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## AP®︎/College Biology

# Surface area to volume ratio of cells

AP.BIO:

ENE‑1 (EU)

, ENE‑1.B (LO)

, ENE‑1.B.1 (EK)

Seeing that the surface area to volume ratio of cells generally decreases as cells get larger, making the exchange of resources, waster and heat more and more difficult.

## Want to join the conversation?

- i don't understand

how is it done the video helped only a bit still don't get(2 votes)- Basically, cells have a limit as to how large they can get. Every cell has a limit of surface area to volume ratio to ensure that the exchange of resources and waste occurs quickly enough for the cell to survive. If cells were too big, diffusion would take an extremely long time, and a cell could die from starvation or poison itself with its wastes.(27 votes)

- 3:24it's stated that the less surface area per unit of volume, the harder it is for these transfers to occur. Why is that? What does surface area have to do with how much energy it takes for these transfers to occur?(2 votes)
- Hi there Savannah. Thank you for your question.

As a cell grows in size, the surface area gets bigger, but the volume gets bigger faster.

Thinking about this as a ratio (division), the volume is the denominator and the surface area is the numerator. If the volume is getting very big, then the ratio itself will be getting very small.

What this translates to is as a cell gets larger, with a larger volume, it has a smaller SA:V. A smaller SA:V means that diffusion of nutrients and diffusion of waste in less efficient (harder to do).

The rate of diffusion doesn't change, but it would take longer for the nutrients to diffuse through to the centre of the cell than a cell with a higher SA:V (a smaller cell - i.e. a cell with a smaller volume)(4 votes)

- where is he getting his numbers?(1 vote)
- He is using the formula for the surface area of a sphere, and the formula for the volume of a sphere.

He then relates 'r' and talks about as the value of 'r' increases, the surface area to volume ratio decreases. (which makes diffusion of nutrients and waste products less efficient)(3 votes)

- If cells need to increase their surface area by folding their shape when they get too big, why are egg cells so huge, but in a spherical shape?(2 votes)
- Is there any video on Khan Academy regarding cell shape, to be specific, spherical cell shape and information about it?(1 vote)
- how does a cell maximize its surface area to volume ration? I know that the ration can never touch zero, but what cellular activities decrease the volume and or increase the surface area.(1 vote)
- Hi there. Thank you for your question.

A cell can maximise it's surface area to volume ratio by being highly folded (such as the internal membranes of mitochondria), by being long and thin (like the projections of a root hair cell), or they my have a spherical shape (alveoli in the lungs).(1 vote)

- how does increasing the surface area will increase the SA:V in the formula? isn't it always 3/r? how to show that SA:V increases in the formula?(1 vote)
- 1. SA increases by the 2nd power. V increases by 3rd power

2. yes

3. You can use Sal's formula(1 vote)

- I still don't get the question I'm trying to answer: will a highly flattened cell or a spherical cell achieve the largest volume?(1 vote)
- A really flat cell isn't going to have a lot of volume (because it's flat). Because there's so little volume, there's more surface area per unit of volume of a cell.

But with a spherical cell, there's a lot more volume, so there's less surface area to go around for each unit of volume.

So in general, a spherical cell will have a larger volume than a flat cell (if they're on a similar scale), but there's a limit to how big a spherical cell can get before it becomes too big.(1 vote)

- I understand that a cell with more folds in its membrane has a higher surface area to volume ration, therefore it can grow larger, but why exactly does it need to grow larger?(1 vote)
- In the last video, you did the volume divided by the surface area, but in this one, it's surface area to volume. How do you know which equation to set up?(0 votes)
- Hi there. Thank you for your question.

To calculate surface area to volume ratio, you will need to do a division (as it is a ratio / fraction).

SA:V = SA/V

e.g. if I have a surface area of 150mm^2 and a volume of 125mm^3, the SA:V is 1.2:1 or 1.2.

This can then be compared to other cells to determine which has the more efficient diffusion of nutrients and removal of wastes.(1 vote)

## Video transcript

- [Instructor] So let's
say that this is a cell. So we know that all sorts
of activity is going on inside of this cell here,
and we will study that in a lot more depth as we go further in our study of biology. But it's important to
realize that this cell and the activity in that cell
is not operating in isolation. That in order to live,
that cell needs resources from the outside world. So resources need to
make their way through that outer membrane of
the cell so it can be used inside that cellular machinery. And as that cell does what it does, it's also going to
generate waste products, and that needs to be released
somehow across that membrane. So you also have waste. And you also have energy that
is going to be transferred either from the inside of
the cell to the outside or from the outside to the inside. A lotta times, we imagine
that all of the activity inside the cell is
generating thermal energy that has to be dissipated somehow, and that is usually the case, not always. And so you have thermal energy
that has to be dissipated. Now you might see something interesting, or maybe you haven't seen it just yet, is that you have all of
this activity operating in the volume of the cell,
but then all of this exchange, all of the resources coming
in, the waste coming out, the thermal energy going
in either direction, it has to be somehow
diffused across this surface, across this two-dimensional surface. So this raises an interesting question. As our volume increases,
what happens to the ratio of our surface to our volume? Because you could imagine,
maybe at some point, the volume gets large
enough that you don't have enough surface area to do
these three things well. And so let's think about this ratio. Let's think about the ratio
of our surface area to volume. And I'm gonna get a little bit mathy here. You don't have to know
the math for the context of a biology course, but you need to know what the conclusion is that
the math is going to give us. So if this is a sphere of radius r, the surface area of this sphere is going to be four pi r squared, and the volume of this sphere is going to be 4/3 pi r cubed. So this pi would cancel with that pi. If we divide the numerator and
the denominator by r squared, we get a one there and
then we just get an r right over here. If we divide both of these
by four, you get a one there, and this is just going to be a 1/3. And so we are going to be left with one over 1/3 r or we could just write this is this is equal to three over r. And so we see at least for
a spherical cell like this, as r increases, as our cell
gets larger and larger, the ratio between our surface
area to volume decreases. So let me write that. As r goes up, then the ratio between our surface area to
volume, surface area to volume, is going to go down. The bigger your denominator,
the lower the value is going to be. And so what that tells us
is is that as the volume of our cell increases,
as our cell gets bigger and bigger and bigger,
we have less surface area per unit of volume. And so it's going to make that
exchange of the resources, the waste, and that energy
harder and harder and harder. And we would get a
similar result if instead of doing a spherical cell, let's
say we did a cuboidal cell. So let's do it like this, a cuboidal cell. You might see this in some plants, something that's roughly cuboidal or rectangular in some way, or rectangular prism I should say. But let's say it's x by x by x. We could do the same exercise. Our ratio of surface area to
volume is going to be what? Well our surface area, you have six faces that are each have an area of x squared, so our surface area's
going to be six x squared. And then our volume is going
to be x times x times x, over x to the third. And so this is going to
be, divide the numerator and denominator by x
squared, you get six over x. So once again, you see
that as x increases, our ratio of surface
area to volume decreases. As our denominator increases, well then that whole expression
is going to decrease. So given this phenomena, it makes it hard for larger and larger cells to exist. Because for all the activity
happening in the volume, they don't have enough surface area to do all of this exchange. Now there are things we
see in biological systems that help cells get further
than what we see here. If you imagine the
two-dimensional cross-section of this cell, one way to increase
the surface area to volume is for the membrane to
look more like this. The more folds you have, the
higher surface area to volume that you are going to have. And you indeed see this
in a lot of biology. Anytime you want a high
surface area to volume, you tend to see things like these folds in the membranes of the cells.