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

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

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.

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.

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.

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.

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

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

Amazing that you can spend 15 minutes explaining this solution.

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

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

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

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

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.

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

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

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.

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

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

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.

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 !

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.

This was a lot of fun!

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.

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!!

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.

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

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

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

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.

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.

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.

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.

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

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.

Incredible how many words used for this

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.

Not looked, so I may be totally wrong, but, in under half a minute, the diagonal of the square is 2 radii, so by Pythagoras I can simply get the side of the square (isosceles triangle!) and we don't even need to square root it since the question wants us to immediately square it back up... 50 ?

Question: the square aera would always be the vircle radius*10?

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.

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.

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

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!

Used: [10Sin(45)]^2

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

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

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.

Pi r square cornbread r round is how I remembered that formula in school

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.

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

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.

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?!

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.

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.

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.

I missed an obvious step, but I still solved the problem in my head.

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.

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.

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.

I got same solution as jazzcats. The 15 minute version is full of waffle but does provide a cure for insomnia.

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

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

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

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.

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.

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.

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?

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

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.

got it 50. simply pyth thoery with hyp of the isos tri = 10 thanks for the fun

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.

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

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

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

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.

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

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.

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

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)

The area of the circle is about 78.5 (pi*r^2) , so 50 is the only reasonable choice.

No comments.but one problem. Data given is tape role start with R1 ends with R2 and member of turns say N. What is the length of the tape.

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

c) 50 Thanks,

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.

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.

I just figured the diameter is 10 and could visually see the square created using the diameter as one side of the square was equal to 4 times the triangle created by the diameter and two sides of the first square. So, the area of the square is 2*((10*10)/4)

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!

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

I'm just going to take a wild guess, it's C. Because if you took the Pythagorean theorem, the radius is 5, therefore, based on Thale's Theorem, the diameter is 10, therefore it's going to be two equal squares that when multiplied together equals 100, so the closest to that is 7. And 7 * 7 equals 49.

Quick guess 50 based on calculating actual circle area then estimating the square/circle area ratio.

*EASIEST SOLUTION* X both diagonals of the square to create FOUR right triangles... 4((5^2)/2)=50

My God! You are very good at massaging.

I solved it differently. Since I knew the diameter was 10, I also knew the sides were equal to 7.07. 7x7=49. Close enough.

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

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

I did it with diameter being 10 then worked backwards with what I remember from trig.

There is only one possible correct answer to choose. If the radius is 5 units, the diameter is 10, and ten is the length of the square's diagonal. if the square's diagonal is 10, its side length is √50 (100^2)/2 so the square of a side is... 50... And... The area of a square inscribed within a circle is 22/28 of the circle's area, and the area of a circle circumscribing a square is 28/22 times the square's area. Precise enough for these problems... Is based upon an approximation of pi at 22/7 Within 4/10,000ths...

Is there another channel like this that doesn't take 15 minutes to answer 1 math problem?

50: radius is 5 so distance between corners of square is 10, side of square is root of 50 so area is 50

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.