What is the area of the small square inside of this larger square?

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Күн бұрын

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@gavindeane3670 7 күн бұрын

Not clear from the diagram whether 6mm is meant to be the whole diagonal of the big square or half the diagonal. Assuming 6mm is the blue bit, i.e. half the diagonal: The big square is two isosceles triangles, each with base 12mm (the full diagonal of the big square) and height 6mm (half the diagonal of the big square). The area of each triangle is therefore ½ × 12 × 6 so the area of the big square is double that, which is 12 × 6 = 72 mm² The small square is a quarter the area of the big square so its area is 72/4 = 18 mm²

@chamberizer 7 күн бұрын

It is not really an equilateral triangles. It has two sides that are the same not three, 45 degree verses 60 degree. But if you take the 6mm diagonal & mirror it using the vertical axis at center of large square, you end up with a triangle with two sides of 6mm. That is from the center to the two upper corners of the large square. The square area we are looking for is equivalent to area of this triangle: Area of triangle = 1/2 base × height = 1/2 ( 6 x6) = 1/2 x 36 = 18

@gavindeane3670 6 күн бұрын

@@chamberizerThanks! That was a careless error. I meant isosceles! I will correct it.

@chamberizer 6 күн бұрын

@gavindeane3670 Nobody is perfect.

@osgubben 7 күн бұрын

Pytagoras is all you need to solve this.

@gavindeane3670 7 күн бұрын

You need less than that. Just knowing that the area of a triangle is half the base × the height is enough. Oh, and recognising that the small square is one quarter the area of the big square.

@btpcmsag 6 күн бұрын

@@gavindeane3670 And the triangle with base = 12 and height = 6 has half area of the big square.

@btpcmsag 6 күн бұрын

@@gavindeane3670 And there are 4 small squares in the big square, but only 2 triangles of base = 12, height = 6 in the big square. One big ⬛ two big triangles, 4 small squares.

@Ayelmar 6 күн бұрын

Solved in my head after listening to the problem, and IF I'm reading it right after that, the answer is that the small square has an area of 18 mm^2. If the diagonal from the center of the large square to one corner is 6mm, we can use that as the hypoteneuse of a right triangle. Using the Pythagorean theorem, a^2 + b^ 3= c^2, so 6^2=36. Half of that, 18, would then be the square of the length of each of the two sides of the small square, but since we want the area of that square, we can stop at 18.

@chamberizer 7 күн бұрын

Since it is 45 degrees: Multiply 6 × .707 to get length of a side. Both sides are equal thus (6 × .707) (6 x .707) = area = 18

@btpcmsag 6 күн бұрын

Where do you get 0.707?

@btpcmsag 6 күн бұрын

The sine or cosine for 45° are the same, 0.70710678... Opposite / Hypotenuse= sin Adjacent / Hypotenuse = cos I was confused when I didn't notice my cheapo phone calculator was set to radians! Every time I restart the phone it opens with Rad, not Deg. Weird.

@panlomito 6 күн бұрын

Sides of the yellow square = a mm then Pythagoras: a² + a² = (6/2)² assuming the diagonal of the bigger square is 6mm so 2a² = 9 or a² = 9/2 mm² and THAT is also the area of the yellow square while Area = a . a = a²

@Stylux-z1p 7 күн бұрын

The diagonal bisect the 90° angle of the larger square into 2 complementary angles of 45° and also creates 2 special right angles triangles The smaller Right-Angle-Triangle on the left formed by the 1/2 diagonal = 6mm ---> Hypotenuse lets call it side c one of the side of the smaller square --> lets call it side b one part of the side of the larger square --> lets call it side a Now we have Right-Angle-Triangle with sides a - b - c and angles 90° - 45 ° - x° Using the triangle theorem , the sum of the inner angles of a triangle is equal to 180° so x° = 180° - (90° + 45°) = 45° , so its a special Right-Angle-Triangle 45°- 45° - 90° with side ratios --> x : x : x√2 side c (hypotenuse) = 6 mm --> x√2 side b (side of the smaller square) = x side a (part of the side of the larger square) = x side b = c /√2 side b = 6 /√2 --> [6/√2] * [√2/√2] = 6√2 / 2 = 3√2 side of the smaller square = 3√2 Area (smaller square) = side * side Area(smaller square) = 3√2 * 3√2 = [3√2]² = 18 mm²☑ (common error [3√2]² = 11 mm² --> 3√2 ≠ 3 + √2 --> 3√2 = 3 * √2)

@dazartingstall6680 7 күн бұрын

When you get to "side b = 6 /√2" you don't need to get rid of the irrational in the denominator, because we know that squaring it will produce 2. And we know that we'll be squaring it because we're looking for an area. So.. Area = (6 /√2)(6 /√2) =36/2 = 18 mm²

@chrisdissanayake6979 7 күн бұрын

Let each side of the big square be X mm. The diagonal of the big square is given to be (2x6)= 12mm According to Pythagorean theorem, X^2+X^2=12^2 2X^2=144 X^2 = 144/2 = 72 The area of the small square in question = 1/2 X * 1/2 X = 1/4 X^2 Substituting X^2 = 72, 1/4 X^2 = 1/4*72=18 Therefore, the area of the small square is 18 square mm.

@btpcmsag 6 күн бұрын

When you know what the diagonal of a square measures, is the area of the square always 3 times the diagonal? Because this time a diag of 6mm crosses an area of 18mm^2. And 3 × 6 = 18.

@tomtke7351 7 күн бұрын

square --> 45° angles --> sides: 1/1/sqrt(2) here 6 relates to sqrt(2) i.e large square side = 6/sqrt(2) small square side = 3/sqrt(2) area small square = (3/sqrt(2))^2 = 9/2 = 4.5 mm^2

@stevendebettencourt7651 7 күн бұрын

Let’s look at the triangle above the yellow square formed by the top edge of the yellow square, the 6mm length, and the part of the big square’s left edge above the yellow square. This forms a right triangle. Also, because the yellow square’s top right corner separates the big square’s diagonal exactly in half, this triangle is a 45-45-90 right triangle, where the two legs are of equal length. Let’s call this length s. Per the Pythagorean Theorem: x^2 + y^2 = z^2 As mentioned, in this case x = y = s and we know z = 6 mm. Substituting in: s^2 + s^2 = (6 mm)^2 2*s^2 = 36 mm^2 (note mm^2 means square millimeters, a measure of area) s^2 = 36 mm^2 / 2 = 18 mm^2 s = +/- sqrt(18 mm^2) = +/- sqrt(9 mm^2 *2) = +/- 3 * sqrt(2) mm Since we are dealing with a real geometric shape, the negative answer is discarded as nonsensical, thus s = 3 * sqrt(2) mm. This is the length of the top edge of the yellow square. And since this is a square, by definition, all its sides will have the same length s. The area of any square is: A = s^2 , where A is the area and s is the length of a side of the square. In our case, we can plug in our value for s: A = (3 * sqrt(2) mm)^2 A = 9 * 2 mm^2 A = 18 mm^2 The area of the yellow square is 18 square millimeters.

@terry_willis 6 күн бұрын

Niggle point/question here: if distance from a corner to center of square is 6mm, what is purpose of the two black hash marks across the diagonal?

@btpcmsag 6 күн бұрын

In geometry, one hash mark is used to identify a length "a" and two hash marks for length "b" (for example). I don't think there is a name for the hash mark or the segment of line they cross, it's just a convention for equating the length of one segment (or side of a polygon) with another segment. The segments do NOT have to be aligned, but these 2 are. In this case, the pair of hash marks mean that one 6mm diagonal has the same length as the other diagonal segment crossed with a hash mark.

@terry_willis 6 күн бұрын

@@btpcmsag Thanks. I did not know that.

@akmmonirulislam3961 6 күн бұрын

a² + b² = c² Here a = b as mentioned So 2a² = 36 Or a² = 18 Area = a² = 18

@jald910 6 күн бұрын

Area of a square with known diagonal is 1/2 d^2 so d of big square is 12 so area is 72. Little square is 1/4 of 72 or 18.

@btpcmsag 6 күн бұрын

So here again, a square with diagonal = 12 has an area which is a multiple of the diagonal, but it's 6 times the diagonal (72 = 6×12), not 3 times as with the 18 sq. mm square. What if the diagonal was length = 1? Would the area of the d = 1 square be 1/2 d^2, meaning 0.5 mm^2?

@dazartingstall6680 6 күн бұрын

@@btpcmsag Yes. The area of a square with a diagonal of length 1 mm is 0.5 mm².

@btpcmsag 6 күн бұрын

4:00 triangle

@darkinstinctful123 6 күн бұрын

Shorter form videos are much more helpful.

@taddoerrvandervoortjr.2943 6 күн бұрын

maybe put your units on the answer?

@dazartingstall6680 7 күн бұрын

a = (6/√2)² a = 36/2 a = 18 mm²

@btpcmsag 6 күн бұрын

18 Six is to the square root of 2, as x is to 1, when x is the side of the square. 6/(√2) : x/1 . . . So x = 6/√2 And the ⬛ area = (6/√2)^2 = 36/2 = 18 (ANSWER) CHECK: Does a base of 12, height of 6 triangle have twice this area, or 36? (1/2) 12 x 6 = 36✓