Cylinder and cones

Shows that the net of a cylinder is made up of two circles and a rectangle. The total surface area of the cylinder will be the total area of the two circles and the rectangle.

Then, fold the rectangle until you make an open cylinder with it. An open cylinder is a cylinder that has no bases. A good real life example of an open cylinder is a pipe used to flow water if you have seen one before

Next, using the two circles as bases for the cylinder, put one on top of the cylinder and put one beneath it.

obviously, the two circles will have the exact same size or the same diameter as the circles obtained by folding the rectangle

You Finally, you end up with your cylinder!

Now, what did we go through so much trouble? Well if you can make the cylinder with the rectangle and the two circles, you can use them to derive the surface area of the cylinder. Does that make sense?

The area of the two circles is straightforward. The area of one circle is π × r², so for two circles, you get 2 × π × r²

To find the area of the rectangle is a little bit tricky and subtle!

Let us take a closer look at our rectangle again.

Thus, the longest side or folded side of the rectangle must be equal to 2 × π × r, which is the circumference of the circle

To get the area of the rectangle, multiply h by 2 × π × r and that is equal to 2 × π × r × h

Therefore, the total surface area of the cylinder, call it SA is:

SA = 2 × π × r²  +   2 × π × r × h

Example

A cylindrical cup has a circular base of radius 7 cm and height of 10 cm. taking the value of π to be 22/7, calculate a. its curved surface area, b. the area of its circular base.

Solution

1. The area of the curved surface of a cylinder of radius r and height h is s 2πrh.

The curved surface area of the cup

= 2 X 22/7 X 7 X 10 cm²

= 440 cm²

1. The area of the circular base of the cup

= 22/7 X 7 X 7cm²

= 154cm²

Example

Find the surface area of a cylinder with a radius of 2 cm, and a height of 1 cm

SA = 2 × π × r²  +   2 × π × r × h

SA = 2 × 3.14 × 2²  +   2 × 3.14 × 2 × 1

SA = 6.28 × 4  +   6.28 × 2

SA = 25.12 + 12.56

Surface area = 37.68 cm²

Example:

Find the surface area of a cylinder with a radius of 4 cm, and a height of 3 cm

SA = 2 × π × r²  +   2 × π × r × h

SA = 2 × 3.14 × 4²  +   2 × 3.14 × 4 × 3

SA = 6.28 × 16  +   6.28 × 12

SA = 100.48 + 75.36

Surface area = 175.84 cm²

Volume of a cylinder

A cylinder with radius r units and height h units has a volume of Vcubic units given by

v = πr²h

Example

Find the volume of a cylindrical canister with radius 7 cm and height 12 cm.

v = πr²h

= 3.142 X 7² X 12

= 1847.50

So, the volume of the canister is 1847.50 cm²

Cones

Surface area of a cone

What is a cone?

A cone is a type of geometric shape. There are different kinds of cones. They all have a flat surface on one side that tapers to a point on the other side.

We will be discussing a right circular cone on this page. This is a cone with a circle for a flat surface that tapers to a point that is 90 degrees from the center of the circle.

Terms of a Cone

In order to calculate the surface area and volume of a cone we first need to understand a few terms:

Radius – The radius is the distance from the center to the edge of the circle at the end.

Height – The height is the distance from the center of the circle to the tip of the cone.

Slant – The slant is the length from the edge of the circle to the tip of the cone.

Pi – Pi is a special number used with circles. We will use an abbreviated version where Pi = 3.14. We also use the symbol π to refer to the number pi in formulas.

Surface Area of a Cone

The surface area of a cone is the surface area of the outside of the cone plus the surface area of the circle at the end. There is a special formula used to figure this out.

Surface area = πrs + πr²

r = radius
s = slant
π = 3.14

This is the same as saying (3.14 x radius x slant) + (3.14 x radius x radius)

Example

What is the surface area of a cone with radius 4 cm and slant 8 cm?

Surface area = πrs + πr²
= (3.14x4x8) + (3.14x4x4)
= 100.48 + 50.24
= 150.72 cm²

Volume of a Cone

There is special formula for finding the volume of a cone. The volume is how much space takes up the inside of a cone. The answer to a volume question is always in cubic units.
Volume = 1/3πr²h

This is the same as 3.14 x radius x radius x height ÷ 3

Example

Find the volume of a cone with radius 4 cm and height 7 cm?

Volume = 1/3πr²h
= 3.14 x 4 x 4 x 7 ÷ 3
= 117.23 cm ³

Things to Remember

Surface area of a cone = πrs + πr²

Volume of a cone = 1/3πr²h

The slant of a right circle cone can be figured out using the Pythagorean Theorem if you have the height and the radius.

Answers for volume problems should always be in cubic units.

Answers for surface area problems should always be in square units.

Mensuration and estimation

Mensuration is the use of measurement, formulae and calculation to find lengths, areas and volumes of shapes. The use of approximate values in mensuration formulae gives results that are sufficiently accurate for most purposes. This section provides practice in using estimation techniques in mensuration problems.

Assessment

• What is the surface area of a cone with radius 8 cm and slant 12 cm?
• Find the volume of a cone with radius 2 cm and height 9 cm?