# Deriving the surface area equation

• iScience
In summary, the conversation discusses the use of vectors a and b to calculate surface area. While it was initially thought that the area formed by these vectors could be treated as a rectangle, it is actually a parallelogram due to the fact that the surface is curved. This is why there is an extra term in the surface area formula.
iScience
consider the following image

(the red is the surface area element and the green is the differential element that I'm integrating over)

when we derived this in class, we treated the area formed by vectors a and b, as the area of a parallelogram. the thing is, a and b should be at right angles implying that rest of the angles should be right angles as well. if that's the case, then why can't i just treat the area formed from a and b, as a rectangle?

well, that's what i attempted to try but failed. i get all the appropriate terms as they do in the surface area formula, except i get an extra term as shown in the link below, implying, perhaps.. that that extra term must go to zero. but i don't know why it should.

if I'm incorrect in treating the shape formed from a and b as a rectangle, why is this incorrect?

calculations:

http://i.imgur.com/6MFO4xI.jpg
(image was too large to put here)

It's very difficult to read that- the colors do not show up on black very well! But to answer your question, the "a" and "b" on the surface are NOT vector IN the surface because it is not flat- they are tangent vectors to a curved surface and so do NOT form a rectangle in the surface. In order to calculate and area we have to project down to the xy- plane and the vectors are no longer necessarily perpendicular there.

## 1. What is the surface area equation?

The surface area equation is a mathematical expression used to calculate the total area of a three-dimensional object. It takes into account all of the individual surfaces that make up the object.

## 2. How do you derive the surface area equation?

The surface area equation can be derived by breaking down the object into smaller, simpler shapes and calculating the area of each individual surface. These areas can then be added together to get the total surface area of the object.

## 3. What is the importance of the surface area equation in science?

The surface area equation is important in science because it allows us to measure and compare the amount of exposed surface an object has. This can be useful in fields such as chemistry, where surface area plays a crucial role in reactions and interactions between substances.

## 4. Can the surface area equation be applied to any three-dimensional object?

Yes, the surface area equation can be applied to any three-dimensional object, as long as the object has distinct surfaces that can be measured and added together. This includes objects such as cubes, spheres, cylinders, and more complex shapes.

## 5. How can the surface area equation be used in real life?

The surface area equation has many practical applications in real life, such as in architecture and construction to calculate the amount of materials needed for a structure, in engineering to design efficient heat exchangers, and in biology to determine the amount of surface area available for gas exchange in the lungs.

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