Problem: Article: The formula is apothem=s2tan⁡(180n){\displaystyle {\text{apothem}}={\frac {s}{2\tan({\frac {180}{n}})}}}, where s{\displaystyle s} equals the side length of the polygon and n{\displaystyle n} equals the number of sides the polygon has. Remember to substitute for the variable s{\displaystyle s}. For example, for a hexagon with a side length of 8 cm, the formula will look like this: 82tan⁡(180n){\displaystyle {\frac {8}{2\tan({\frac {180}{n}})}}}. A hexagon has 6 sides. Remember to substitute for the variable n{\displaystyle n}. For example: 82tan⁡(1806){\displaystyle {\frac {8}{2\tan({\frac {180}{6}})}}}. You are finding the degrees you will use to calculate the tangent. For example, 1806=30{\displaystyle {\frac {180}{6}}=30}, so the formula now look like this: 82tan⁡(30){\displaystyle {\frac {8}{2\tan(30)}}}. To do this, use a calculator or a trigonometry table. For example, the tangent of 30 is about .577, so the formula will now look like this: 82(.577){\displaystyle {\frac {8}{2(.577)}}}. This will give you the length of the apothem of your hexagon. For example:apothem=82(.577){\displaystyle {\text{apothem}}={\frac {8}{2(.577)}}}apothem=81.154{\displaystyle {\text{apothem}}={\frac {8}{1.154}}}apothem=6.93{\displaystyle {\text{apothem}}=6.93}So, the apothem of a regular hexagon with 8-cm sides is about 6.93 cm.
Summary: Set up the formula for finding the apothem of a regular polygon. Plug the side length into the formula. Plug the number of sides into the formula. Complete the calculation in parentheses. Find the tangent. Multiply the tangent by 2, then divide the side length by this number.

Problem: Article: Retinoids are vitamin A derivatives that may improve fine wrinkles, splotches and skin roughness. Administering a topical retinoid that your doctor prescribes may help improve the appearance of your neck skin and help improve its elasticity.  Tretinoin and tazarotene are two types of retinoids your doctor may prescribe.  You need a prescription for a retinoid from your doctor, so consult with her to make sure this is a viable option for you.  When applying retinoids to the face for palliation of fine wrinkles, apply a pea-size amount of cream once daily at bedtime or in the evening. Be aware of sensitivity to UVA light when using this medication.  Minimize exposure to sunlamps or sunlight. Be aware that some insurance companies will not cover retinoids used for cosmetic purposes. There are some skin creams available over-the-counter that have low grade retinoids in them. Be aware that these are not as powerful as prescription retinoids and may not improve your skin over the long-term.  Retinoids can cause redness, dryness, and burning on your skin. Using treatments derived from laser, light sources, or radiofrequency can stimulate the growth of new collagen in your skin. Undergo one of these treatments to help firm your neck skin.  Laser and light source treatments destroy the outer layer of the skin and heat the underlying layer to stimulate collagen growth. As your wounded skin heals, it forms smoother and tighter skin.  It can take several months to fully heal from light source or laser resurfacing and it runs risks such as scarring, or lightening or darkening of your skin.  Consider nonablative laser treatments for skin that is less saggy.  Radiofrequency treatments are also nonablative to consider. Although you won’t achieve the same results as with laser or light source therapy, you may still notice mild to moderate skin tightening.  Be aware that most insurance companies will not cover these treatments for cosmetic purposes. There are less invasive treatments that can peel away layers of the skin. Chemical peels, dermabrasion and microdermabrasion remove the outer layer of your skin and may help not only improve elasticity, but also appearance.  A chemical peel requires that your doctor apply an acid to the top layer of your skin. This will burn away the skin on the affected area along with some wrinkles, fine lines, and freckles. It can take several weeks to heal from a chemical peel and may require several treatments to see results.  Dermabrasion will sand away the surface layer of your skin with a rotating brush. This will stimulate the production of a new layer of skin, which may make your neck skin firmer. It can take months to see results and fully heal from the procedure.  Microdermabrasion is similar to dermabrasion, but only remove a small layer of skin. It may take several treatments to see results with dermabrasion, but your overall healing time will be lower than with alternatives. Microdermabrasion also only produces modest results.  Be aware that most insurance companies will not cover these treatments for cosmetic purposes. Botox, which is the Botulinum toxin type A, can keep muscles from contracting, making the skin appear smoother and less wrinkled. Get Botox injections for mild skin sagging to help firm your neck skin.  Botox lasts for three to four months and needs repeat injections to maintain your results.  One of the side effects of Botox is an inability to move your face and neck muscles. Be aware that this can limit how well you can express your emotions. Be aware that most insurance companies will not cover Botox injected for cosmetic purposes. There are different types of fillers for soft tissues, including fat, collagen, and hyaluronic acid. Have these injected into your neck area to help plump up and firm your neck skin.  You may have some swelling, redness and bruising from soft filler injections.  Like Botox or microdermabrasion, you may need to get repeat injections because most fillers only last for a few months.  Be aware that most insurance companies will not cover filler injections for cosmetic purposes. If you have extremely saggy skin on your neck, surgery is an option. This is the most extreme form of treatment to tighten your skin and you should only consider it if absolutely necessary or other options don’t work.  As with all cosmetic surgery, make sure that you are fully aware of the risks involved and consult a trusted surgeon and company. A facelift will require removing excess skin and fat from your neck and then tightening the muscle and connective tissue underneath. It can take a long time to heal from a facelift, and you may bruising and swelling for many weeks after the procedure.  Results can last for five to 10 years.  After surgery try to wear comfortable clothing that you can pull over you head very easily and comfortably.  Have pillows available to prop your head and neck up in a good position. Have someone stay with you at least 24 hours after surgery.   Quit smoking completely and stop taking blood thinners (with the guidance of your doctor) before surgery. Smoking cessation is necessary for proper healing, and blood thinners increase your risk of bleeding during surgery.  Be aware that most insurance companies will not cover face lifts for cosmetic purposes.
Summary:
Administer topical retinoids. Undergo laser, light source or radiofrequency therapy. Peel away layers of skin. Get Botox injections. Inject soft tissue fillers. Consider a surgical facelift.