Surface Area to Volume Ratio (information)
Introduction: We have discussed Cell Theory; the idea that all living things are
composed of cells. Cells are limited in size for various reasons. To understand the
limitations in size, we must first understand some basic mathematical concepts
before we understand the biological constraints.
We will treat cells as regular solids. This means that the lengths of the sides are
measurable and the volume can also be derived.
To calculate the surface area of a rectangular solid, you simply add the areas of
each side. Here is an example:
The lengths of the sides are 3cm, 4cm, and 5cm.
So we know that the six rectangles that make up the exterior of the solid are:
2 of 5.0 cm x 3.0 cm
2 of 4.0 cm x 3.0 cm
2 of 4.0 cm x 5.0 cm
Adding these, we find the surface area to be 94cm2.
Notice the units of surface area. The units are squared.
To calculate the volume of a rectangular solid, you multiple the length by the
height by the width. In this case, you multiply:
3.0 cm x 5.0 cm x 4.0 cm= 60cm3.
To calculate the surface area to volume ratio, you simply divide the surface area by
the volume. In our case, it is:
94 cm2/60cm3= 1.56 cm2/cm3, or with significant figures, 1.60cm2/cm3.
To calculate the surface area of a sphere, use the following equation:
Surface area= 4* π *r2 with r being radius.
To calculate the volume of a sphere, use the following equation:
Volume= (4/3) * π * r3.
We will treat most cells as spheres.
Let us consider the following spheres:
Radius of sphere 1= 1cm
SA: 4*3.14*(1.0cm)2= 12.56 cm2
V: (4/3)*3.14*1.0cm3= 4.18cm3
SA/V= 3.0 cm2/cm3
Radius of sphere 2=2cm
SA: 4*3.14*(2.0cm)2=50.24cm2
V: (4/3)*3.14*(2.0cm)3=33.5 cm3
SA/V= 1.5cm2/cm3
As you can see, when we doubled the radius, we cut the surface area to volume
ratio in half.
Objectives (SOL Bio. ):
TSWBAT:
1. Calculate the surface area of regular solids
2. Calculate the volume of regular solids
3. Calculate the surface area : volume ratio of the solids
4. Understand the implications for cell size
Activities: Do Now, Water polo and tennis balls, discussion
Essential Vocabulary:
1. Nutrients
2. Waste
3. Surface Area
4. Volume
5. Ratio
6. Daughter
Critical Learning: Cells are limited in size due to biological restraints. As they
increase in size, cells need more and more nutrients, and produce more waste. All
resources and waste need to pass through the membrane, or else the cell will die.
As the cell gets bigger, the volume of the cell increases faster than the surface
area, so the needs of the cell exceed the ability of the cell to transport the
materials across the membrane. Once cells reach a critical size, they will divide to
form two daughter cells.
SOL Sample Questions:
Which of the following structures has the highest surface area to volume ratio?
10cm
Radiusa=5cm
Radiusb=10cm
5cm
8cm
8cm
4cm
2cm
Sphere
A
A) Box A
B) Box B
C) Sphere A
D) Sphere B
Sphere
B