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Your job is to work with the costume designer to repair and/or make the costumes that the designer has chosen. Depending on your time, skills, and interest, as well as the needs of the costume designer, this can look like anything from sewing on a loose button to making whole costumes from scratch. You should know (or be willing to quickly learn) how to use a needle and thread, and it's helpful if you also know how to use a sewing machine.
Timeline
At or around the first prod meeting, you should talk with the costume designer and make sure you're on the same page about 1) your sewing skills and experience 2) how much time you're willing and able to put towards this position and 3) what the costume designer expects from you. It's perfectly fine to tell the designer that you're not comfortable making clothes from scratch. If you do agree to make costume pieces, keep in mind how much time this will take. Depending on the pattern and your experience level, making a dress, for example, will take about 10-20 hours.
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During Tech Week, you should be available to do emergency repairs when necessary. It's helpful to bring the Oh Shit box to the dressing room during Put-In, as it has supplies for costume emergencies (it's probably worth checking first though that it actually has things like needles and thread in the important colors).
20 Foot Rule
Remember, the audience will be viewing these costumes from much farther away than you. As a general rule, if it looks good from 20 feet away, you're fine!
Sewing Help
Fit and Alterations
It's your job to make sure that everyone's costumes fit. (Although you should check with the costume designer on this, as occasionally ill-fitting clothes are intentional.) Here are some common things to check:
Hems
This will depend on the character, era, and costume, but generally pants hems should end just above the top of a person's shoes. This is a good guide to hemming pants (feel free to skip some of the steps if you're rushed for time and know what you're doing - 20 foot rule!) Skirt lengths vary a lot depending on the style, so check with the designer. This guide is helpful for hemming skirts, although the process is pretty similar to hemming pants.
Dress Backs
The Ensemble has a lot of dresses made for very skinny people, which often fit an actor pretty well but don't zip up all the way in the back. One way to solve this is to add a few sturdy ribbons to either side of the back panels to tie the back of the dress together. If a lot of skin will be showing, consider adding a cloak or something to cover the back. You can also add a back panel of the same or coordinating fabric, although this is much more time-consuming.
Waists
Sometimes you'll need to take things in at the waist, particularly dresses and waistcoats. The first part of this guide gives a good method for that. This is an alteration where you'll want the actor around to try the costume on as you work.