles that make two squares of side 5 = 25 •2

[@vanderpoolfarmsl.l.c.9983]

WOW!! Took the long way around! Draw the diagonal. Now you have two triangles. What is the area of a triangle? "Half times the base times the height" That gives you the area of half of the square. Multiply times 2. Took 2 seconds.

[@KAF128]

Nice angle, no pun intended

[@KAF128]

Or, when I see your route, I guess, the square could be considered as 4 triangles of 0.5 x r x r, and 2 r squared is 50. Neat !

[@lesaustin3462]

I enjoy the challenges but there is too much needless talk and repetition.

[@Kellen81]

Divide the square into 4 triangles each with height and base of 5. Triangle has area of 1/2*bh, or 12.5 for each triangle. 12.5*4=50

[@mitchbogart8094]

c) 50 Draw both diagonals. It forms 4 isosceles right triangles each with 2 sides each of length radius 5 and hypotenuse side S. S = Sqrt (5^2 + 5^2) = Sqrt(50) Area of square = s^2, which is (Sqrt(50))^2, which by definition of square root is 50

[@ItsNotAllRainbows_and_Unicorns]

Some good answers below. When I saw the problem I immediately drew lines from one corner of the square to the other. Repeat with the other corners. Now the square is in 4 segments. Taking a segment, the radius gives a and b, and the hypotenus is the outer edge. So, a=r, and b=r, therefore a^2 + b^2 is 2r^2=hypotenus^2 Hypotenus^2 , as you know will give the area of the square. Always these problems by drawing it out into something that makes sense to you.

[@erembald2927]

a is the side of the square, 10 is the diameter of the circle and diagonal of the square, 10 = (2a^2)^0.5, square both sides;100 = 2a^2; divide both sides by 2, 50 = a^2 = area pf square

[@Kleermaker1000]

The diagonal of the square is also the diameter of the circle: 2 x the radius = 2 x 5 = 10. Then we have two rectangles who are identical. So 10^2 = 2X^2 (c^2 = a^2 + b^2). X is of course the side of the square. So 2X^2 = 100 => X^2 = 50, which is also the area of the square (all sides are X and the formula for the area of a square is: length x width; the length of each side of the square is of course sqr. rt. 50 or 5 x sqr. rt. 2).

[@churchillsmythe4284]

Simply, area of 1 quadrant of the square is 1/2 base x height = 5x2.5 = 12.5 units². So area of square is 4x 12.5 = 50 units²

[@bruce_c_in_nz]

That is how I did it (before reading the comments!). Much simpler than the video representation and the other suggestions made prior to yours.

[@JosephWood1941-iz6mi]

2 identical triangles. Base 10, height 5. Area of each triangle half base times height. 5×5=25. Two triangles, so area of square is 50.

[@Wilkins_Micawber]

The diagonal forms 2 triangles. Lay the hypotenuse on one triangle to form a the base of a triangle. Base will be 10 the hight will be 5. Half the base 10/2 = 5. 5 x 5 = 25. The second triangle is identical, therefore 25 x 2 = 50. This math teachers method is only useful to ascertain the lengths square’s sides.

[@DdDd-ss3ms]

Two triangles with base of 2r and night of 1r A triangle is base x half night 10x 2.5 is 25 each 2 triangles is 2 x 25 is 50

[@davidmurphy563]

If you cut the square into quarters along the diagonal then the length of both the opposite and adjacent are 5. Therefore the length of the side is the hypotenuse: sqrt(5^2+5^2). We can remove the sqrt because we need the area. So whatever 5^2*2 is. Um. 50.

[@terry_willis]

That's what I did too. ;)

[@JeanPierreWhite]

Same here. Much easier.

[@robeigner4390]

I might be confusing your notation but I agree the easiest way is to divide the square into quarters making each side next to the right angle equal to 5. Since this is your typical right triangle with each equal sides, the sides being "" and the hypotenuse being "", then the sides of the circumscribed square are all . Use distributive property to multiply two sides (to find area of square) giving you 5x5 and √2 x √2, or 25 x 2 =50. (I used the square root emoji. Hope this works.)

[@TigerP1]

me too

[@robertbailey5239]

This guy always takes a 90 second or less problem and spends 15+ minutes to explain it.

[@JeanPierreWhite]

Monetization.

[@angelarhule4239]

And bcuz of that I don’t understand bcuz he talks a lot & not get to the point, he’s confusing me

[@richardhole8429]

Because he is teaching. Those who could understand the one minute version would not need to hear even that. Yes, he does introductory stuff forvthe newcomers. Just drag the slider to skip what you don't need

[@marknieuweboer8099]

This qualified and experienced teacher math and physics doesn't take 15 minutes to explain this.

[@AllDogsAreGoodDogs]

If he's too slow, go somewhere else. YT is free.

[@teofilodeguzman7348]

The side x of the square can also be found using trigonometry Cos 45= x/10 X=10,xcos 45=10*sqr root of 2/2 ,x=5squste of2 X square=50 The area of square

[@LuisdeBritoCamacho]

Diagonal of any given Square = Side * sqrt(2). R = 5 ; 2R = 10 (Diagonal = 10) So, 10 = S * sqrt(10) S = 10 / sqrt(2) S = (10 * sqrt(2)) / 2 S = 5 * sqrt(2) S^2 = Area = [5 * sqrt(2)]^2 Area = 25 * 2 = 50

[@buckets212]

Amazing that you can spend 15 minutes explaining this solution.

[@louisd95714]

You're correct on that, but I wish I had a Math teacher like this when I was in HS. He appears to be very through, which is far away from my Math teacher who always sped through the lessons as if by rote.

[@darylfindley7551]

A 45- 45- 90 triangle has a ratio of sides of 1:1:sqrt(2). Apply this to the similar triangle given in this problem with hypotenuse of 10 (the diameter of the circle) using an equal ratios equation gives you the sides of the square as 10/sqrt(2). Apply the area of square formula (s^2) gives (10/sqrt(2))^2 = 100/2=50.

[@paulmasgalajian8102]

I could have imparted this knowledge and solution within 60 seconds with more clarity than you did in 15 minutes.

[@llywyllngryffyn8053]

C Split the Square into 4 right triangles with side = r. Area of each is 1/2 r^2. So 4 * (1/2 r^2) = 2 r^2. 4=5. 4^2 = 25, so 50 is the answer.

[@Eleuthero5]

I divided the square into two right triangles via a diagonal. That diagonal's length is 10. Then you simply use Pythagoras' Theorem (a*2 + b^2 = c^2). Since a == b here each side is sqrt(50). Thus 50 is correct.

[@elmarbaumann7081]

d = 2xr - [Diagonal of Square == d] - diagonal = 1.414 x s - s = d / 1.414 A = s x s ~ 50

[@richardhole8429]

Recognizing the half-square is a 45-45-90 right triangle the two sides are 1/sqrt(2) x hypotenuse, 10/sqrt(2) Area of the square is [10/sqrt(2)] × [10/sqrt(2)]= 100/2= 50 Square units.

[@jald910]

My method also

[@sheeladatar4005]

Join the corners of the square diagonally, we get four right angled triangles with one corner of the right angle being the centre of the circle. Area of each triangle is 1/2 x base into height I.e. 1/2x5x5. Area of 4 triangles will be 4x 1/2x5x5= 50. Easy solution.

[@jerryeldridge1690]

I asked Copilot, a large language model, this question "In this figure, there is a circle with inscribed square. The circle has radius 5. What is the area of the inscribed square?" and I created a picture with Microsoft Paint of a circle, an inscribed square, and a line segment denoting the radius. Copilot answered this to my surprise but perhaps it used my natural language description instead of the picture. I do not think this is an easy task. For example, one could let one's mind wonder and think about calculating square roots with Euler's Method, system of non-linear recurrence relations, and solving with Groebner bases, but Copilot did not do that. It seemed to optimize some hidden cost function which led it directly to the goal.

[@geoffroberts1126]

IF we consider the diameter as 10 (a radius of 5 x 2 =10) And consider it an equilateral triangle with a hypoteneuse of Length 10, with 2 equal remaining sides (have to be equal since it's a square). . We should get the right answer with Pythagoras,, square of hypoteneuse =10 squared so 100 So the sum of the squares of the other two sides must be 100. Both sides are equal in length so therefore half that must be 50 so we take the square root of 50 and we get 7.071067811865475 which should be the length of each side of the triangle and also the length of two sides of the square, so the others sides, since it's a square, must also be the same length. Area of a square is length x breadth so basically 7.071067811865475 x itself, so squared, which gives 50. So C.

[@apparentlybrian]

Exactly but recognize you don't need to take the square root. Just stop at side squared. :))

[@geoffroberts1126]

@@apparentlybrian Um... ah I see what you mean. Yes. Thank you. Basically doing an extra step. Thanks.

[@JimmyD806]

While in high school, my geometry teacher would have us derive the formula for the ratio of the areas for various shapes (inscribed) nested inside other shapes. Circles inside triangles, hexagons inside circles, etc., for extra credit. On this one, the formula for the ratio of the two areas is 2/pi. If the circle was inside the square, the ratio would be pi/4. But, the answer is 50.

[@niranjanchakraborty1139]

Ans =50. Let side of the spare is x. Given r=5 ,so d=10, now 2x^2=200, now x^2=50. Area is also x^2=50.

[@icedriver2207]

2 Triangles. The height of each triangle is 5 (the circles radius). The base of the triangle is 10 (2*radius). 1/2 base * height = the area of 1 triangle. (1/2 *10 )*5= 25. Two triangles total 50.

[@kennethwright870]

The diameter of the circle, 2r or 2*5=10, which is also the diagonal of the square. The Area of a square using a diagonal is (d^2)/2. So, 100/2=50.

[@ubermo1182]

@kennethwright1870 A lot of people here going about it the hard way. Your way is best. Solving using plane geometry, instead of trigonometry, gives the exact answer without needing a calculator. And the calculator is rounding anyway.

[@Dutch-vj2eg]

Less than 10 seconds to figure. Diagonal = 2* radius = 100, apply pythagoras to find side of square is sqrt(50). Area is product of two sides; sqrt(50) * sqrt(50) = 50.

[@briansignorelli7090]

Talk about taking the long Road I figured this out in about 5 seconds in my head

[@wreckim]

While it is commendable that you are at a certain level higher than others in your math skills, it is often better, IMO, to think of how would you solve this problem step by step, no matter what the radius might be? I appreciate this channel because he takes the time to slowly let those that don't come up with answers in their head into the game. 2c.

[@panlomito]

And the much easier way to solve this: the diagonal of the circle equals the diagonal of the square is 10. So the area of the square is 2 times the area of the triangle with base = 2 x radius = 10 and height = 1 x radius = 5 Asq = 2 x Atr and Atr = ½ . base . height = ½ . 10 . 5 so Asq = 2 . ½ . 10 .5 = 50 units²

[@petergeorge2716]

It's a really easy problem, just break it up into 4 right angled triangles, the area of a triangle is half the base x height, so area of the square is 1/2x5x5x4 = 50. I did it in my head in less than 30 seconds and I am definitely no math wizz.

[@laurencelim5078]

I did it in 5 seconds. The diagonals are 10 and at 90 degrees. Using Geometry = Area of triangle = half height x base = (2.5x10)=25. Square =25x2 =50.

[@deograciousuwiragiye8429]

S squared + S squated =d Squareed 2s squared= 10 Squaree 2s squared= 100 S squared =50 S= square root of 50 Area of A square=s×s Area = square of 50 × square room of 50 Area= 50 I chose C.

[@erikamdi5840]

r=5 2xR=hypotonuse in a triangel the hight of the triangel=r so hight X hypotonuse x ½ is one triangel so x 2 (or just leave the half out)

[@jarvisfamily3837]

In the square, draw the diagonal lines. Those are each 10 units long. At the center of the square these lines meet, forming four triangles with sides of 5 and an included angle of 90 degrees. 1/2 * 5 * 5 * sin(90) = 12.5 * 4 = 50. QED

[@danielmadden9691]

Solve using theorem of Pythagoras ,5^2+5^2 =50 implies area =√50×√50

[@georgejohnson1498]

That is how I worked it out before watching. You do not need a calculators or even pen and paper! √50×√50 equals 50!

[@johnyarbrough502]

@@georgejohnson1498 Yep. 8th grade. Miss Sparks would be proud.

[@John_Lyle]

The Diameter is 10 (2xRadius) The line of the diameter bisects the square into two 90° triangles with the hypotenuse being 10. The sides of the 45,45,90 triangle are 0.707x the hypotenuse so the area is 7.07x 7.07 which is ≈50

[@chenks54]

The square can be divided into 4 equal right angled isosceles triangles where two of the sides are each equal to the radius of the circle. Since the area of a right angled triangle = half base x height, the area of each triangle = 5/2 x 5 which =12.5. Since there are 4 triangles of equal area making up the square, the area of the square = 4 x 12.5 = 50

[@MichaelPaoli]

50 - a do it in my head problem: radius is 5, so square's diagonal is 10, or half that is 5. Now make a triangle, start from center of square to the upper corners, the lower sides are 5, and that bottom angle is a right angle. Now reflect that triangle across the upper line and attach it to the one below, that's a 5x5 square thus of area 25. Now split it through that upper horizontal line. Take the upper half, translate it down until it forms triangle in the square with base at the bottom of the square. Those two triangles cover exactly half the area of the square, and still have the same total square area of 25. Double that (e.g. think add a rotation of 90 degrees about the center) to cover all and only exactly the entire square, so double that area of 25 to 50, and we've got exactly the entire square covered, so that's our answer: 50 What calculator? I used wetware. ;-) Additional hint/tip: sometimes math problems give you additional information that's not at all needed or useful. E.g. the circle and radius - mostly superfluous. All we really need/used from that, is distance from center of square to corner of square - which is the radius of the circle, so circle and radius is otherwise totally irrelevant. Could have just as well had the information that distance from center of square to corner is 5, and there'd be no need for mention of circle or radius. Also don't need Pythagorean theorem at all here - often there are simpler ways of doing things geometrically. In fact to calculate square's area or perimeter or side length or diagonal length or half of that (center to corner) we only need any one measurement on the square - any given known length between two well defined points, or a well defined area relative to the square.

[@silverknight1966]

the radius = 5, the diagonal of the square =10 that is the hypotenuse of the 2 triangles. the sides of the square are equal so the pythothagorean equation is a^2 +a^2 = 10^2. that simplifies to 2a^2=100, a^2 = 50. a^2 is the area of the square so answer is 50

[@johnplong3644]

All you need to know is some basic geometry and trigonometry You need to know what the definition of a square .What a diameter is what a radius is You know this you can solve this problem There are a bunch of ways of doing this problem using trigonometry You should do it the fastest and easiest way .Trigonometry students would solve this quickly using Cosine .usually a person learns the Pythagorean theorem first So I can understand why John chose this approach

[@apveening]

Using the Pythagorean theorem also avoids usage of calculators or cosine tables.

[@paulanizan6159]

John, could you maybe make videos about calculus and/or differential equations?

[@rtktom]

Why to complicate everything so much? It could be quickly solved with just couple of arithmetical operations: Two diagonals create 4 triangles, area of each is: (5x5)/2 So area of the square is sum of these 4 triangles: 4 x ((5x5)/2) = 2 x (5x5) = 50

[@uweherdel8496]

1 the area must be fewer than pi 5^2 but much more than half of it ~> so the only solution should be 50. or exactly : the diagonal of the square is equal to the diameter of the circuit = 10 2* a^2 = 100 ( Pythagoras) -> a^2 =50 q.e.d

[@michaelkouzmin281]

A square is a rhombus. Rhombus area = product of its diagonals/2. Here d(iagonal)=d(iameter)=2*r=2*5=10; Area of the square = d^2/2 = 10^2/2=100/2=50 sq units.

[@ArnoldWilliams-ge6jg]

I used 4 equal triangles. these are 45-45-90 triangles,I used the square root of 2 times 5 to get 7.071 to get one side of a triangle. then I squared 7.071 to get 49.999.This is pretty close to 50. If I am incorrect in my thinking on this,please correct me.Thanks!!

[@philipkudrna5643]

If r=5, then the whole diagonale is 10. The side is therefore 10/sqrt(2). If you square this, you get the area of the square as 100/2 or simply 50. Thus: c is correct.

[@WernerKerschbaumer]

I’m intrigued to know what age group this is aimed at, because this channel’s videos keep popping up on my feed with silly maths problems that I learned to solve more than 45 years ago. And my teachers sure didn’t take 15 minutes to explain it! There was way too much work to cover to waste so much time on such easy stuff.

[@whomigazone]

Area = A of any square inscribed in a circle with radius = r will be A = 0.5*(2*r)^2, SO 0.5*(2×5)^2 = 0.5*(10)^2 = 0.5*100 = 50

[@andyr9492]

Find the area of the circle. Pi x radius squared. Say 3 and a bit x25 = 75 and a bit. Square is more than half the circle but not as much as 75 so it’s got to be 59. in multiple choice Qs you haven’t got time to mess about with Pythagoras or even triangles calcltns.

[@Ctaylo19.007]

Never thought to apply some algebra and solve for x squared, I'm used to solving only for x, gratitude for exposing me to the thought of solving for something besides a plain old X

[@lientran1808]

Side of the square = root of( 5²+5² ) = square root of50 Surface of square = side× side=square root of 50×square of 50 =50 Answer is 50

[@Matt1969-wh4mu]

Given this was multiple choice, the quick way was to calculate the area of the circle. Call pi a bit more than 3, you get the area of the circle being a bit more than 75. The area of the square is somewhere between that and half of it. the only possible answer was 50

[@nuclearmedicineman6270]

LoL, that's what I would have done if I didn't happen to know the answer by heart. Got to love multiple choice questions, you know one is correct, so just eliminate the wrong answers and you nail it. That r=5 circle is a bit of a special case; the smallest square you can fit a r=5 circle inside is A=100, and the largest square that fits inside a r=5 circle is A=50.

[@davidfabien7220]

c to the square equals a to the square plus b to the square. c to the square = 100, therefore a to square = 50 and b to the square = 50. The square root of 50 is roughly 7.07 a x b = 7.07 x 7.07 = 49.98

[@DavidWChrisman]

Okay, I had a stroke in 2020 and this looks like a good metal exercise for me. - I have to write down as I go as I forget where I am at and what I am doing. First like using a slide rule, what is the approximate value of the calculation. The area of the square is smaller than the area of the circle as it is inside the circle. The area of the circle is over 75 using just 3 for pie and pie times r squared. I could guess at the answer thinking that it is less than 75 and being multiple choice. I think the center of the square is the center of the circle no matter how you rotate the square, but I can’t remember the trig needed to prove that. The area of a rectangle would be the unit Height or Vertical, (H or V), times the unit Length or Horizontal, (L or H). Not to confuse H with H, I will use V and L. I will use H later. From the top left Vertical to the center is one Radius (R) and from the bottom right Length to the center is also one Radius (R) in length, where R is 5 undefined units. A line from these two corners of the square creates two triangles. Think of this line as the Hypotenuse of the triangles. Now I will use Pythagorean Theory and H for Hypotenuse. Next, I will use the theory that VV + LL = HH. If R = 5, then H = 2R =10, and HH = 100. Now I have VV + LL =100. Being a square, V = L, replacing L for V, I now have LL +LL =100. Okay LL +LL = 2LL. Next 2LL = 100, and LL = 50. The area of the square is VL and VL is the same as LL and LL = 50. - Thanks for the puzzle!

[@dk3480]

Since diameter equals diagonal here, area of square will be half of square of diagonal, ie 10*10 :-2=50 sq.units. Am I right Sir?

[@dilippatil3909]

Yes ,You are right.❤

[@BertGrink]

This is dead simple: the diagonal of a square is SQRT(2) times the side, and since we know that the diagonal is 2R (10) we simply divide 10 by SQRT(2), and multiply that number by itself. Case closed.

[@nancyblockcolsky1387]

The area of the square is 50. If the radius is 5, the diagonal of the square is 10. The square of the hypotenuse = the sum of the square s of the other two sides. 100 = 50 + 50, and the square root of 50 is 7.05. The area of any rectangle is height times width, so we wind up multiplying 7.07 times 7.07, or 7.07 squared, which brings you back to 50. It seems that the area of the square will always be the diameter squared divided by 2.

[@sudipamaitycreation3612]

Radious=5;diameter= 10;each side of the square =a, so a^2+a^2= 10^2 or,2a^2=100 or, a^2=50 so area= 50

[@Dr_piFrog]

Area of Square = 50 | Area of Circle = 25 x pi | Area of Space between Circle and Square = 28.54

[@billcape9405]

There are about 6 ways to calculate this. Not all that tough, but you said it right... you have to know what is a radius. Just use the triangles if you get stuck.

[@DaManCave123]

Just draw a diameter using 2 oppo point on the square, which is 10 units. Then use Pythagoras theorem to find the side which are equal=root(50) , therefore area=side*side=root(50)*root(50)=50!

[@DaManCave123]

My question is, why did it take that long to do?

[@Ayelmar]

At the title card (0:01), my answer is c) 50. Since the radius is 5, the length of a line from the center of the square to any corner is likewise 5. We can divide the square diagonally into 4 triangular quadrants, and any two of those quadrants can be fitted together to form a suare with sides 5 units long, so the 4 quadrants form two 5 x 5 squares, each with area 25, and 25 * 2 = 50.

[@wesleyc.4937]

Yea, that's better.

[@BachirRita]

Ohhhh! Marvelous pythagoras... So elementary... What a genius explanation!!!! 50...

[@RobWhittlestone]

This took me 30 seconds or less. Area = (Cos 45⁰ x 10 )^2 You can use Sin too. I did it in my head. I went to school in the 1970's so I memorized the answer of Cos 45⁰ and Sin 45⁰. This is the kind of problem i could solve aged 13 or 14. English school system.

[@Spudz76]

C because ((5*2)^2)/2 as squaring (area of a square) the square root (side length) can just be skipped

[@normanhansen730]

500

[@petesam1911]

My God! You are very good at massaging.

[@SoulLeister]

Ho-hum, ur bragging about being many yeared teacher; I expected more than Pythagoras 101. Not being a teacher let me elucidate (seeing as u asked for my thoughts): ...the 5x5 square fits diagonally inside a 10X10 square (which is literally makes it 50% of 100), NO PYTHAGORAS NEEDED. FUNFACT, for ANY square (with a diagonal = radius x 2) its area will be .637% of the circle's area. ...conversely, any circle (with a diameter = side of the square it fits in) its area will be 78.5% of the square's area. ...rounding for brevity... 1974, I was kicked out of a high school algebra class for showing "MY" work (required on a test), I had arrived at the correct answer, but the teacher made his case to the principle (even though my answer was correct) I hadn't done it the way he had instructed us to solve such problems!?! I pointed out to the principle that the test instructions said, "SHOW YOUR WORK, WHICH I DID" not his work dutifully regurgitated over the ages. THE PROBLEM IN QUESTION WAS: WHAT WAS THE AREA A CIRCLE WITH A DIAMETER OF 7. ...the teacher expected me to regurgitate the textbook: pi * radius squared. ...I DIRECTLY/CORRECTLY USED THE DIAMETER: DIAMETER SQUARED * PI/4 (REENGINEERING THE EQUATION)

[@apparentlybrian]

Don't need side length! Diag length 2r = 10. Square's area = s^2. s^2 + s^2 = 2s^2 = 100. s^2 = 100/2=50. That's IT

[@nolongerhave-couth4771]

I solved it by realizing that the triangles, inside the circle have a relationship where the sides are in this ratio: the two shorter side are equal to 1and the hypotenuse is equal to the square root of two. Therefore, if the diameter is 10 units,(2×r), then the length of the hypotenuse is 10÷√2,(10÷1.414), or approximately 7. Multiply the sides, approximately 7×7=49. The closest answer is 50.

[@thomasharding1838]

Also, The area of a SQUARE is equal to one-half the square of the diagonal. In this case, the diagonal is 10 and 10 squared is 100 and one-half of 100 is 50.

[@terryfox9344]

I solved it by constructing a slightly different right triangle, making the hypotenuse one of the sides to the square and the other 2 sides to the triangle 2 of the radii. Using the Pythagorean theorem the resulting calculation was the same.

[@themister3865]

I used the cos function to calculate the length of the opposite side of a 45-degree angle and a hypotenuse of 10. Then, 7.071 X 7.071 = 50

[@jayrussell3796]

I just took half of 5 (radius) times 10 (diameter) equals 25(area of triangle) and doubled it in my head.

[@jayrussell3796]

Your way is definitely the way to do it though...sin, cos, tan....those are just so hard to remember lol

[@pitiedvod]

I did the same using sin. This problem is really simple with some basic trigonometry and logic.

[@ubermo1182]

@themister3865 But using the cosign doesn't yield 50. It approximates 50. Your equation actually yields 49.999041, which is definitely not 50. Using Euclidean geometry and simple algebra will give an exact answer. The square is composed of two triangles, height 5 and width 10. The area of a triangle is 1/2 h × w. The formula for the area of the square would be area=2×([5×10]÷2). Remove the outer brackets and the twos cancel each other. 5×10=50=area, which is the exact answer. Trigonometry is a great tool, but the above method gives the true whole number answer and, frankly, can be used to solve the problem in your head without a calculator.

[@jayrussell3796]

@@ubermo1182 That's how I did it. Geometry works way better for me, especially in my head. I think it has something to do with trig using approximation that makes it difficult for me to really grasp...idk

[@kenodaro2466]

One side of the square= 10xSin(45) Area of Square= (10xsin(45))^2 = 50

[@glasshalffull2930]

I just did A=Pi * r^2 A=78.5 So the area of circle is 78.5. Therefore, the answer can’t be 74 and it isn’t 36 because that’s half the area and that’s obviously not the answer. So by a little math and logic, the answer is 50. A classic SAT style question that can be answered in seconds.

[@panlomito]

Diagonal of square = 2 . radius = 10 units = side of square . V2 units So the side of the square = 10/V2 units And the area of the square is (10/V2)² = 100/2 = 50 units² No calculator needed, no paper needed, only delicious brainpower for 1 second...

[@CypherSpectre]

Used: [10Sin(45)]^2

[@Timmmmartin]

A circle of diameter 7 has circumference approx. equal to 22 and area approx. equal to 77/2. Furthermore, the inscribed square has side lengths equal to 7xsqrt(2)/2, perimeter equal to 7x2xsqrt(2) and area equal to (7^2)/2. How neat is that?!

[@rajdhar327]

1. Join any two points of diagonal of the square whose value is 10 (evidently seen). Diagonal of the sqaure=diameter of the circle=2xRadius=10 2. Now apply Pythagorus Theorem 10^2=a^2+a^2 [where a is the side of the sqaure. 3. 100=2 a^2 or a^2 = 100/2 = 50 = Area of the square [Formula: Area of square= side^2] Seedhi baat. No bakwaas !!! Clear hai?

[@PWingert1966]

As a test of my ability to craft a suitable prompt for this physical problem, I fed this to the copilot using the following prompt and received the attached answer. The answer does not match any of the answers provided. what is the area of an inscribed square if the radius of the touching circle is 5 Copilot When a square is inscribed within a circle, the length of each side of the square is equal to the diameter of the circle. Let’s denote the side length of the square as (a) and the radius of the circle as (r). The radius of the circle is equal to half the length of the square’s side: (r = \frac{a}{2}). The diameter of the circle is twice the radius: (d = a). The circumference of the circle is given by: (C = d \cdot \pi = \pi a). The area of the square is given by: (A = \frac{\pi a^2}{4}). Now, let’s apply this to the given problem where the radius of the touching circle is 5: We have (r = 5). From the relationship (r = \frac{a}{2}), we can solve for (a): [a = 2r = 2 \cdot 5 = 10]. The area of the inscribed square is: [A = \frac{\pi a^2}{4} = \frac{\pi \cdot 10^2}{4} = \frac{100 \pi}{4} = 25 \pi]. Therefore, the area of the inscribed square is approximately (25 \pi) square units. Did I phrase the prompt incorrectly?

[@PWingert1966]

I did not think that the length of the sides of the square being equal to the diameter from a visual inspection of the problem statement made sense, so I rephrased the prompt as follows: Given a circle of radius 5 and a square inscribed within a circle such that the corners of the square touch the circle and are equidistant from the center of the circle find the area of the square within the circle expressed in units. I still get the same result of 25pi^2 units. The new results do eliminate that first statement about the diameter of the length of the square as being equal.

[@duderino1950]

This could have been easily explained in about three minutes or less. If the radius of the circle is five, the diameter is 10 which is equal to the diagonal of the square. The diagonal of a square is 1.414 x the side. Therefore 10= 1.414x. 10/1.414=7.07. Square that, you get 50. Correction. Takes about one minute.

[@danolson1467]

I love this channel! I do wish though that we could start pronouncing "pythagorean" with an N sound at the end. It's not a terribly big deal, but we're math students after all.

[@stevencraven4897]

Easier way. If you stay with the radius = 5 and make BOTH diagonals you are left with 4 right isoceles TRIANGLES with base and height 5. 4*(1/2(5*5)) = 4*(1/2(25)) = (4*1/2)*25 = 2*25 = 50.

[@georgiydoroshko5977]

Диагональ квадрата - - диаметр окружности, т. е. равен двум радиусам - 10. Диагональ в то же время является гипотенузой прямоугольного равнобедренного треугольника. со стороной х По теореме Пифагора квадрат гипотенузы равен сумме квадратов катетов 100 = 2*х*х . х в квадрате = 50. С другой стороны, сторона треугольника является и стороной квадрата и поэтому площадь квадрата со стороной ж будет х в квадрате, т.е. 50.

[@pandurangaraonimmagadda9966]

2a sqr=100, a sqr=50 ,a = root 50 area=a sqr=50

[@joanmackie1735]

I would assume that any student who is expected to know pythagoras theorem is already confident about the properties of squares and circles. Just one diagonal gives you a right triangle with two equal sides. If you start there, it is less confusing.

[@lynnwood7205]

Which of four answers to select? if we gave each side a value of five and multiplied two sides to obtain the area of a rectangle, of which a square is one form, you obtain an area of 25 square feet. The only answer of the four answer choices that would apply is 19.5 because we know the length of each side is less than 5.

[@thomasharding1838]

I just found the same puzzle posted by TCMath except the radius is 4. What would be a great puzzle would be to find the area of the circle outside of the square.

[@johnward5102]

I take 10, the diameter, divide by root 2, so 1,412, to get 7.08, the side of the square, and then square it to get (nearly) 50.