Cone Volume: Generatrix 15cm, Angle 288° - Solve It!

Nov 13, 2025 by Admin 53 views

Let's dive into the fascinating world of cones and their volumes! This article will guide you through calculating the volume of a cone, given that its generatrix (the slant height) measures 15 cm, and when you unfold its lateral surface, it forms a circular sector with a central angle of 288°. Sounds like a geometry puzzle? Absolutely! But don't worry, we'll break it down step by step.

Understanding the Cone and Its Properties

Before we jump into calculations, let's make sure we're all on the same page with the basics of a cone. A cone, in its simplest form, is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. Imagine an ice cream cone, and you've got the basic idea.

Generatrix (l): This is the slant height of the cone, the distance from any point on the edge of the circular base to the apex. In our problem, the generatrix is given as 15 cm.
Radius (r): The radius of the circular base of the cone. We'll need to find this.
Height (h): The perpendicular distance from the apex to the center of the circular base. We'll also need to determine this.
Central Angle (θ): The angle of the circular sector formed when the cone's lateral surface is unfolded. In our case, it's 288°.

Why is understanding these properties crucial? Because they are interconnected and vital for calculating the volume of the cone. The formula for the volume (V) of a cone is given by:

V = (1/3) * π * r^2 * h

So, our mission is clear: find 'r' (radius) and 'h' (height) using the given information (generatrix and central angle) and then plug those values into the formula.

Decoding the Circular Sector

The problem states that when the lateral surface of the cone is unfolded, it forms a circular sector with a central angle of 288°. This is a key piece of information! The circular sector is essentially a "slice" of a circle. The radius of this sector is equal to the generatrix (l) of the cone, which we know is 15 cm. The arc length of this sector corresponds to the circumference of the cone's base. Let's break this down further:

Relationship between arc length and circumference: The arc length (s) of the sector is a fraction of the entire circumference of a circle with radius 'l' (the generatrix). The fraction is determined by the central angle (θ) divided by 360° (a full circle). s = (θ/360°) * 2 * π * l
Arc length equals base circumference: The arc length 's' of the sector is equal to the circumference of the cone's circular base, which is 2 * π * r.

Therefore, we can equate these two expressions:

(θ/360°) * 2 * π * l = 2 * π * r

Now, we can simplify and solve for 'r' (the radius of the cone's base):

r = (θ/360°) * l

Plugging in our values (θ = 288° and l = 15 cm):

r = (288°/360°) * 15 cm = (4/5) * 15 cm = 12 cm

Excellent! We've found the radius of the cone's base: r = 12 cm.

Finding the Height of the Cone

Now that we know the radius (r) and the generatrix (l), we can use the Pythagorean theorem to find the height (h) of the cone. The height, radius, and generatrix form a right-angled triangle, with the generatrix as the hypotenuse.

l^2 = r^2 + h^2

Rearranging to solve for 'h':

h^2 = l^2 - r^2

h = √(l^2 - r^2)

Plugging in our values (l = 15 cm and r = 12 cm):

h = √(15^2 - 12^2) = √(225 - 144) = √81 = 9 cm

Great! We've found the height of the cone: h = 9 cm.

Calculating the Volume of the Cone

Finally, we have all the pieces of the puzzle! We know the radius (r = 12 cm) and the height (h = 9 cm), so we can plug these values into the formula for the volume of a cone:

V = (1/3) * π * r^2 * h

V = (1/3) * π * (12 cm)^2 * (9 cm)

V = (1/3) * π * 144 cm^2 * 9 cm

V = (1/3) * π * 1296 cm^3

V = 432 * π cm^3

So, the volume of the cone is 432π cm^3. If you want a numerical approximation, using π ≈ 3.14159:

V ≈ 432 * 3.14159 cm^3 ≈ 1357.17 cm^3

Therefore, the volume of the cone is approximately 1357.17 cubic centimeters.

Summary and Key Takeaways

Let's recap what we've done. We started with a cone with a generatrix of 15 cm and the information that its lateral surface, when unfolded, forms a circular sector with a central angle of 288°. Our goal was to find the volume of the cone. We achieved this by:

Understanding the relationship between the circular sector, the cone's radius, and the generatrix.
Calculating the radius of the cone's base using the central angle and generatrix.
Determining the height of the cone using the Pythagorean theorem.
Applying the formula for the volume of a cone to find the final answer.

This problem highlights the importance of visualizing geometric shapes and understanding their properties. It also demonstrates how seemingly different pieces of information (like the central angle of a sector) can be connected to solve a problem. Remember the key formulas and relationships:

r = (θ/360°) * l (Radius of cone's base in terms of central angle and generatrix)
h = √(l^2 - r^2) (Height of cone in terms of generatrix and radius)
V = (1/3) * π * r^2 * h (Volume of a cone)

Practice Problems

Want to test your understanding? Try these similar problems:

A cone has a generatrix of 20 cm, and its lateral surface unfolds into a sector with a central angle of 216°. What is the volume of the cone?
The volume of a cone is 500π cm^3. The radius of its base is 10 cm. What is the generatrix of the cone?

Geometry problems like these can seem daunting at first, but with a systematic approach and a solid understanding of the fundamentals, you can conquer them! Keep practicing, and you'll become a cone-volume-calculating pro in no time! Remember, geometry is all about visualizing and connecting the dots – or, in this case, connecting the angles, radii, heights, and volumes!

Advanced Tips and Tricks

For those who want to delve deeper, here are a few advanced tips and tricks that can help you solve even more complex cone-related problems.

Using Similar Triangles: In some problems, you might not be given the central angle directly. Instead, you might have information about the ratio of the arc length of the sector to the circumference of the entire circle. In such cases, remember that the ratio of the arc length to the circumference is equal to the ratio of the central angle to 360°. This allows you to find the central angle indirectly.
Working with Frustums: A frustum is the portion of a cone or pyramid that remains after its upper part has been cut off by a plane parallel to its base. Calculating the volume of a frustum involves finding the difference between the volumes of the original cone and the smaller cone that was removed. This requires careful application of similar triangles and the volume formula.
Surface Area Calculations: While we focused on volume in this article, don't forget about surface area. The total surface area of a cone is the sum of the area of the base (πr^2) and the lateral surface area (πrl). Understanding these formulas is essential for a comprehensive understanding of cones.
Spherical Geometry Connections: Believe it or not, cones can be related to spherical geometry! Consider a cone formed by rotating a line segment about an axis. The intersection of this cone with a sphere can create interesting geometric figures and relationships. Exploring these connections can lead to fascinating insights.

Conclusion

Calculating the volume of a cone given its generatrix and the central angle of its unfolded lateral surface is a classic geometry problem. By understanding the relationships between the cone's properties, applying the Pythagorean theorem, and using the volume formula, you can confidently solve these types of problems. Remember to break down the problem into smaller, manageable steps, visualize the shapes involved, and practice regularly. So, go ahead and impress your friends (or your math teacher!) with your newfound cone-calculating skills!

Keep exploring the wonderful world of geometry, and you'll discover endless fascinating concepts and challenges!