I have recipes for chocolate cake, Maderia cake, lemon drizzle and champagne cake.
I want to increase the ingredients to make a bigger cake. Do I still use the same baking time and temperature?
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The following rule of thumb comes from week 6 in a Harvard University physics course called "Science and Cooking". I took it as a MOOC in the website edx (https://www.edx.org/node/8086).
And the rule is: double the thickness, quadruple the time. If it takes a minute to cook the first outer 0.5cm of the cake, it will take four minutes to cook the first whole cm.
So if you double your cake's size, try to start with multiplying the time by four, without raising the temperature. If the change is smaller, then it might take some rough calculations. But that's the general rule.
Here's the video of the specific part of the lesson that regards discusses the matter: https://youtu.be/fqyVZ5Ifmio
When baking, try to keep the proportions of the cake's height, width and depth the same with the big cake as with the original recipe. I would start by increasing the oven temp just 25 ° F, and start checking for done-ness at the normal time, knowing that you'll probably need to go longer.
However, if you want a 2x sized cake, I recommend just making two layers of the cake, and sticking them together with whatever frosting or icing you'll use on the outside of the cake. You can remove a lot of the complexity by just baking two "normal" sized cakes.