$解:(1)優(yōu)弧\widehat {AB}的長(zhǎng)為\frac {{(360-135)π×24}}{{180}}=30π(\ \mathrm {cm}),$
$優(yōu)弧\widehat {CD}的長(zhǎng)為\frac {{(360-135)π×12}}{{180}}=15π(\ \mathrm {cm}),$
$至少需要花邊的長(zhǎng)度為30\pi +15\pi =45\pi (\ \mathrm {cm}).$
$(2)燈罩的側(cè)面積=S_{陰影}$
$=(\pi \times 24^2-S_{扇形OAB})-(\pi \times 12^2-S_{扇形OCD})$
$=\frac {{(360-135)π×{{24}^2}}}{{360}}-\frac {{(360-135)π×{{12}^2}}}{{360}}$
$=360π-90π$
$=270π({c{\ \mathrm {m^2}}}).$