The angle of a sector of a circle with radius 42cm is 120 degree.if the sector is cut out and used to make a cone,what is the curved surface area of the cone?
Well, well, well! Looks like we have a sector going rogue and becoming a cone. That's quite the transformation! But fear not, my friend, I'm here to help with your math-y conundrum.
To find the curved surface area of the cone, we need to first calculate the slant height of the cone. Lucky for us, we know the radius of the sector, which is 42cm. However, we don't have the height of the cone directly, so we'll have to calculate it using some mathematical wizardry.
To find the height of the cone, we can use some nifty trigonometry. The angle of the sector is 120 degrees, and we know that the radius forms the hypotenuse of a right-angled triangle (with the height of the cone as the other side). So, by using the sine function, we can say:
sin(120) = height / radius
Now, solving for the height:
height = sin(120) * 42cm
Once we have the height, we can move on to finding the slant height of the cone using the Pythagorean theorem:
slant height = √(radius^2 + height^2)
slant height = √(42cm^2 + (sin(120) * 42cm)^2)
Now that we have the slant height, we can finally calculate the curved surface area of the cone using the formula:
curved surface area = π * radius * slant height
curved surface area = π * 42cm * slant height
And there you have it! Just plug in the values we calculated and you'll have your answer. Keep in mind that I'm a clown bot, not a calculator, so I'll leave the number crunching to you. Have fun!
To find the curved surface area of the cone, we need to determine the slant height of the cone first.
1. The angle of the sector is given as 120 degrees, and the radius of the circle is 42 cm.
We know that a full circle has an angle of 360 degrees. Therefore, the fraction of the circle represented by the sector is:
Fraction of circle = Angle of sector / Angle of full circle
= 120 degrees / 360 degrees
= 1/3
2. Since the sector is cut out and used to make a cone, the arc length of the sector is equal to the circumference of the base of the cone.
Arc length = Circumference of base
Let 'C' be the circumference of the base.
Fraction of circle = Arc length / Circumference of base
1/3 = Arc length / C
3. The arc length can be found using the formula:
Arc length = 2πR * (Angle of sector / Angle of full circle)
= 2π * 42 cm * (120 degrees / 360 degrees)
4. Simplify and calculate the arc length:
Arc length = 2π * 42 cm * (1/3)
= 28π cm
5. As the base of the cone is formed from the sector, the radius of the base is equal to the original radius of the circle, which is 42 cm.
The slant height (l) of the cone can be found using the Pythagorean theorem:
l^2 = r^2 + h^2
The height (h) of the cone can be calculated using the formula:
h = (Arc length * r) / circumference of the base
= (28π cm * 42 cm) / (2π * 42 cm)
= 28 cm
Now, substituting the values in the Pythagorean theorem:
l^2 = (42 cm)^2 + (28 cm)^2
l^2 = 1764 cm^2 + 784 cm^2
l^2 = 2548 cm^2
l ≈ 50.48 cm
6. Finally, we can calculate the curved surface area (CSA) of the cone using the formula:
CSA = πrl
= π * 42 cm * 50.48 cm
≈ 6685.77 cm^2
Therefore, the curved surface area of the cone is approximately 6685.77 cm^2.
To find the curved surface area of the cone, we first need to find the slant height of the cone.
The angle provided, 120 degrees, corresponds to the central angle of the sector. Since the sector is cut out and used to make a cone, the slant height of the cone will be equal to the radius of the circle.
Given that the radius of the sector is 42 cm, the slant height of the cone will also be 42 cm.
Now, we can use the formula to find the curved surface area of the cone:
Curved Surface Area = π * radius * slant height
Substituting in the known values:
Curved Surface Area = π * 42 cm * 42 cm
Curved Surface Area = 1764π cm²
Therefore, the curved surface area of the cone is 1764π cm².
So in effect the third of our circle becomes the curved surface area of the cone
( 120° = (1/3) of 360°)
area = (1/3)π 42^2 = 588π cm^2