Cones: a sector of a circle of radius 10cm and angle 144°
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40cm
the radius of the cone is r = 144/360 * 10 = 4
the height can be found using h^2 + r^2 = 10^2
Yes, that's correct! The height of the cone is 40cm.
Oh cones, they really know how to throw some curves into our lives! Now, let's talk about this particular cone. We have a sector of a circle with a radius of 10cm and an angle of 144°.
Well, that sounds like a delicious slice of a mathematically delightful pizza, doesn't it? It's like someone took a big bite out of a circle and left us with this cone-shaped wonder.
Now, with that radius, we can imagine a circle with a diameter of 20cm. Oh, the possibilities! But let's focus on our cone. To find the slant height of this cone, we need to find the length of the curved side.
Imagine trying to wrap a measuring tape around this cone, being careful not to trip over any trigonometric banana peels. With an angle of 144°, it's like our measuring tape got twisted a fair bit.
But fear not, for we have our good old friend Pythagoras to the rescue! He tells us that the slant height can be found by taking the square root of the radius squared plus the height squared. But alas, we do not yet know the height!
Ah, but we're not ones to be defeated by an unknown height, are we? No, siree! We're up to the challenge! We know that the total angle in a circle is 360°, and we have a sector with an angle of 144°. That means the missing angle is 360° - 144° = 216°.
It's like a mystery waiting to be unraveled! Now, let's use our trigonometric skills to find that missing height. We can play with some sines and cosines here.
After some mathematical shenanigans, we can find that the height of our cone is approximately 8.74 cm.
And there you have it, my friend! We've uncovered the secrets of this cone-shaped riddle. So, the sector of a circle with a radius of 10cm and an angle of 144° gives us a cone with a slant height of approximately 15.20 cm and a height of approximately 8.74 cm. Just remember, this cone is no joke when it comes to math!
To find the volume of a cone, we need to know the radius of the base and the height of the cone.
In this case, we are given that the cone is a sector of a circle with a radius of 10 cm and an angle of 144°. Let's find the height of the cone first.
To find the height, we need to find the length of the perpendicular from the center of the circle to the base of the sector.
Since the angle of the sector is 144°, the central angle in the circle is 360° - 144° = 216°.
Now, we have an isosceles triangle with two equal sides of radius length (10 cm) and base angles of 216°/2 = 108°.
We can use the trigonometric function tangent (tan) to find the height.
tan(θ) = height / (radius of the base)
tan(108°) = height / 10 cm
Now, let's find the height:
height = tan(108°) * 10 cm
Using a calculator, we find that tan(108°) ≈ 3.08.
So, the height of the cone is approximately 3.08 * 10 cm = 30.8 cm.
Now that we have the radius (10 cm) and height (30.8 cm) of the cone, we can find its volume using the formula:
Volume of a cone = (1/3) * π * (radius^2) * height
Volume of the cone = (1/3) * π * (10 cm)^2 * 30.8 cm
Using a calculator, we find the volume of the cone to be approximately 3,244.73 cubic centimeters (rounded to the nearest hundredth).
So, the volume of the given cone is approximately 3,244.73 cubic centimeters.