*Int. Math. Res. Not. (2015), no. 19, 9751–9774.* Also available on the arXiv. We study the dynamics of a collection of families of transcendental entire functions which generalises the well-known exponential and cosine families. We show that for functions in many of these families the Julia set, the escaping set and the fast escaping set are all spiders’ webs of positive area. This result is unusual in that most of these functions lie outside the Eremenko-Lyubich class $\mathcal{B}$. This is also the first result on the area of a spider’s web.