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POTW #21 was a little tricky at first, but I noticed that once students took the time to re-read it and think critically about what the question was asking, they were able to solve it.
Well, the sides of the square are 6 cm. The base of the triangle is 6 cm. The height of the semi-circle is probably 1.5 cm. Therefore, the triangles height would be around 4.5 cm. So the triangles area is: 6 * 4.5 / 2 = 13.5 cm2. Now to find the semi-circles area. I looked up the formula, and there was a bunch of symbols I don't know about, so I guessed. The area of BOTH of the semi-circles is ABOUT 12 cm2. 12 + 13.5 = 25.5 cm2. Therefore, the amount of space I have to write my message is ABOUT 25.5 cm2.
First i used the knowledge given in the question to write down everything that i knew and could see.
Pi:3.14 Triangle area: bh/2 Circle area : 3.14r2 r=3cm b=4*3=12 h=12-3=9
With this info i can fill out the formula: Triangle area: 12*9/2 = 54cm2 Circle area: 3.14*3squared =3.14*9= 28.26cm2
Now its simply a matter of adding the area of the circle and the triangle. 54cm2+28.26cm2= 82.26cm2 Therefore, you have 82.26cm2 worth of space to write a message on the card.
First, I find the area of a circle with radius 3 (which is the same as the area of two semicircles with radius 3).
I know that the formula for the area of a circle is π*r squared (where r= radius of the circle).
So, I substitute the radius of each semicircle (which is 3).
Area= (3.14 * 3squared) Area = 3.14 * 9 Area = 28. 26
Therefore, the area of the semicircles is 28.26 cm squared.
Now, focusing on the triangle. For this part, I see that the top of the triangle will align with the bottoms of the semicircles. So, the length of the top side of the triangle would be the sum of the diameters of the semicircles.
Allow the diameter to be represented by D, the radius by r and the length of the top of the triangle (which is essentially the base) by B.
D = 2r B = 2D r = 3 cm
Finding D: D = 2r = 2*3 = 6 cm
Knowing the value of D, I can now find B: D = 6 cm B = 2D = 2*6 = 12 cm
So, the base of the triangle (or the upper side) is 12 cm long.
Now, I must find the height of the triangle. I can determine this from the image given. I can see that the section of the heart featuring the semicircles is 3 cm tall (as that is the radius of the semicircles). Therefore, I can subtract this from the height of the red paper in order to find the height of the triangle.
Let H represent the height of the triangle, P represent the height of the entire paper, and r represent the radius of each semicircle/ the upper part of the heart.
P = 12cm H = ? r = 3 cm
H = P - r H = 12 - 3 H = 9 cm
Now knowing the base and height of the triangle, I can decipher its area. The formula for area of a triangle is A = BH/2 (where B is the base of the triangle, A is the area and H is the height).
A = BH/2 cm squared A = 12*9 / 2 A = 54 cm squared
Adding the area of the triangle to the area of the semicircles (in order to find the total area of the heart) :
Total area = 54 + 28.26 = 82.26 cm squared
The total surface area of the heart (the white space) is 82.26 cm squared.
Here's how I solved it: So first the first thing I did was find the area of one semicircle. The formula to find the area of a semi circle is: Area = (π (R * R) )/2 Let π equal pi (roughly 3.142) Let R equal radius.
So I first wrote what I was to do in this situation and got; (3.142(3 * 3))/2 Then following out the operation these are the results (3.142 * 9 )/2 28.278/2 14.139cm2 So the area of one semicircle is 14.139cm2, but since there is two semicircles in this question I multiplied 14.139cm2 by 2 and got 28.278cm2 for both semicircles.
Next I found the area of the triangle by doing bh/2 Let b equal base Let h equal height Due to the fact that this triangle looks like a equilateral triangle I did: (12 * 12)/2 = area of triangle In the end I got 72cm2 for the triangles. The final area of the heart is 28.278cm2 + 72cm2 So the area in which you have to write the message is 100.278cm2.
The heart is made out of 1 triangle, and a circle split in half. Therefor, the way to find the area of the heart is to add the triangle and circle.
Starting with the triangle: b x h /2 The square is 12cm by 12cm paper. Since the circle is cut right to the edge, and has the radius of 3cm, we know the space it's taking up is 3. 12 - 3 = 9cm = height The base is 12 since it takes up the entire side length. 12 x 9 = 108 / 2 = 54
For the circle: πr2 = (π x r x r) radius = 3 π x 3 x 3 = 9π or 28.2743338823
54 + 9Ï€ = 82.2743338823cm2 is the area of the heart
The 2 semi circles add up to a circle with a radius of 3 cm. Area= 9 pi cm 2 (pi r 2= 3 pi 2= 9 pi cm 2) On the other hand, to calculate the are od the triangle I used Heron's formula which essentially states that the area of a triangle = √s(s-a)(s-b)(s-c) when s equals the semi perimeter and a, b, and c are the 3 side lengths of the triangle. In this case I assumed that the triangle was equilateral as that is the only way this formula can work. s=12 x 3/2=18 √18 x (18-12) x (18-12) x (18-12) =√18 x 6 x 6 x 6 =√3888 =36√3 Therefore, the total amount of area available is (36√3 + 9 pi) cm 2.
The area of the circles together is 28.87 cm To get the area of the triangle multiply 9x12 to get 108. Then divide this by two since it's a triangle to 54 Add 28.87 + 54 which equals 82.87cm YOU have 82.87 cm to write your message
To find the area of the two semi-circles, I already knew that the radius is 3 cm, so I used the formula. A = pi*(r to the power of 2) = 3.14159...*(3 to the power of 2) = 3.14159...*9 = 28.27 cm2 Because there are two semi-circles, I didn't divided the area by two. This is because I would still need to multiply it by two again, to find the area of the two semi-circles. To find the area of the triangle, I knew that the radius of the semi-circles is 3 cm, so I subtracted 3 cm from 12, which is the height/length of the red paper. The 3 cm that were subtracted are from the top of either one of the semi-circles. I also knew that because the red paper is a square, the width is also 12 cm, so I used the formula: A = (b*h)/2 = (12*9)/2 = 54 cm2 To find the total area of the heart, I added 54 to 28.27. This resulted in 82.27, and therefore, the area of the heart is 82.27 cm2.
I need to calculate the area of the Valentine. To do so I'd need to find the area of the semi circles; (3.14*(radius*radius))/2 then multiply that by two because there are 2 congruent semi circles, then find the area of the triangle; (Base*Height)/2. Semi circles: 3.14*(3*3)= 28.26/2= 14.13 14.13*2= 28.26
Triangle: 12-3=9 3*2=6 6*9=54/2=27
27+28.26= 55.26
The total amount of space available for my greeting is 55.26 cm squared.
I doubt my answer is correct but I got 82.26 cm2 as the total area of available for the greeting. So I went on Google and searched up how to find the area of a semi-circle, (Pi*Radius^2)/2. So Pi is 3.14 and the radius is 3cm. So it's (3.14*3^2)/2 or 28.26/2. Why I didn't do the divide by 2 is because the formula I said earlier was for the area of a semi-circle, since there are 2 semi-circles, I don't divide by 2 because I'll have to multiply by 2 anyways. So the area of the 2 semi-circles combined is 28.26 cm2. Now find the area of the triangle. The paper was a 12cm by 12cm, but the circles have a radius of 3cm. So the length of the square is reducted by 3 (12-3) making it 9cm long. The rest of the paper has an area of 12cm by 9cm, 108 cm2 now. The formula for finding the area of a triangle is (base*height)/2. The base of the triangle is as long as the width of the square (12cm), and the height is as long as the distance from the base of the semi circle to the bottom of the square (9cm). So the numbers in the formula would be (12*9)/2, which is 54 cm2. Now to find the total area, add the area of the 2 semi-circles and triangle, 54+28.26 is 82.26, the total area is 82.26 cm2.
To find the area of the triangular white part of the valentines day card, first, I thought that if the radius of each circle is 3cm, then the 12 X 12 paper would now be a 9 X 12 paper ( = 108cm squared) . The way to calculate area of a triangle is B X H / 2, but since that triangle is half of the rectangle anyways, then I simply divided 108cm squared (the area of the rectangle) by 2 = 54cm squared. The area of the space to write the message on is 54cm squared.
Well, the sides of the square are 6 cm. The base of the triangle is 6 cm. The height of the semi-circle is probably 1.5 cm. Therefore, the triangles height would be around 4.5 cm.
ReplyDeleteSo the triangles area is: 6 * 4.5 / 2 = 13.5 cm2.
Now to find the semi-circles area. I looked up the formula, and there was a bunch of symbols I don't know about, so I guessed. The area of BOTH of the semi-circles is ABOUT 12 cm2.
12 + 13.5 = 25.5 cm2.
Therefore, the amount of space I have to write my message is ABOUT 25.5 cm2.
First i used the knowledge given in the question to write down everything that i knew and could see.
ReplyDeletePi:3.14
Triangle area: bh/2
Circle area : 3.14r2
r=3cm
b=4*3=12
h=12-3=9
With this info i can fill out the formula:
Triangle area: 12*9/2 = 54cm2
Circle area:
3.14*3squared
=3.14*9= 28.26cm2
Now its simply a matter of adding the area of the circle and the triangle.
54cm2+28.26cm2= 82.26cm2
Therefore, you have 82.26cm2 worth of space to write a message on the card.
First, I find the area of a circle with radius 3 (which is the same as the area of two semicircles with radius 3).
ReplyDeleteI know that the formula for the area of a circle is π*r squared (where r= radius of the circle).
So, I substitute the radius of each semicircle (which is 3).
Area= (3.14 * 3squared)
Area = 3.14 * 9
Area = 28. 26
Therefore, the area of the semicircles is 28.26 cm squared.
Now, focusing on the triangle. For this part, I see that the top of the triangle will align with the bottoms of the semicircles. So, the length of the top side of the triangle would be the sum of the diameters of the semicircles.
Allow the diameter to be represented by D, the radius by r and the length of the top of the triangle (which is essentially the base) by B.
D = 2r
B = 2D
r = 3 cm
Finding D:
D = 2r = 2*3 = 6 cm
Knowing the value of D, I can now find B:
D = 6 cm
B = 2D = 2*6 = 12 cm
So, the base of the triangle (or the upper side) is 12 cm long.
Now, I must find the height of the triangle. I can determine this from the image given. I can see that the section of the heart featuring the semicircles is 3 cm tall (as that is the radius of the semicircles). Therefore, I can subtract this from the height of the red paper in order to find the height of the triangle.
Let H represent the height of the triangle, P represent the height of the entire paper, and r represent the radius of each semicircle/ the upper part of the heart.
P = 12cm
H = ?
r = 3 cm
H = P - r
H = 12 - 3
H = 9 cm
Now knowing the base and height of the triangle, I can decipher its area. The formula for area of a triangle is A = BH/2 (where B is the base of the triangle, A is the area and H is the height).
A = BH/2 cm squared
A = 12*9 / 2
A = 54 cm squared
Adding the area of the triangle to the area of the semicircles (in order to find the total area of the heart) :
Total area = 54 + 28.26 = 82.26 cm squared
The total surface area of the heart (the white space) is 82.26 cm squared.
Here's how I solved it:
ReplyDeleteSo first the first thing I did was find the area of one semicircle.
The formula to find the area of a semi circle is:
Area =
(Ï€ (R * R) )/2
Let π equal pi (roughly 3.142)
Let R equal radius.
So I first wrote what I was to do in this situation and got;
(3.142(3 * 3))/2
Then following out the operation these are the results
(3.142 * 9 )/2
28.278/2
14.139cm2
So the area of one semicircle is 14.139cm2, but since there is two semicircles in this question I multiplied 14.139cm2 by 2 and got 28.278cm2 for both semicircles.
Next I found the area of the triangle by doing
bh/2
Let b equal base
Let h equal height
Due to the fact that this triangle looks like a equilateral triangle I did:
(12 * 12)/2 = area of triangle
In the end I got 72cm2 for the triangles.
The final area of the heart is 28.278cm2 + 72cm2
So the area in which you have to write the message is 100.278cm2.
The heart is made out of 1 triangle, and a circle split in half. Therefor, the way to find the area of the heart is to add the triangle and circle.
ReplyDeleteStarting with the triangle: b x h /2
The square is 12cm by 12cm paper. Since the circle is cut right to the edge, and has the radius of 3cm, we know the space it's taking up is 3.
12 - 3 = 9cm = height
The base is 12 since it takes up the entire side length.
12 x 9 = 108 / 2 = 54
For the circle: πr2 = (π x r x r)
radius = 3
Ï€ x 3 x 3 = 9Ï€ or 28.2743338823
54 + 9Ï€ = 82.2743338823cm2 is the area of the heart
The 2 semi circles add up to a circle with a radius of 3 cm.
ReplyDeleteArea= 9 pi cm 2 (pi r 2= 3 pi 2= 9 pi cm 2)
On the other hand, to calculate the are od the triangle I used Heron's formula which essentially states that the area of a triangle = √s(s-a)(s-b)(s-c) when s equals the semi perimeter and a, b, and c are the 3 side lengths of the triangle.
In this case I assumed that the triangle was equilateral as that is the only way this formula can work.
s=12 x 3/2=18
√18 x (18-12) x (18-12) x (18-12)
=√18 x 6 x 6 x 6
=√3888
=36√3
Therefore, the total amount of area available is (36√3 + 9 pi) cm 2.
The area of the circles together is 28.87 cm
ReplyDeleteTo get the area of the triangle multiply 9x12 to get 108. Then divide this by two since it's a triangle to 54
Add 28.87 + 54 which equals 82.87cm
YOU have 82.87 cm to write your message
To find the area of the two semi-circles, I already knew that the radius is 3 cm, so I used the formula.
ReplyDeleteA = pi*(r to the power of 2)
= 3.14159...*(3 to the power of 2)
= 3.14159...*9
= 28.27 cm2
Because there are two semi-circles, I didn't divided the area by two. This is because I would still need to multiply it by two again, to find the area of the two semi-circles. To find the area of the triangle, I knew that the radius of the semi-circles is 3 cm, so I subtracted 3 cm from 12, which is the height/length of the red paper. The 3 cm that were subtracted are from the top of either one of the semi-circles. I also knew that because the red paper is a square, the width is also 12 cm, so I used the formula:
A = (b*h)/2
= (12*9)/2
= 54 cm2
To find the total area of the heart, I added 54 to 28.27. This resulted in 82.27, and therefore, the area of the heart is 82.27 cm2.
I need to calculate the area of the Valentine. To do so I'd need to find the area of the semi circles; (3.14*(radius*radius))/2 then multiply that by two because there are 2 congruent semi circles, then find the area of the triangle; (Base*Height)/2.
ReplyDeleteSemi circles:
3.14*(3*3)= 28.26/2= 14.13
14.13*2= 28.26
Triangle:
12-3=9
3*2=6
6*9=54/2=27
27+28.26= 55.26
The total amount of space available for my greeting is 55.26 cm squared.
I doubt my answer is correct but I got 82.26 cm2 as the total area of available for the greeting.
ReplyDeleteSo I went on Google and searched up how to find the area of a semi-circle, (Pi*Radius^2)/2. So Pi is 3.14 and the radius is 3cm. So it's (3.14*3^2)/2 or 28.26/2. Why I didn't do the divide by 2 is because the formula I said earlier was for the area of a semi-circle, since there are 2 semi-circles, I don't divide by 2 because I'll have to multiply by 2 anyways. So the area of the 2 semi-circles combined is 28.26 cm2.
Now find the area of the triangle. The paper was a 12cm by 12cm, but the circles have a radius of 3cm. So the length of the square is reducted by 3 (12-3) making it 9cm long. The rest of the paper has an area of 12cm by 9cm, 108 cm2 now. The formula for finding the area of a triangle is (base*height)/2. The base of the triangle is as long as the width of the square (12cm), and the height is as long as the distance from the base of the semi circle to the bottom of the square (9cm). So the numbers in the formula would be (12*9)/2, which is 54 cm2.
Now to find the total area, add the area of the 2 semi-circles and triangle, 54+28.26 is 82.26, the total area is 82.26 cm2.
To find the area of the triangular white part of the valentines day card, first, I thought that if the radius of each circle is 3cm, then the 12 X 12 paper would now be a 9 X 12 paper ( = 108cm squared) . The way to calculate area of a triangle is B X H / 2, but since that triangle is half of the rectangle anyways, then I simply divided 108cm squared (the area of the rectangle) by 2 = 54cm squared.
ReplyDeleteThe area of the space to write the message on is 54cm squared.
First I found out the area of the triangle.
ReplyDelete12-3=9
12x9/2=54
Then I found out the Area of the 2 semi circles.
Area of a circle is equal to pi x radius squared.
pi*3squared
pi*(3*3)
pi*9
3.1415*9=28.2735
54+28.2735=82.2735