The document discusses the formulas for calculating the volumes of different geometric solids. It provides the formulas to calculate: 1) The volume of a cylinder as πr2h, where r is the radius and h is the height. 2) The volume of a hollow cylinder as πh(R2 - r2), where R is the outer radius and r is the inner radius. 3) The volume of a horizontal cylindrical segment as sl, where s is the area of the circular segment and l is the length of the cylinder. 4) The volume of a truncated cylinder (one with two nonparallel cut planes) as r2π(h1+h2)/2

1. Volume of Cylinder and
Sphere

2. Cylinder 1
V=𝜋𝑟2
ℎ
A cylinder is a solid that has two parallel faces which are
congruent circles. These faces form the bases of the cylinder.
The cylinder has one curved surface. The height of the
cylinder is the perpendicular distance between the two bases.
The volume of a cylinder is given by the formula:
Volume = Area of base × height
where r = radius of cylinder and h is the height or length of cylinder.

3. How to remember: Volume = pizza
Imagine you just cooked a pizza.
The radius is "z", and the thickness "a" is the same everywhere ... what is the volume?
Answer: pi × z × z × a
(we would normally write "pi" as π, and z × z as z2, but you get the idea!)
Z

4. Cylinder 2
Volume of hollow cylinder:
= πR2 h – πr2 h
= πh (R2– r2)
Example:
A hollow cylinder copper pipe is 21cm long. Its outer and inner diameters are 10cm and 6cm respectively. Find the volume of copper used in making the pipe.
Solution:
Given that:
Height of cylindrical pipe, h=21cm=210cm
∴ External radius, R=
10
2
=5cm
Internal radius, R=
6
2
=3cm
Volume of the copper used in making the pipe
= Volume of external cylinder − volume of internal cylinder
=π𝑅2
h−π𝑟2
ℎ=π(𝑅2
− 𝑟2
)h=227[52−32]×210
=227×16×210=22×16×30=10560𝑐𝑚3

5. Cylinder 3
-Horizontal cylinder segment
So as a formula the volume of a horizontal cylindrical segment is Where
Volume=sl
s = the area of the circle segment forming the end of the solid, and
l = the length of the cylinder.
The area of the circle segment can be found using it's height and the radius of the circle.
S=
𝑅2
2
(
𝜋
180
𝐶 − 𝑠𝑖𝑛𝐶) where:
C is the central angle in DEGREES
R is the radius of the circle of which the segment is a part.
π is Pi, approximately 3.142
sin is the trigonometry Sine function.

6. Cylinder 4
A cylindrical segment (truncated cylinder) is the solid cut from a circular cylinder by two planes.
A common special case is given by one of these planes being perpendicular to the axis of the
cylinder (the base of the cylinder) and the other plane being tilted without cutting the base of
the cylinder.
The volume V of the truncated cylinder is given by
V = r2π(h1+h2)/2

7. Sphere
To find a sphere's volume, the amount of space inside a three-dimensional
object, we use this formula:
𝑉 =
4
3
𝜋𝑟3