Maths
from Scratch for Biologists| MicrobiologyBytes: Maths & Computers for Biologists: Areas & Volumes | Updated: October 19, 2004 | Search |
Further information on this topic can be found in Chapter 5 of:
Maths
from Scratch for Biologists
Numerical ability is an essential skill for everyone
studying the biological sciences but many students are frightened by
the
'perceived' difficulty of mathematics, and are nervous about applying
mathematical skills in their chosen field of study. Maths from Scratch
for Biologists
is a highly instructive, informal text that explains step by step how
and why you need to tackle maths within the biological sciences.
(Amazon.co.UK)
Geometry is the branch of mathematics that deals with the properties of space and objects. It is one of the oldest branches of mathematics, named from Greek for "Earth measurement".
Trigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations in geometry.
| TWO-DIMENSIONAL SHAPES: | PERIMETER: | AREA: |
|---|---|---|
| Square:
|
4 x | x2 |
| Rectangle:
|
2 (x + y) | x y |
| Circle:
|
2 p r
(p * d) |
p r2 |
| Ellipse:
|
p (1.5 (x + y) -  (x * y)) |
p x y |
| Triangle (any):
|
x + y + z | 0.5 z h |
| Triangle (right-angled):
|
x + y + z | 0.5 x y |
| THREE-DIMENSIONAL OBJECTS: | SURFACE AREA: | VOLUME: |
| Cube:
|
6 x2 | x3 |
| Cubeoid:
|
2 (x y) + 2 (x z) + 2 (y z) | x y z |
| Sphere:
|
4 p r2 |
4 p r3 / 3 or 4/3 p r3 |
| Ellipsoid:
|
No simple formula! | 4 p r x y / 3 |
| Cylinder:
|
2 p r h + 2 p r2 | p r2 h |
| Cone / Pyramid:
|
0.5 p h + b | b h / 3 |
Calculations involving simple geometric shapes are usually straightforward (except for ellipsoids!) - use the formulae above.
The pain-free way to explore geometry - and come out smiling.
This friendly guide explains it all in easy-to-understand, no-nonsense
language. Explore
postulates and theorems, calculate areas, circumferences, and more, solve
linear equations, determine ratios, nderstand trigonometry. Includes
a handy glossary of geometric terms for easy reference. (Amazon.co.UK)
There are few straight lines in biology. In biology, most shapes are complex but are either:
|
Example 1:
A tissue is composed of cells which are roughly spherical in shape and about
45 µm in diameter.
Calculate the total number of cells present in 1 cm3 of this tissue.
Strategy:
a) Volume of a sphere = 4/3 p r3
Volume of each cell = 4/3 p 22.53 µm3 = 47713 µm3
b) 1cm = 10,000 µm
1 cm3 = 104 * 104 * 104 = 1012 µm3
c) Number of cells in 1 cm3 = 1012 / 47713 = 20958648 = 2.09*107
Example 2:
Fibroblast cells like the one below are cultured on circular coverslips 10 mm in diameter, and have a doubling time of 24 hrs.
If
you have a coverslip with 1000 cells on it, how long will it take for these cells
to multiply to give a confluent culture covering the whole of the coverslip?
Strategy:
b) Area of cell = 0.5 z h = 0.5 * 45 * 15 = 338 µm2
c) 7.85*107 / 338 = 2.32*105 cells required to cover the coverslip.
d) By repeatedly multiplying 1000 by 2, it would take about 9 days for the original 1000 cells to cover the coverslip:
|
Day 1:
|
Day 2:
|
Day 3:
|
Day 4:
|
Day 5:
|
Day 6:
|
Day 7:
|
Day 8:
|
Day 9:
|
|
1*103
|
2*103
|
4*103
|
8*103
|
1.6*104
|
3.2*104
|
6.4*104
|
1.28*105
|
2.56*105
|
Are all the assumptions made valid?
Example 3:
An
Arabidopsis seedling has a root system which weighs 1.5 grams. The
average diameter of the roots is 0.2 mm.
Calculate the total length and surface area of the root system.
Assume: The roots have the same density as water, each root is a cylinder
with a constant radius and the root tips have a negligible volume.
Strategy:
= 1.5 g / 1 g mL-1 = 1.5 mL
b) Volume of a cylinder: V = p r2 h
Therefore length of a cylinder: h = V / p r2
h = 1.5 / p * 0.01 cm2
= 47.8 cm = 0.0478 m
c) Surface area of a cylinder = 2 p r h (ignoring the ends)
= 2 p (0.01 cm2) 47.8 cm
= 3 cm2
Are all the assumptions made valid?
© MicrobiologyBytes 2007.
Geometry
For Dummies