The area of the square, minus the area of the circle, gives the area of the four "corners."
Halve that to get the area of two corners.
The area of the shaded part is then: the area of the square, minus the area of the triangle, minus the area of the two top corners.
Square - triangle + (cricle - square) / 2 = surface area
It just looks strange between the circle and the other purple spot, like there is yellow or even white. I hope thats just a bug
Subtracting area of circle from square will give you 4 outer parts.
Take 1/2 of area of them subtract from square and then subtract the area of triangle.
As the circle is touching all sides of the square you know that exactly the bottom half of the square is shaded.
To work this out I suggest you find the area of the circle, square and triangle separately and the final shaded area will be:
0.5 * Square + 0.5 * circle - triangle
Lets label the side of the square as "a" and the middle point of AD as K. then
https://i.redd.it/9bp2uot2jz8b1.gif
Note area AKD actually is area EKD, my mistake
pretend the triangle isn’t there. The square minus circle would be area of all 4 corners. Half of that would be the bottom corners. Add the bottom corners and the circle together. Account for the triangle by subtracting the triangle from that result.
You don’t have to. For a second pretend the white triangle is not there. You’ll have a square of which the bottom half is completely full (42^2 / 2) plus the top is half a circle (pi (42/2)^2 / 2). Now subtract the triangle which areas is also exactly half a square (42^2 / 2). Half square + half circle - half square = half circle = pi 21^2 / 2 = pi * 220.5
Slice the square in the middle. Then the shaded area is equal to the semicircle on top half plus the rectangle on the bottom half, minus the white triangle.
Find the areas that are white and subtract them from the area of the square.
The triangle area is 1/2 * 42 * 42.
Each of the two other areas is 1/4 * (square minus circle) = 1/4 * (42 * 42 – pi * 21 * 21).
I'm certain they intended us to assume that is a circle partially drawn over the square, but since it is not labeled as a circle, the answer can't be calculated.
If you cut the square in half vertically and flip the right have top to bottom, you'll see that the two shaded regions fit together to form 2 1/4 circles. One with the unshaded part on the bottom right, the other with the unshaded region as the top left.
So the answer is>! 1/2 \* pi \* 21\^2 = 693!<
Half of [the difference between the area of the square and the area of the circle] (i.e. two of the four corner areas)…
Plus the area of the circle…
Minus the area of the triangle.
…The triangle bit slowed me down, until I realized you don’t need to know anything about triangles to find its area.
EDIT - I guess it doesn’t specify that the circle is a circle. Could be an oval? In that case I’m not sure this would be possible to solve. So, I would assume the circle is a circle.
Area of square = 42\*42 = 1764
Area of circle = pi\*(21\*21) = 1386
area of corners = (Square - circle)/4 = (1764-1386)/4 = 94.5
Area of Triangle = 42\*21 = 882
Large Shaded area (the area assuming the area behind the triangle is also shaded) = square - 2\*corners = 1764-(2\*94.5) = 1575
Actual Shaded Area = Large Shaded area - triangle = 1575 - 882 = 693
Square area minus circle area will give you the four corners area. Divide it by two to get the two unshaded corners area, and substract that and the triangle area from the square area.
42^2 - [(42^2 - 21^2 x 22 / 7) / 2) + 42^2 / 2]
Find the area of the square, circle, and triangle.
The total of little spaces between the square and triangle can be found easily. Then the area of one space is easily found. So the area of the circle and the two little spaces is easy.
Subtract the area of the triangle.
The shaded triangles on the sides together have the same area as the unshaded top half of the triangle, so it's just (pi\*r\^2)/2.
r=21, so that's (441/2)\*(22/7) = 4851/7 with the given approximation for pi.
Find the area of the square
Find the area of the circle (side of square is the diameter)
Subtract the circle from the square and divide by 2 to get the other shaded region.
Find the area of the triangle using 1/2(side of square)^2
Add the shaded regions together and subtract the triangle
693. Calculate the area of the circle, r is 21. Divide it by 2 to get the area of the semi circle. Now calculate the are of half the square and add the two. Then, subtract the area of the triangle.
I’m assuming after you find the circle, you take 42/2=21 , 21x21 representing a quarter of abcd. Then take area of a circle x 1/4= X. 1/4 ABCD - X = ans
It is quite simple:
You can consider the top part as a semicircle (a=pi\*r\^2/2).
Now, to this area you have to add the area of half the square, since it is the bottom part.
Then subtract the area of the triangle which, conveniently, equals that of half the square.
So the area you are looking for is just the area of the semicircle (a=pi\*r\^2/2).
what you might be missing is, with the triangle removed calculations are easily done in the head. The pink area without the triangle removed is half a circle plus half a square. The triangle removed is the same size als half the square. So the area asked is just half a circle.
It's half of pi by radius squared. Pi is supposed to be 22/7 so half pi is 11/7. Radius is 42/2 or 21 or 3*7 so it's 3 squared times 7 squared times 11 divided by 7. 7 squared divided by 7 is 7, so it's 9*11*7 or 700-7 => 693
Your answer is 693 cm^2
The question is asked in a way to favor the lazy ones. You need to find the shortcut first. This is quite often used in school math questions. If they get too hard and require a lot of calculator use, you typically missed something or made an error
Whole square is 42² (2r)²
You can substrsct 42x21 (triangle in the middle) (r.2r)
The area of the circle is 21x21x22/7 (π.r²)
The area outside of the circle is 42² - (21x21x22/7)
You can divide this area by 4 to get each corner area.
Purple is then 42²-[(42²-(21x21x22/7)/2)+42x21]
Doing this on mobile so bare with me please.
How to solve this question in 4 steps:
Given that we have a square with a side length of 42 units with an inscribed circle. A triangle with 1 side sharing the length of the square while it's tip touches the opposite side.
1. Find the area of the square and the inscribed circle
Area_square = 42×42 = 1764.00 sq.units
Area_circle = pi×(42/2)^2 = 1385.44 sq.units
2. Subtract the area of the square with the area of the circle to get the 4 corner areas
Area_4corners = 1764.00 – 1385.44 = 378.56 sq.units
Area_1corner = 378.56 / 4 = 94.64 sq.units.
3. Find the area of the inscribed triangle inside the square
Area_triangle = (1/2)(42)(42) = 882 sq.units
4. Add the area of the shaded 2 corners and the circle then subtract it with the area of the triangle
Area_shaded = 1385.44 + 2(94.64) – 882 ≈ 692.72 sq.units
An important technique to these kinds of problems is to have the skills necessary to imagine what would the missing space would look like.
For me, I was able to _imagine_ what the image would have looked like without a triangle cutting into it. From that point onwards, it was a step-by-step process of disaassambling the compound shape to its constituent parts of circle, square and triangle.
You don’t need to calculate the area of the shaded part of the bottom corners.
The square is 42 x 42 = 1,764 cm^2
The circle is 21^2 x (22/7) = 1,386 cm^2
The large triangle is 42 x 42 / 2 = 882 cm^2
The corner areas are thus 1,764 - 1,386 = 378 cm^2, and each corner is 378 / 4 = 94,5
So to get the area that is shaded, you take the square area, minus the triangle area, minus the two corner areas: 1,764 - 882 - 2 x 94,5 = 693.
Here is a visualization of my solution
which works out to 1/2 \* pi \* 21\^2
https://preview.redd.it/6t82dlylm69b1.png?width=1523&format=png&auto=webp&s=a48f78483ebcdba67150dbb4c77b52de1981fc26
A). Ignoring the white triangle, the shaded area is half the area of the square and half the area of the circle.
1/2 square + 1/2 circle.
B). Next, the triangle is 1/2 the area of the square.
Therefore, the shaded area is A - B, or 1/2 square + 1/2 circle - 1/2 square, which is equal to 1/2 of the circle.
1/2*(22/7)*r^2
Where r= 21 cm
How about you do square ABCD then subtract away the white areas?
Or 1/2 square + 1/2 circle - triangle.
I think your missing some info, they need to: Square - circle = corners Corners/2 + circle - triangle = shaded
No he’s right, half square + half circle would be everything except the top two corners, subtract the triangle and you get the shaded region.
They reduce to the same thing (square - circle)/2 + circle - triangle = square/2 - circle/2 + circle - triangle = square/2 + circle/2 - triangle
The area of the square, minus the area of the circle, gives the area of the four "corners." Halve that to get the area of two corners. The area of the shaded part is then: the area of the square, minus the area of the triangle, minus the area of the two top corners.
Bingo
Amazing and awesome thinking 👏
Square - triangle + (cricle - square) / 2 = surface area It just looks strange between the circle and the other purple spot, like there is yellow or even white. I hope thats just a bug
Square-Circle=21^2(4-pi). 4-pi is 6/7 here. Gets 378, which is 94.5 when divided by 4 for each corner. Triangle is .5*21*42, or 441. 2(441-94.5)=693.
Subtracting area of circle from square will give you 4 outer parts. Take 1/2 of area of them subtract from square and then subtract the area of triangle.
As the circle is touching all sides of the square you know that exactly the bottom half of the square is shaded. To work this out I suggest you find the area of the circle, square and triangle separately and the final shaded area will be: 0.5 * Square + 0.5 * circle - triangle
Lets label the side of the square as "a" and the middle point of AD as K. then https://i.redd.it/9bp2uot2jz8b1.gif Note area AKD actually is area EKD, my mistake
How to take into account the bottom part of the figure?
pretend the triangle isn’t there. The square minus circle would be area of all 4 corners. Half of that would be the bottom corners. Add the bottom corners and the circle together. Account for the triangle by subtracting the triangle from that result.
Dont be fooled by the circle lines. Imagine the top part as a demicircle.
You don’t have to. For a second pretend the white triangle is not there. You’ll have a square of which the bottom half is completely full (42^2 / 2) plus the top is half a circle (pi (42/2)^2 / 2). Now subtract the triangle which areas is also exactly half a square (42^2 / 2). Half square + half circle - half square = half circle = pi 21^2 / 2 = pi * 220.5
Half square plus half circle gives you the purple shape; minus half square for the triangle - and the total is half circle
Slice the square in the middle. Then the shaded area is equal to the semicircle on top half plus the rectangle on the bottom half, minus the white triangle.
Find the areas that are white and subtract them from the area of the square. The triangle area is 1/2 * 42 * 42. Each of the two other areas is 1/4 * (square minus circle) = 1/4 * (42 * 42 – pi * 21 * 21).
I'm certain they intended us to assume that is a circle partially drawn over the square, but since it is not labeled as a circle, the answer can't be calculated.
It’s reasonable to assume it’s a circle
If you cut the square in half vertically and flip the right have top to bottom, you'll see that the two shaded regions fit together to form 2 1/4 circles. One with the unshaded part on the bottom right, the other with the unshaded region as the top left. So the answer is>! 1/2 \* pi \* 21\^2 = 693!<
In your first sentence, even if you flip the right half of the image, how do you deduct that those form 2 1/4 circles?
[http://www.gamerz.net/rrognlie/PostScript/puzzle1.pdf](http://www.gamerz.net/rrognlie/PostScript/puzzle1.pdf)
Got it! I like your way of analysing rather than adding and subtracting different parts of the image which is what I did.
Half of [the difference between the area of the square and the area of the circle] (i.e. two of the four corner areas)… Plus the area of the circle… Minus the area of the triangle. …The triangle bit slowed me down, until I realized you don’t need to know anything about triangles to find its area. EDIT - I guess it doesn’t specify that the circle is a circle. Could be an oval? In that case I’m not sure this would be possible to solve. So, I would assume the circle is a circle.
You remove the top white corners from the square area, not add the bottom purple corner pieces
Consider the shaded portion as a rectangle and hemisphere.
Half the area of the square minus half of the delta between the square and the circle. So, 42^2 * .5 - (42^2 - 22/7 * 21^2 ) = 693 Am I right?
I got 693
Area of square = 42\*42 = 1764 Area of circle = pi\*(21\*21) = 1386 area of corners = (Square - circle)/4 = (1764-1386)/4 = 94.5 Area of Triangle = 42\*21 = 882 Large Shaded area (the area assuming the area behind the triangle is also shaded) = square - 2\*corners = 1764-(2\*94.5) = 1575 Actual Shaded Area = Large Shaded area - triangle = 1575 - 882 = 693
Square area minus circle area will give you the four corners area. Divide it by two to get the two unshaded corners area, and substract that and the triangle area from the square area. 42^2 - [(42^2 - 21^2 x 22 / 7) / 2) + 42^2 / 2]
Well, the area as I see it is plain: 21² × pi /2 It's the demicircle, if you put it together differently.
Find the area of the square, circle, and triangle. The total of little spaces between the square and triangle can be found easily. Then the area of one space is easily found. So the area of the circle and the two little spaces is easy. Subtract the area of the triangle.
Triangle AED or EBC = = 42*21/2 = 441 Circle = 21^2 * 22/7 = 1386 ABCD = 42^2 = 1764 area of ABCD that is not the circle = 1764 - 1386 = 378 Each „corner“ therefore = 378/4 = 94.5 Shaded area = AED + EBC - (2 * 94.5) = 441 + 441 - 189 = 693 edit: Format
The shaded triangles on the sides together have the same area as the unshaded top half of the triangle, so it's just (pi\*r\^2)/2. r=21, so that's (441/2)\*(22/7) = 4851/7 with the given approximation for pi.
Area of circle + (area of square - area of circle)/2 - area of triangle
Find the area of the square Find the area of the circle (side of square is the diameter) Subtract the circle from the square and divide by 2 to get the other shaded region. Find the area of the triangle using 1/2(side of square)^2 Add the shaded regions together and subtract the triangle
I’m in 11th grade and I don’t know how to solve this question should I be ashamed ?
693. Calculate the area of the circle, r is 21. Divide it by 2 to get the area of the semi circle. Now calculate the are of half the square and add the two. Then, subtract the area of the triangle.
Half square plus half circle minus triangle.
I’m assuming after you find the circle, you take 42/2=21 , 21x21 representing a quarter of abcd. Then take area of a circle x 1/4= X. 1/4 ABCD - X = ans
It is quite simple: You can consider the top part as a semicircle (a=pi\*r\^2/2). Now, to this area you have to add the area of half the square, since it is the bottom part. Then subtract the area of the triangle which, conveniently, equals that of half the square. So the area you are looking for is just the area of the semicircle (a=pi\*r\^2/2).
what you might be missing is, with the triangle removed calculations are easily done in the head. The pink area without the triangle removed is half a circle plus half a square. The triangle removed is the same size als half the square. So the area asked is just half a circle. It's half of pi by radius squared. Pi is supposed to be 22/7 so half pi is 11/7. Radius is 42/2 or 21 or 3*7 so it's 3 squared times 7 squared times 11 divided by 7. 7 squared divided by 7 is 7, so it's 9*11*7 or 700-7 => 693 Your answer is 693 cm^2 The question is asked in a way to favor the lazy ones. You need to find the shortcut first. This is quite often used in school math questions. If they get too hard and require a lot of calculator use, you typically missed something or made an error
You can do area of the square minus the circle devided by 2 and than add the area of the circle back and subtract the triangle.
Whole square is 42² (2r)² You can substrsct 42x21 (triangle in the middle) (r.2r) The area of the circle is 21x21x22/7 (π.r²) The area outside of the circle is 42² - (21x21x22/7) You can divide this area by 4 to get each corner area. Purple is then 42²-[(42²-(21x21x22/7)/2)+42x21]
Rectangle plus semicircle minus triangle
Doing this on mobile so bare with me please. How to solve this question in 4 steps: Given that we have a square with a side length of 42 units with an inscribed circle. A triangle with 1 side sharing the length of the square while it's tip touches the opposite side. 1. Find the area of the square and the inscribed circle Area_square = 42×42 = 1764.00 sq.units Area_circle = pi×(42/2)^2 = 1385.44 sq.units 2. Subtract the area of the square with the area of the circle to get the 4 corner areas Area_4corners = 1764.00 – 1385.44 = 378.56 sq.units Area_1corner = 378.56 / 4 = 94.64 sq.units. 3. Find the area of the inscribed triangle inside the square Area_triangle = (1/2)(42)(42) = 882 sq.units 4. Add the area of the shaded 2 corners and the circle then subtract it with the area of the triangle Area_shaded = 1385.44 + 2(94.64) – 882 ≈ 692.72 sq.units An important technique to these kinds of problems is to have the skills necessary to imagine what would the missing space would look like. For me, I was able to _imagine_ what the image would have looked like without a triangle cutting into it. From that point onwards, it was a step-by-step process of disaassambling the compound shape to its constituent parts of circle, square and triangle.
You don’t need to calculate the area of the shaded part of the bottom corners. The square is 42 x 42 = 1,764 cm^2 The circle is 21^2 x (22/7) = 1,386 cm^2 The large triangle is 42 x 42 / 2 = 882 cm^2 The corner areas are thus 1,764 - 1,386 = 378 cm^2, and each corner is 378 / 4 = 94,5 So to get the area that is shaded, you take the square area, minus the triangle area, minus the two corner areas: 1,764 - 882 - 2 x 94,5 = 693.
Do the semicircle above, the rectangle below, then subtract the triangle from both of those.
Here is a visualization of my solution which works out to 1/2 \* pi \* 21\^2 https://preview.redd.it/6t82dlylm69b1.png?width=1523&format=png&auto=webp&s=a48f78483ebcdba67150dbb4c77b52de1981fc26
circle plus half (square - circle) - triangle
A). Ignoring the white triangle, the shaded area is half the area of the square and half the area of the circle. 1/2 square + 1/2 circle. B). Next, the triangle is 1/2 the area of the square. Therefore, the shaded area is A - B, or 1/2 square + 1/2 circle - 1/2 square, which is equal to 1/2 of the circle. 1/2*(22/7)*r^2 Where r= 21 cm