You can easily roll up a triangle or semicircle to create a cone and if you start with a larger piece of material you can adjust the height and width of the cone by hand. If you need to make a cone of a precise shape, there are online calculators or mathematical formulas you can use to determine the size of the shape you need: a circle with a cut out segment.
Steps
Method 1 of 3: Create a Paper Cone Using a Semicircle
Step 1. Draw a semicircle on the card
Place a sheet of paper or card on a flat surface if you want the cone to be stronger. Place the tip of a compass on the edge of the paper, then use the pencil to draw a semicircle. The width of the cone will be double the distance between the two points of the compass.
- If you don't have a compass, use another method, such as tracing a cup.
- Set the compass distance at 23-25cm for a medium-sized hat.
- To obtain a cone of width "l", create a semicircle of diameter "l" x π.
Step 2. Cut out the semicircle
Use scissors or a utility knife to cut out the semicircle from the paper.
Step 3. Roll the paper into a cone shape
Lift the two corners of the semicircle and join them. Pull them slightly over each other so that the paper overlaps, creating a closed cone shape.
Step 4. Use glue or tape to secure
Apply the adhesive along the edge where the paper overlaps, then press the two flaps together. You may have to hold the paper for a minute or two for the glue to set. Alternatively, you can use tape on the inside and outside of the cone.
Method 2 of 3: Create a Cone Using a Paper Triangle
Step 1. Cut out a rectangular or square piece of paper or cardboard
You can start with a rectangle, but with a square you can create a predictable cone shape, not too squashed or thin. Use a ruler to measure a square on the paper, then cut it out. If you don't have a ruler, you can fold a corner of the paper over itself to make a square, then draw a line where you will need to trim the excess paper.
- Do not create a mark when you fold the paper.
- If you want a cone with width "l", create a square with side "l" / 0.45, or slightly longer (this calculation is based on the Pythagorean theorem and the formula for the circumference of the circle).
Step 2. Cut the paper in half diagonally
Cut the paper along the diagonal of the square with scissors or a utility knife. The diagonal of the square will become the base of the cone.
Step 3. Apply tape to one side of the cone
Lift one corner of the triangle, adjacent to the longer side, and bring it to the corner between the two short sides to form a cone. Use glue, tape, or staples to hold it in place.
You can adjust the "tapering" of the cone by moving the angle of the triangle to a different point instead of aligning it with an angle
Step 4. Close the cone
Roll the cone over the remaining paper to complete it. Use tape or glue to pin the edges where they meet.
Method 3 of 3: Create a Cone of Exact Proportions
Step 1. Use an online calculator if you want to create a funnel
If you need a template for a cone-shaped funnel, with openings on both sides, an online calculator will save you time and reduce the likelihood of an expensive math mistake. Enter your desired aspect ratio on i-logic.com or craig-russel.co.uk to find the shape and size you need. If you want to create a complete cone (with an opening and a tip), with the following steps you can calculate the measurements yourself.
- If you don't care about explanations, here are the complete formulas for a cone:
- L = √ (h 2 + r 2), where h is the height of the cone (with the tip) and r is the radius of its opening.
- a = 360 - 360 (r / L)
- You can create a cone from a circle of radius "L", after having cut out and discarded a segment with angle "a".
Step 2. Create the shape you need
To create a cone with precise proportions, you will need to use a circle of a specific radius, after removing a "slice" of a specific angle. To create a funnel instead, you will need to cut out a second circle from the first, to create the smaller opening.
- This guide describes the cone as if it were standing on the larger base, with the tip up.
- You can cut out "slices" of more than half the circle to make very narrow cones.
Step 3. Calculate the apothem of the cone
Imagine the complete cone (ignore the openings above for now). The apothem runs from tip to base and is the hypotenuse of a right triangle. The other two sides of the triangle are the height of the cone ("h") and the radius of the lower opening ("r"). We can use the Pythagorean Theorem to calculate the apothem ("L") based on the desired cone size:
- L 2 = h 2 + r 2 (Remember, use the radius, not the diameter!)
- L = √ (h 2 + r 2).
- As an example, a cone of height 12 and radius 3 has an apothem of √ (122 + 32) = √ (144 + 9) = √ (153) = approximately 12, 37.
Step 4. Draw a circle with the apothem as a radius
Imagine cutting and unfolding the finished cone to roll it out. You would get a circle with a radius equal to the "L" apothem just calculated. Once you have found the radius, continue to the next step to calculate the "slice" of the circle to be cut.
Step 5. Calculate the base circumference
This measurement is the length of the perimeter of the base of the cone (the largest opening). You can calculate it based on the desired radius of the opening ("r"), using the formula for the circumference ("C") of the circle:
- C (base of the cone) = 2 π r
- In our example, a cone of radius 3 has a circumference of 2 π (3) = 6 π = approximately 18.85.
Step 6. Calculate the circumference of the total circle
We now know the circumference of the cone, but the circle itself has a larger circumference when opened (before any parts are cut out). We can use the same formula to find this number, but this time the radius would be the apothem of the cone (L).
- C (full circle) = 2 π L.
- Our example cone with apothem 12, 37 has a circumference of the complete circle equal to 2 π (12, 37) = approximately 77, 72
Step 7. Subtract the two circles to measure the slice to be removed
The full circle with no cut parts has circumference C (full circle). The material we need for the cone has a circumference C (base of the cone). Subtract one value from the other, and you will get the circumference of the missing "slice":
- C (full circle) - C (cone base) = C (slice)
- In our example, 77.72 - 18.85 = C (slice) = 58.87
Step 8. Find the angle of the slice (optional)
You can cut out a circle, then measure its circumference with a measuring tape. For almost everyone, however, it is easier to calculate the angle of the slice instead and use a protractor to measure it, starting from the center of the circle. Just a couple more calculations:
- Calculate the ratio of the missing segment to the full circumference: C (slice) / C (full circle) = Ratio. In our example: 58, 87/77, 72 = 0.75. We found that the "slice" represents 75% of the circle in our case.
- Use this ratio to find the angle. The same ratio applies to the angles. A circle has 360 °, so you can find the angle of the slice ("a) with the formula Ratio = a / 360º, or a = (Ratio) x (360º). That is 0.75 x 360º = 270º in our example.
Step 9. Cut out your model and roll it up
If you have machinery that can do the job for you, you can have templates printed in specific sizes. Otherwise, draw a circle with a compass, or with a pencil tied to a pin by a string as long as the radius of the circle. Use a protractor to draw the angle of the "slice" that will not be part of the cone, and use a ruler to extend the mark from the center to the circumference. Cut out the rest of the circle and roll up the cone.
It's a good idea to cut out a circle that is slightly larger than you need to overlap the paper when joining the two sides
Advice
- If you need a round-tipped cone, you can use half a plastic egg, half a ping pong ball, or a rubber ball.
- The mathematical formulas shown in the guide are applicable to all units of measurement, as long as they are constant throughout the operation.
