Party Hat construction help needed!

October 10, 2019 1:50 AM

I need to make a party hat out of construction paper with very specific diameter and vertical length measurements. The instructions I have found online don't seem to allow one to perform such fine-tuning. The party hat I require should have a diameter of 7 1/2 in. (radius 3 3/4 in.) and a vertical length (or slant height) of 15 13/16 in. What's the best and/or easiest way of doing this and ensuring that the measurements are spot on?
posted by tenderly to Sports, Hobbies, & Recreation (14 answers total)

Is this a cone shape? Draw a circle on a piece of paper to the right dimensions. Have a piece of card as wide as you need the hat to be. Roll into cone shape using your circle base, and using the width of the card to create the height.
posted by b33j at 2:52 AM on October 10, 2019 [1 favorite]

And then cut the bottom to be straight.
posted by b33j at 2:53 AM on October 10, 2019

Best answer: Using this template-maker might do the trick?
posted by 73pctGeek at 3:46 AM on October 10, 2019 [1 favorite]

This is really just a geometry problem. I'm assuming that by "slant height" you mean "distance from the brim to the tip" rather than "height of the cone when it's sitting on a table".

You need a "wedge" of a circle such that the radius is r = 15 13/16" and the arc-length along the brim is L = π*(7 1/2") (since this arc becomes the circumference of the brim.) The length of an arc of radius r which spans an angle θ (in degrees) is L = 2*π*r*θ/360. In your case, this means that (solving for theta) θ = 360° * (7 1/2") / (15 13/16") = 170.75°.

So draw a half-circle with a radius of 15 13/16" and use a protractor to remove a small wedge of 9.25° (or as close as you can get) from it. Then join together the two straight edges.
posted by Johnny Assay at 4:19 AM on October 10, 2019 [6 favorites]

Seconding Johnny Assay's method above. The angle calculation may seem messy but its just proportions.

Thinking about construction, it may be good to leave some paper overlap for the glue/tape/assembly process (so not cut at exactly 9.25 degrees.)
posted by Wulfhere at 5:00 AM on October 10, 2019 [1 favorite]

Or just mark off that nine degree wedge and use all of it as an overlap for gluing.
posted by flabdablet at 5:05 AM on October 10, 2019

Best answer: Not remotely overthinking this, I ran Johnny Assay's solution through OpenSCAD and made this: party_hat.pdf.

Code is:

// party hat for tenderly - scruss, 2019-10
in = 25.4;
r = (15 + 13 / 16) * in;
w = 9.25;

union() {
    difference() {
        // semicircle
        difference() {
            circle(r = r, $fn = 128);
            translate([0, r])square(2 * r, center = true);
        }
        // cut out notch
        rotate(-w)square(2 * r);
    }
    // add glue tab
    translate([-r / 2, 0])square([r / 2, in], center=true);
}

(To truly overthink this to Douglas Adams and the megapode nest level, you could let OpenSCAD solve for the height/angle too. Then you could add Customizer support for point and click customization and … where did today go?)
posted by scruss at 5:42 AM on October 10, 2019 [4 favorites]

Something about Johnny Assay's formula didn't sit quite right with me so I'm working it out again.

The finished brim will be a circle with a radius of 3¾". When it's still in its flat sheet form, the brim will form part of a larger circle with a radius equal to the finished hat's tip-to-brim distance of 15¹³⁄₁₆". So the pie-slice wedge we need to make will account for (3¾ / 15¹³⁄₁₆) of the larger circle, not (7½ / 15¹³⁄₁₆) of it. 7½" is a diameter, not a radius.

Which means that the angle the pie wedge needs to be cut at is not 360° * (7½ / 15¹³⁄₁₆) but 360° * (3¾ / 15¹³⁄₁₆). This comes out to a smidgen over 85°, half the value Johnny Assay got.

So if you start with a square of construction paper 15¹³⁄₁₆" on a side, then cut along an arc centred on one of the corners to make a shape like a quarter of a pizza, then glue the result into a cone with a smidgen under 5° of overlap, that will get you there.
posted by flabdablet at 5:51 AM on October 10, 2019 [1 favorite]

If you haven't got a set of compasses that can open out wide enough to trace an arc with that radius, you can improvise one with a drawing pin and a length of string.
posted by flabdablet at 5:55 AM on October 10, 2019

Get 2 pieces of paper so you can make and adjust a practice hat first, and then once you have the dimensions right on that on, take it apart to use as a pattern to make the second good hat.
posted by 5_13_23_42_69_666 at 10:20 AM on October 10, 2019 [1 favorite]

Best answer: I think 73pctGeek's link to TemplateMaker gets it right if you plug in the numbers 0, 7.5 and 15.361 (=sqrt(15.8125² - 3.75²))
posted by scruss at 2:08 PM on October 10, 2019

Looks to me like the constant-width glue tab that TemplateMaker designs in is going to cause problems at the tip when gluing up a complete cone as pointy as the one you're contemplating. I think you'd have an easier time with the gluing if you just cut out a quarter circle and marked off a 5° wedge for your glue.
posted by flabdablet at 3:59 AM on October 11, 2019

Something about Johnny Assay's formula didn't sit quite right with me so I'm working it out again.

You're right, I dropped a factor of 2. It should be θ = 360° * (7 1/2") / (15 13/16") / 2 = 85° and change.
posted by Johnny Assay at 4:21 AM on October 11, 2019

Best answer: Time to break out the Royal Doulton, réchauffer les haricots and polish up your great-great-grandmother's Bohnenpinzette …

Here is a print template, and the same but posterized onto US Letter paper. Make sure you don't scale when printing.

The easiest way to make these might be to knot a loop at each end of a piece of string such that a drawing pin at one end and a pen at the other are 401.6 mm / 15.81 inches apart. Put the drawing pin at one corner of a sheet of construction paper (smallest side more than the length you just made) and mark out a quarter circle. Measure 32.4 mm / 1.27 inches in from the edge along one of the arcs (doesn't matter which side) and mark a straight line from that point to the arc centre. That'll give you near-as-dammit your desired angle of 85.4°

If you're worried about glue tab angles from the template maker output, set the glue angle to something smaller like 30°. You may find, however, that gluing a full paper cone at this sharp an angle is incredibly difficult: at the tip, you're rolling the paper in an infinitesimal radius, and real paper (especially pulpy construction paper) won't stand for that. Unless the dimensions have incredibly specific occult/aerodynamic significance¹, I'd make what you can make and like it.

Revised OpenSCAD code to do the above for many values of slant height and base is:

// pointy party hat mk2 for tenderly - scruss, 2019-10

in = 25.4;
r = (15 + 13 / 16) * in;            // slant height
b = 7.5 * in;                       // base diameter
a = 360 * (b * PI) / (2 * r * PI);  // arc of cone
bh = sqrt(r * r - b * b / 4);       // built height of cone
c = r * cos(a);                     // inset from y axis

echo(str("Arc radius:   ", r, " mm / ", r / in, " inches"));
echo(str("Arc angle:    ", a, " degrees"));
echo(str("Y-axis inset: ", c, " mm / ", c / in, " inches"));
echo(str("Built height: ", bh, " mm / ", bh / in, " inches"));

union() {
    difference() {
        // semicircle
        intersection() {
            circle(r = r, $fn = 128);
            translate([0, 2 * r])square(4 * r, center=true);
        }
        // cut out notch
        rotate(a)translate([0, 2 * r])square(4 * r, center=true);
    }
    // add glue tab
    translate([r / 2, 0])square([r / 2, in], center=true);
}

which outputs helpful numbers:

Arc angle:     85.3755 degrees
Y-axis inset:  32.3821 mm /  1.27489 inches
Built height: 390.18   mm / 15.3614 inches

---
¹: as a nosecone for Naruto-running a marathon, perhaps, and thereby shredding Kipchoge's sub-2 hour time.
posted by scruss at 7:53 AM on October 12, 2019 [1 favorite]