A folded surface has more surface area than a flat surface. Imagine an accordion, which looks small when it’s bunched up, but actually can be stretched out due to its folds. So,how does this relate to the mitochondria?

Inside the mitochondria, chemical reactions occur which produce energy for the cell. One of these reactions happens at the folded surface of the inner membrane. Before the reaction can occur, charge is transported across this membrane, similar to charging a battery. This charge then allows the energy-producing reaction to happen. Since the reaction happens at the membrane surface, a folded membrane with increased surface area can produce more energy for the cell.

They increase the surface area of the inner membrane, allowing there to be more proteins and more exchange of ions across the membrane, allowing the mitochondrion to do its job more quickly.

The function of the folds within the mitochondria is to increase the surface area of the mitochondrial membrane. Basically, the physical form of a structure can increase or decrease its surface area to volume ratio. For instance, a sphere has the minimum surface area to volume ratio, whereas a pyramid has a larger surface area to volume ratio. For example, if you had a fixed volume of clay and made it into a sphere or a pyramid, the pyramid would have a larger surface area.

The same concept applies to mitochondria that by having a highly folded structure, they increase the surface area of their membrane.

The reason mitochondrial membranes need a high surface area is because the process of using oxygen to make energy is carried out on this membrane. The larger the surface area of this membrane, the more room for the membrane-bound cellular machinery that uses oxygen to make energy. Therefore, for a cell to make energy at a high rate, the folds are necessary.

The function of the folds in mitochondria is to increase the surface area. This inner folded part of the mitochondria (the inner membrane) is responsible for cell respiration (the process of breaking down carbohydrates (sugars) to make energy). Part of cell respiration happens by transferring molecules across the inner membrane, so by adding folds, a longer piece of membrane can be stuffed inside the mitochondria. This increases the amount of molecules that can be transferred across it. Increasing surface area can be seen in many other parts of the body too. For example, absorption of nutrients in our digestive system happens in the small intestine. Nutrients have to be moved across the wall of the small intestine to be absorbed. We increase absorption by making the intestine really long, but on the inside it is also very folded (in structures called "villi") which increases the surface area so more absorption can occur.