“Go down deep enough into anything and you will find mathematics.” - Dean Schlicter
Paper cones have various uses. From cooking to piñatas and Christmas decorations, you can make a paper cone for various things. That said, there are a few mathematical rules you need to know before you can make one.
Ready to start making paper cones? Want to know more about what cones are in mathematics?
Here’s Superprof’s quick guide to conical geometry.
How to Make a Right Circular Cone
The right circular cone belongs to the pyramid family. Pyramids are defined by their base.
The right circular cone is made by rotating a right-angle triangle around one of the axes by the right angle. For those out of the loop, here’s a quick reminder.
A right circular cone has:
- A circular base.
- Sloped surfaces.
- An apex.
- An axis of rotation that passes from the summit of the cone to the centre of the circular base.
- A height defined by the distance from the centre of the circular base to the apex.
Now that you know what constitutes a right circular cone, let’s see how you can make one. You’ll need to make a base, a circular segment for the top of the cone, and the mathematical know-how to make sure it all fits together.
Keep in mind that the radius of the circular base will be equal to that of the slant height and that the length of the circle arc has to be equal to the perimeter of the circular base.
Before you can create the paper patterns, you’ll need to know the radius of the circular sector and the angle. Drawing a circle for the base shouldn’t be a problem as you can just use a compass. To draw the larger circle segment for the slant, you’ll need to calculate the slant length and the angle.
You can use Pythagoras’ theorem for this. Remember that for a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides.
The arc of the circle segment needs to be the same as the perimeter of the circular base. Once you’ve worked this out, you can start drawing out your cone. The final step is to fold the circular segment so that it can fit atop the circular base.
Here’s the formula you’ll need to work out the height of your right triangle.
A triangle ABC with C as the hypotenuse will be AB² = AC² + BC².
From there, you can calculate the volume of your cone if you want.
How to Make a Paper Cone Starting with a Circle
This method is simpler because it’s not as precise. In this case, the radius of the circle will determine the height.
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Here’s how you can do it.
Step 1: Make a Paper Circle
Determine the radius of your circle and make a mark in the centre. The rounder your circle, the better the surface of your cone will be.
Step 2: Draw Diagonal Lines from the Centre of the Circle
With a ruler, draw two segments emanating from the centre. This will give you a somewhat triangular segment on the circle. The smaller your triangle, the wider the cone will be.
Step 3: Cut Out the Triangle
Cut out the triangle with some scissors. This will be your basic shape.
Step 4: Join the Edges
Bring the two edges together with them slightly overlapping. Make sure they line up to form a base. With a bit of tape, join the edges together.
You should have equal slopes. As you can see, this is pretty easy.
This is a particularly good method for young learners. You can use this for things other than maths.
“Mathematics is the most beautiful and most powerful creation of the human spirit.” - Stefan Banach
Make a Cone with the Folding Technique
This is probably the easiest technique. You just have to draw a triangle and you’re done.
If you’re not that interested in making a cone the mathematical way, this is an easy way to do it.
Step 1: Cut Out a Triangle
Draw and cut out a triangle. Any triangle will technically work. Of course, how you draw the triangle will affect the resulting cone.
Step 2: Fold the Edges
Fold two of the edges inwards and then fold the overlapping segments outwards. You’ll end up with a cone. Make sure the folds hold your cone together.
Step 3: Adjust the Cone
You can tweak the folds to ensure that the whole thing has a more uniform shape.
Step 4 (optional): Trim the Edges
If you have any overlapping edges after folding, you can always cut off the overlap.
Step 5: Refold the Triangular Part
You should still have one part of your original triangle sticking out. You can fold and tape this part down. Get creative with it!
Uses for Paper Cones
Now you can make paper cones, you need to work out what you can do with them.
You can use these paper cones for everything from cooking to arts and crafts.
Here are some of the things your paper cones can be used for.
- Paper Christmas trees
- Pointy hat
- Paper dolls
- Cones for sweets or chips
- Birthday piñatas
- And many other things...
You just need to find the right paper for your projects.
Kids might enjoy decorating the paper before making their cones. You can find plenty of ideas for arts and crafts on sites like Pinterest and passionate users regularly post ideas for rainy-day activities. It’s a good way to keep kids entertained while they’re off school.
Find Projects on YouTube
We’ve suggested just how to make paper cones, but there are plenty of projects out there. On YouTube, you can find tonnes of different activities and ideas.
For visual learners, these are great as you can see exactly what you’re supposed to be doing and there are projects for all levels, too.
If you want to learn more about the mathematical side of things, there are also plenty of maths tutorials on geometry. Whether you want to calculate the volume of a cone, the height, or the surface area, you can learn how to with YouTube tutorials.
There are even tutorials on how to make paper cones with paper and materials suitable for cooking. You can also use cones to replace icing bags. Come Christmas, there are plenty of projects for making decorations, too.
If you'd like to learn more about maths, consider getting help from one of the many talented and experienced tutors on the Superprof website. You can find tutors specialising in maths for all levels from primary school to university. There are different ways to learn from a private tutor so make sure you choose the type of tutoring that works for you, how you like to learn, and your budget.
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