A circle with an inscribed square - can you find the area of the square?

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Күн бұрын

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@SeanKennedy-b4o 2 ай бұрын

I find the negative comments regarding this man’s explanations condescending. Sure.. be proud if you wish to be for solving in seconds, but keep your thoughts to yourself. The explanations in this video and others are exactly right for me to help out my grand children advance. My congratulations.

@gavindeane3670 2 ай бұрын

A couple of ways... The diagonal of the square = the diameter of the circle = 8 From a bit of Pythagoras (or from knowing that the sine and cosine of 45 degrees is 1/√2), the diagonal of a square is √2 × the side length, so the side length is 8/√2 So area = (8/√2)² = 64/2 = 32 Or... Drawing the diagonal of the square makes two triangles, each with base length = diameter of the circle = 8 and triangle height = radius = 4. Triangle area = ½ × base × height = ½×8×4 = 16. But we have two such triangles so total area = 16+16 = 32.

@robertstuart6645 2 ай бұрын

Your first way of solving is the simplest, most straightforward way of solving the problem. I essentially solved the problem that way, except I converted 8 to "2 cubed," divided by sort. 2, then squared what was left.

@gavindeane3670 2 ай бұрын

@@robertstuart6645 Interesting. I think I found the second way simpler. In my head I didn't actually do ½×base×height and then double it. I just did base×height.

@petersearls4443 2 ай бұрын

@@gavindeane3670how about this solution. An easy way to calculate it is as follows. 1. Draw two diagonals creating 4 small triangles. 2. Label the top edge of the square C and the two legs of the small triangle, A and B. 3. We know that A=4. B=4 4. Using a^2 + b^2 = c^2 5. 4^2 + 4^2 = c^2 = 32 6. We also know that c^2 is the area of the square. Area = 32.

@arthurcrown3063 2 ай бұрын

Sides are x, diagonal of square =8; For the triangle x^2 +x^2 = 8^2; 2x^2 = 8^2; x^2 = 32, area of square. Done.

@tamjarvie6021 2 ай бұрын

Area of triangle = .5 x base x height = 4 x4 x2 = 32

@panlomito 2 ай бұрын

You mixed up some things... Area of triangle Atr = ½ . base . height with base = 2 . r = 2 . 4 = 8 and height = r = 4 So Atr = ½ . 8 . 4 = 4 . 4 = 16 (not 32) But the area of the square is asked with Asq = 2 . Atr = 2 . 16 = 32

@argonwheatbelly637 2 ай бұрын

32 squ. in about 10 seconds in my head. It's a circumscribed square, so finding the area of one the right triangles made by drawing lines from the center to a corner of the square is easy enough. Actually, the area is 2r², where r is the radius of the circle.

@panlomito 2 ай бұрын

This one is really easy again... rule of Pythagoras a² + b² = c² with for the square a=b so the area of the square A = a² and diagonal of the square c = 2r = 8 a² + a² = 8² so 2a² = 64 and a² = 64/2 = 32 and that is the area of the square.

@louf7178 2 ай бұрын

Very abstract. Good one.

@jimkoss3318 2 ай бұрын

Yes! Don’t even need to know the sides.

@JoLowden-oz9no 2 ай бұрын

Thank you for the videos. You are awesome at explaining everything.

@petersearls4443 2 ай бұрын

An easy way to calculate it is as follows. 1. Draw two diagonals creating 4 small triangles. 2. Label the top edge of the square C and the two legs of the small triangle, A and B. 3. We know that A=4. B=4 4. Using a^2 + b^2 = c^2 5. 4^2 + 4^2 = c^2 = 32 6. We also know that c^2 is the area of the square. Area = 32.

@dazartingstall6680 2 ай бұрын

Diagonal of the square = 2r = 8 Let the side of the square be x and the area of the square be a. 2x² = 8² 2x² = 64 x² = 32 ∴ a = 32 units²

@louf7178 2 ай бұрын

Very abstract. Good one.

@dazartingstall6680 2 ай бұрын

@@louf7178 Thank you, but it's just Pythagoras.

@louf7178 2 ай бұрын

@@dazartingstall6680 I see it plain as day now, but seems evasive as a chemical double reaction.

@gavindeane3670 2 ай бұрын

@@dazartingstall6680It is just Pythagoras, but it rather neatly solves for x² directly, rather than getting to x² by solving for x first.

@dazartingstall6680 2 ай бұрын

@@gavindeane3670 On the other hand, I realised a while back that the half × base × height method yields a nice little proof. If we inscribe a circle within a square, and a second square within the circle, the outer square will always have twice the area of the inner. As noted elsewhere in the thread, the area of the inner square simplifies to the diameter × the radius. The area of the outer square is d × d We can restate that as 2 × r × d QED.

@hexbinoban6170 2 ай бұрын

2 x radius = the diameter of the circle = 8 The diameter of the circle = one of the 2 diagonals of the square = 8 The diagonal split the square in 2 congruent right angle triangles with angles 45° 45° 90° ---> 1 : 1 :√2 given : Hypotenuse = 8 Calculate the other side --> 8 / √2 since it's a 45° 45° 90° with the sides ratio of 1 : 1 : √2 the length of the sides are 8 / √2 : 8 / √2 : 8 The area of a square is side x side = (8 / √2 ) x (8 / √2) = (8 / √2 )² = (64 / 2) = 32 some square units

@rufflycorrect 2 ай бұрын

This sound much more complicated than it is. Using the KISS principle (Keep It Simple Stupid) Here's the simplest explanation of the simplest solution - If a radius is drawn from the centre of the circle to each corner, the square becomes divided into four right-angle triangles each with two sides of 4. These can be simply rearranged into a rectangle with an area of 4 x 8. Job done.

@EjazNabi-l5f 2 ай бұрын

This man has taken 12 minutes to explain something that any 14 year old should be able to do in 12 seconds on his head! Boy, maths lessons must be long in his school.

@thomasharding1838 2 ай бұрын

Divide the square into two triangles by bisecting it from corner to opposite corner. The bisect will pass through the center of the circle is a diameter and equals radius x 2 or 8 and it is the hippo of the new triangle. Square that to 64 and then half the 64 to 32 and since that 32 is the square of one of the sides of the triangle, it is the area of the square. As Pythagoras said 'The squaws of the hippopotamus is equal to the sum of the squaws of the other two hides.

@KaB-d3n 2 ай бұрын

2r^2 should do it, unless I'm missing something.

@pippodeclown 2 ай бұрын

you missed some school lessons?

@KaB-d3n 2 ай бұрын

@@pippodeclown I am sure I did. Let me explain my reasoning. r^2=4*4=16 This gives you a square area that covers half the area. If you want to visualize, you can draw it starting with one corner in the centre of the circle. Then half the r^2 square covers 1/4 of the target square. If you then cut the r^2 square diagonally and put the half that was outside inside the target square. You now cover half the target square. Now you have to multiply by two to get the answer. 2x16=32 So 2r^2 should always give you the largest square area you can put into a circle.

@pippodeclown 2 ай бұрын

@@KaB-d3n Great one 😃

@blitzkrieg316 2 ай бұрын

You did this way more complex than needed... Radius is 4. We know it's a square so the radius can be rotated 90° making a nice right angle triangle with the hypotenuse set up as X = 4x4 + 4x4. Root 32 is ~5.66 making the answer 32... Took me 20 seconds

@xzxz214 2 ай бұрын

Why so long, I needed 5 secs.

@dazartingstall6680 2 ай бұрын

Come to think of it, if we draw two diagonals we have four right-angled triangles, each with base 4 and height 4, so: 4(2 × 4) = 32

@smylesg 2 ай бұрын

New poster here. Anyone else impressed by the chalk dust on the board? Nice effect.

@pandurangaraonimmagadda9966 2 ай бұрын

Diameter=8= diagonol of square ,2a sqr=64,a sqr=32=Area ofsquare

@Bizmyurt 2 ай бұрын

Yeah I saw it directly and solved it from my head!

@gregwork49 2 ай бұрын

The consensus seems to be that this solution was much simpler for the class than Mr. M expected it to be. Agreed. Before the presentation was finished I began exploring the difference between total area of the circle and the square, easy enough. 50.272 - 32 = 18.272. Then the ratio, which will be constant, and the circumference. Child's play so far, right? Now wondering what is the area between the intersections of one side of the square and the arc between them. Does this require calculus? And before anyone answers 18.272/4 = 4.568, that is not what I'm asking. ed: bad student here. I started from the rabbit hole we were told not to go down. 8/1.414 to solve for a or b.

@gregwork49 2 ай бұрын

Or is it quadratic?

@johnleeson6946 2 ай бұрын

I didn't even think of a^2 + b^2 = c^2. I got it this way: Since the other two angles are 45 degrees, I took the SINE of 45 degrees and multiplied it by 8 and got 5.65685..., squared it,and got 32. The COSINE works, too. Yep, overthinking a problem as usual...

@DanoshTech 2 ай бұрын

Or you could just use multivariable calculus for the volume of solid revolution for the square and circle as both there functions are easy to determine (well the circles function is x^2+y^2=r^2 so x^2+y^2=16

@danharold3087 2 ай бұрын

The radius can be used as 2 known sides of a triangle with one side of the square being the 3rd. 4² + 4² = side² 32 = side² we know that the area of a square is the side squared so we have the answer 32

@ottonormalo4638 2 ай бұрын

What is the formula for the ratio between the radius of a circle and the side of an inscribed square? Is it side = r√2 ?

@Ayelmar 2 ай бұрын

At the thumbnail, solved in my head, haven't gone back to check on paper or with calculator, I think the answer is 32 units. My thinking: The radius of the circle is 4, so the diagonal of the inscribed square is twice that, 8. Working the Pythagorean theorem backwards for a triangle forming half the square, and plugging 8 in for C, we get c^2 = 64. A^2 and B^2 are each half of C^2, or 32. We could then get the square root of 32 for A & B, which would be 5.?????, but htis isn't necessary, since we can figure the area of the square as A * B, or in this case A^2 or B^2 since they're equivalten, so the result takes us back to 32.

@DavidRobinson-rj2sp 2 ай бұрын

Diameter = 8 = hypotenuse. Square so bisected angles are 45. 8 Cos 45 = 4 root 2. 4 root 2 squared = 32. Using COS is quicker.

@MrMousley 2 ай бұрын

If you draw a line from each corner of the square to the opposite corner you then have four triangle shapes that can simply be arranged to make two squares and these two squares are both 4 x 4 4 x 4 = 16 and 16 x 2 = 32 The square has an area of 32 sq cm

@rbpeirce 2 ай бұрын

Rats! I used r to get 32 via the smaller triangle. D would have been so much easier.

@markchapman9591 2 ай бұрын

Interesting, without all the complicated maths I thought a square that fits exactly inside a circle is half the area of the circle. Radius 4 or Diameter 8 x 3.14 ÷ 2 Correct me if I'm wrong please.

@hannahpreece3651 Ай бұрын

Plus one for doing it the much faster area of a triangle way. Given that we know both the base and the height.

@lunar-e.6620 2 ай бұрын

32 sq. units area = a² c = √a² + a² 8 = a√2 a = 5.6569 Area, (5.6569)² = 32 qr. units

@chrisdissanayake6979 2 ай бұрын

Let’s draw 2 lines, each joining the opposite corners of the square, passing through the center of the circle. Then, the square has 4 right angled triangles inside it. Each of the right angled triangle has two equal sides of 4 units each. In a right angled triangle, a^2+b^2= c^2, where a and b are two sides and c is the hypotenuse. a = b = 4 So, a^2 = 4^2= 16 b^2 = 4^2 = 16 a^2+b^2= 16+16=32 Therefore, c^2=32 c = each side of the square Area of the square = c^2 = 32 units.

@dottymansfield832 2 ай бұрын

Parse que c = 32^(1/2)

@pippodeclown 2 ай бұрын

I thought of this first as well. 🤓

@scout13fox 2 ай бұрын

That's the way I did it too! I think it's simpler and more elegant.

@Janie-y6e 2 ай бұрын

I got it correct 4 sides x 8 diameter = 32. Radius is 1/2 of a diameter, thus 2 radii =8. That is how i figured it out. Did i work it the right way?

@abinam2255 2 ай бұрын

Pssst .... too much talk and too much explanation causes confusion. You only needed about 2 minutes to adequately explain the method and solution.

@12baumarobi 2 ай бұрын

For real, as soon as I looked at the thumbnail, I knew that since the radius is 4, the diameter is 8, then turn the square into two triangles, and the hypotenuse is 8. Then use Pythagrian Theorum to find the sides of the square. Side times side equals area of the square

@user-Orkb186-3 2 ай бұрын

I did this in my head in about 30 seconds

@mjtunstall1976 2 ай бұрын

too easy and you are on track!

@debasismajumdar8634 2 ай бұрын

In India it happens to be an oral exercise ( viva voce) and 12+ age children solve it in less than 30 seconds. What's a big deal? The r=4, d=8, being a square, just apply Pythagoras 2xsquare=64, so x square =32. Less than 30 seconds.

@lj6079 2 ай бұрын

Wow you guys are so clever, wtf are you watching this for

@mn1233 2 ай бұрын

this looked complicated. the area of the triangle is 8*4/2 or 16 and the square has two triangles.

@donaldnorton3502 2 ай бұрын

Radius(sqRoot2)×Radius(SqRoot2)=32

@gopherspace8571 2 ай бұрын

OMG great lesson 👍 Wow 😂 thanks Mr J 👍👏🙏💪😎🌎

@bobkrol5939 2 ай бұрын

I did this problem differently. I multiplied COS 45 degrees * 8 and got 5.656 = length of one side. So I squared the to get the area of the square and got 31.9903. I don't have a scientific calculator, but know the SIN or COS of a 45 degree angle is approximately 0.707. So, my answer is essentially the same, and was easy.

@davoblahblah 2 ай бұрын

Draw a diagonal and you've got two trangles with a base of 8 and a height of 4 ...

@markchapman9591 2 ай бұрын

If I'm correct that means a square that fits exactly around the circle will be double the area of the square inside the circle.

@Citizen-pg8eu 2 ай бұрын

Sin (45 degrees) x hypotenuse

@tomtke7351 2 ай бұрын

the square's CORNERS touch the circle. Therefore the square's DIAGONAL = 2r. A square's diagonal is 45° to its side. Therefore the 45/45/90 triangle has sides of ratio: 1/1/sqrt(2). Or the square's side = (1/sqrt(2))×Diagonal = (1/sqrt(2))×2R = (1/sqrt(2))×8 Area of square = S × S = [(1/sqrt(2))×8]^2 = (1/2)(64) = 32 units^2

@PaulCoomber-g7h 2 ай бұрын

Why, oh why, do these videos take so long to explain something so simple? The average high school student would solve problems like this in a few seconds. Why so much repetition and unnecessary verbiage? Does KZbin reward such stuff?

@russelllomando8460 2 ай бұрын

got 32 Diam (Hypot) = 8 Pyth is 64 = 2X^2 so 32 = X^2 so side = sqrt 32 so B x H = 32 thanks for the fun

@terryurquhart2413 2 ай бұрын

Cos 45 x 8 = 5.6568 Ans squared = 32

@ceeemm1901 2 ай бұрын

Dang! I got 31.99....I used too many decimal places. Mr Mathman is that still a pass??

@carstenlarsen8144 2 ай бұрын

diameter in the square = 8, means the ara must be 64/2....= 32 why show us all the s a unness math ? like reputation of being a gig-fessor ? take the easy problems and make them diiiiiificult.

@philbrooke-little7082 2 ай бұрын

I did half base x height for area of one triangular half of square so half 8=4 height is radius too so 4 so half the square is 4x4=16 add the other half also 16 so tot area 32

@dazartingstall6680 2 ай бұрын

A nice reminder that we don't need to automatically reach for Pythagoras when we see a right-angle triangle. I like this.

@fred_2021 2 ай бұрын

A=((2r)^2)/2

@markchapman9591 2 ай бұрын

Area of square 25.12

@josephlaura7387 2 ай бұрын

32sq.units

@johnkashka803 2 ай бұрын

I just made 4 right triangles bxh/2 then multiply times 4

@akbarak3031 2 ай бұрын

S=32

@rodfulford4306 2 ай бұрын

@chadleonard2800 Ай бұрын

@Surreal_Wizard 2 ай бұрын

I come up with thirty two.

@markchapman9591 2 ай бұрын

Sorry 12.56😊

@MichaelEastman-e8v 2 ай бұрын

16+16=32

@JoeA1974 2 ай бұрын

I got 32.0096

@benfranz5812 2 ай бұрын

I like that you are challenging people to use their brains to solve problems but I have never been able to watch a single one of your videos to the end. The condescending style is like nails on a chalkboard to me. Is it really necessary to talk to your audience as if they are 6 years old?

@terry_willis 2 ай бұрын

I change video speed to 2X and it is still very understandable. So half the regular time. It's like removing a band-aid by pulling it off quickly. Doesn't hurt as much.

@davidloveday8473 2 ай бұрын

Maybe you're not the target audience

@t.s.maheswaran6494 2 ай бұрын

This channel is famous for too much talk for even mental sums....ask him to show three sums in one lot !

@EliasVelazquez-t9g 2 ай бұрын

Please share to the point. Too much yak!

@clivewilliams3661 2 ай бұрын

I lost the will to live after just 1 minute. I do hope the narrator doesn't teach maths anywhere because his students will be old men by the time he gets to a testable level.

@SeanKennedy-b4o 2 ай бұрын