How to make a cone for measuring the volume

How to make a cone for measuring the volume
For teachers
In teaching elementary and high school geometry, we derive the formula for the volume of
a cone by comparing the volume of a cylinder with a volume of a cone by using rice or
water. In this topic, teachers are required to make a cone and a cylinder accurately in order
to get an exact result. Several mathematical methods of making a cone by using a paper
(Board paper) are introduced in this handout. You can make a cone based on the size of a
cylinder accurately!
► Utilizing special right triangles
We have already known the following special right triangles that the ratio of three lengths of
sides is represented by whole numbers. (You can confirm it by using Pythagorean theorem.)
We are going to utilize it to make a cone.
5
4
5 2 =4 2 +3 2
25=16+9
13
12
13 2 =12 2 +5 2
169=144+25
3
5 4
3
× 2
10
6
8
5
13
5
12
(Example) we are going to make this cone from a shell
13
6
5
(Circumference of a base)
= Diameter×circular constant
12×π=12×3.14 =37.68= (arc length of a sector)
↑Not necessary to compute
(The central angle of the sector)
The arc length of the sector =
x
20× π × = 12×π
360
diameter ×π ×
20 × x = 12 x = 18×
12 = 216°
360
( central)
angle
360

Therefore,
The procedure of making a cone is below.
○1 Compute the circumference of the base
○2 Compute the central angle of the sector
○3 Draw a line which length is 10cm
○4 Measure the central angle that is 216° or
144°(360°-216°)
○5 Draw a line which length is 10cm
○6 Create an arc with the use of a compass
(If you need a base of a cone, draw a circle
whose radius is 10cm. Also if you need an over
lapping edge, you can create it.)
○7 Cut the sector and make it
(Try exercise)
►General method
If you'd like students to use several different cones in the activity, you can also make any
sizes of cones. (But you might need to use a calculator.)
Cylinder
x
2πx
x
y
y
Slant height
2 2
x + y y
x
2
x +
a
y
2
2
2x π = 2×
π × x + y ×
360
2 a
Circumference of the base = arc length of the sector
x
Cone
The central angle (a) =
x × 360
x
2
+ y
2
○1 Compute the slant height of a cone by using Pythagorean theorem
○2 Compute the circumference of a base
○3 Computer the central angle of the sector
○4 Draw a line which length is equal to the slant height
○5 Measure the central angle of the sector (○2 )
○6 Draw a line which length is equal to slant height
○7 Create an arc with the use of a compass

► Example
We are going to use a wafer stick container as a cylinder.
○1 Measure the sizes of this container (approximately).
4.7 cm
16 cm
Based on this cylinder, we are going to make a cone whose base and height are
congruent to the corresponding parts of the cylinder respectively.
○2 Compute the size of the shell of a cone.
4.7 cm
16 cm
2
2
16 + ( 4.7) = 256 + 22. 09
= 278.09
= 16.7(
appoximately)
cm
4.7 cm
a
2× 4.7 × π = 2×
16.7 ×
360
4.7 cm
16.7 cm
9 .4×
360
a =
33.4
=101° (approximately)
○3 Form a cone by using scotch tape
Ateneo De Davao University - Regional Science Teaching Center