Central Angle Calculation: Unfolding A Cone's Surface

Hey guys! Let's dive into a cool math problem today where we'll figure out how to calculate the central angle of a disc sector. This sector is formed when we unfold the lateral surface of a right circular cone. We've got a cone with a radius of 5 cm and a generator (that's the slant height) of 18 cm. Sounds like fun, right? This problem combines geometry and a bit of spatial reasoning, so let's break it down step by step.

Understanding the Problem

Before we jump into calculations, let's make sure we understand what's going on. Imagine you have a cone, like an ice cream cone. Now, if you carefully cut along the side and flatten it out, you'll get a sector of a circle. This sector is what we call the lateral surface of the cone. The central angle of this sector is what we're trying to find. Think of it like this: the curved surface of the cone, when flattened, forms a pizza slice shape. We want to know the angle of that slice.

Keywords to keep in mind: central angle, disc sector, right circular cone, lateral surface, generator, and radius. These terms are crucial for understanding the problem and searching for relevant formulas or concepts. We'll be using these keywords throughout the explanation to make sure everything is crystal clear.

The key here is to connect the dimensions of the cone to the dimensions of the sector. The radius of the original cone becomes related to the arc length of the sector, and the generator of the cone becomes the radius of the sector. This is a crucial concept to grasp, so let's reiterate: the generator of the cone (18 cm in our case) becomes the radius of the sector when the cone is unfolded. The circumference of the base of the cone (which depends on the cone's radius, 5 cm) becomes the arc length of the sector. This connection is what allows us to solve for the central angle. So, understanding this relationship between the cone's properties and the resulting sector's properties is the cornerstone of the solution. We'll build upon this understanding as we move forward.

Setting Up the Formula

The trick to solving this problem lies in understanding the relationship between the cone's dimensions and the sector's dimensions. The arc length of the sector is equal to the circumference of the base of the cone. Why? Because when you fold the sector back into a cone, that arc forms the circular base. The formula for the circumference of a circle is 2πr, where r is the radius. In our case, the cone's radius is 5 cm, so the circumference (and thus the arc length) is 2π * 5 = 10π cm.

Now, let's think about the sector itself. The radius of the sector is the same as the generator of the cone, which is 18 cm. We're looking for the central angle, let's call it θ (theta). The formula that connects arc length (s), radius (R), and the central angle (in radians) is: s = Rθ. Remember, this formula works when the angle is in radians, so if we want the angle in degrees later, we'll need to convert it. It's like speaking a different language – we need to make sure we're using the right units to communicate effectively in math!

In our problem:

• s (arc length) = 10π cm
• R (radius of the sector) = 18 cm
• θ (central angle) = ? (This is what we want to find!)

So, we have all the pieces of the puzzle. The next step is to plug these values into our formula and solve for θ. Think of it like fitting the pieces together in a jigsaw puzzle – each piece (value) has its place, and once they're all connected, we'll see the complete picture (the central angle). We're essentially using the formula as our roadmap, guiding us from the known values to the unknown one. So, let's get ready to substitute and simplify!

Solving for the Central Angle

Okay, guys, let's put those numbers into the formula we talked about: s = Rθ. We know s = 10π cm and R = 18 cm. So, we get:

10π = 18θ

Now, we need to isolate θ. To do that, we'll divide both sides of the equation by 18:

θ = (10π) / 18

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

θ = (5π) / 9 radians

Awesome! We've found the central angle in radians. But sometimes, angles are easier to understand in degrees. So, let's convert radians to degrees. The conversion factor is:

1 radian = (180/π) degrees

To convert (5π)/9 radians to degrees, we multiply:

θ (in degrees) = [(5π) / 9] * (180 / π)

Notice that π cancels out, which makes our calculation simpler:

θ (in degrees) = (5 * 180) / 9

θ (in degrees) = 900 / 9

θ (in degrees) = 100 degrees

There we have it! The central angle of the sector is 100 degrees. It's like we've navigated through a maze of formulas and calculations and finally reached our destination – the answer! Remember, the key was understanding the relationship between the cone and the sector, applying the right formulas, and carefully doing the math. So, give yourselves a pat on the back for making it this far!

Final Answer and Conclusion

So, to wrap things up, we've calculated the central angle of the disc sector formed by unfolding the lateral surface of our right circular cone. The cone had a radius of 5 cm and a generator of 18 cm. We found that the central angle is 100 degrees. That's our final answer!

Let's quickly recap the steps we took:

1. Understood the problem: We visualized unfolding the cone and identified the key relationships between the cone's dimensions and the sector's dimensions.
2. Set up the formula: We used the formula s = Rθ, where s is the arc length, R is the radius of the sector, and θ is the central angle in radians.
3. Solved for the central angle in radians: We plugged in the values and found θ = (5π) / 9 radians.
4. Converted to degrees: We converted the angle from radians to degrees, getting θ = 100 degrees.

This problem is a great example of how geometry connects different shapes and concepts. By understanding the relationships between the cone and its unfolded lateral surface, we could apply the right formulas and find the solution. Remember, math is like a puzzle – each piece fits together to create a beautiful picture. So, keep practicing, keep exploring, and you'll become a math whiz in no time! You guys rock!