Cones – Part 2

Slideshow 48, Mathematics

Mr Richard Sasaki

Room 307

Objectives• Review the properties of a cone and

formulae we have learned about their properties

• Be able to make calculations about attributes of cones regarding both their bases and lateral surfaces

Answers

𝑏=9

𝑎=5

𝑎=3𝑏=

1𝜋𝑜𝑟 𝜋−1

𝑏=1𝑏=

450𝜋

𝑎=300𝜋

The Net for a Cone

Let’s have a quick review.

Sector(s)

Radius / Radii

Central angle(s)

Arc Length

𝑟 𝑟𝑎𝑜

𝑎𝑜 (Area)

(Area)𝑙𝑙

Lateral Surface

Base

The Net for a Cone

The net for a cone is formed by two things.A and a .circle sectorIf we have information regarding both…what is one common piece of information with the same name? The radius!!Oh no! It’s crazy and confusing so we will only consider the cone as a shape, not as its net.

The Cone

When the cone is produced, what is the name of the radius of the sector?

Slant height - 𝑎𝑜

𝑟h

The slant height!Let’s review the properties of a cone.Two pieces of information (not labelled) are identical. What are they? The circumference of the

base and arc length of the sector!

Finding Things…Have a look at this cone.

5cm

2cmWe’re given two pieces of information.𝑟 −The radius of the base.𝑙−The slant height.

How could we calculate the circumference at the base?

2𝜋 ∙2¿ 4𝜋𝑐𝑚This is also

.

the

lateral surface’s

arc length

Finding Things…5cm

2cm

4𝜋𝑐𝑚

How about the angle at the apex formed by the lateral surface?

𝑎𝑜

Do you remember the formula we need?We need the formula to find the circumference of the base.

𝐶=2𝜋 𝑙𝑎360

4𝜋=2𝜋 ∙5 𝑎360⟹4 𝜋=

𝜋 𝑎36⟹144𝜋=𝜋 𝑎⟹𝑎=144𝑜

Answers (Section 1)

C = cm

𝐶=2𝜋 𝑙𝑎360

⟹ 8𝜋=2𝜋 ∙5𝑎360

⟹2880𝜋=10𝜋 𝑎⟹𝑎=288𝑜C = cm

𝐶=2𝜋 𝑙𝑎360

⟹14𝜋=2𝜋 ∙13𝑎360⟹5040𝜋=26𝜋 𝑎⟹𝑎=( 252013 )

𝑜

𝐴=𝜋 ∙42=16𝜋 cm2C=2𝜋 ∙ 4=8𝜋 cm

𝐶=2𝜋 𝑙𝑎360

⟹ 8𝜋=2𝜋 ∙8𝑎360

⟹2880𝜋=16𝜋 𝑎⟹𝑎=180𝑜𝑆= 𝜋 𝑙2𝑎360

= 𝜋 ∙82 ∙180360

=32𝜋𝑐𝑚2

+ =

Shortcuts…We know and and these are fine to use to calculate the surface area when we are given and . But it’s a long process, right?

Let’s check our process with Q3.

𝐴=𝜋 𝑟2Surface Area = A + S

𝑆= 𝜋 𝑙2𝑎360

𝐶=2𝜋 𝑙 𝑎360𝐶=2𝜋 𝑟⟹𝑎=¿

360𝐶2𝜋 𝑙

Shortcuts…

𝐴=𝜋 𝑟2Surface Area = A + S

𝑆= 𝜋 𝑙2𝑎360

𝐶=2𝜋 𝑟𝑎=360𝐶2𝜋 𝑙

𝑎=360 ∙2𝜋𝑟2𝜋 𝑙

¿360𝑟𝑙

𝑆= 𝜋 𝑙2

360( 360𝑟𝑙 )¿

𝜋 𝑙 ∙360𝑟360

¿𝜋 𝑙𝑟

Surface Area (cone) =

ExampleKnowing shortcuts makes things easier! But of course you have to remember them.

ExampleA cone has radius 5cm and slant height 8cm. Calculate its surface area.

𝑆 . 𝐴=𝜋 ∙52+𝜋 ∙8 ∙5𝑆 . 𝐴=25𝜋+40𝜋¿65𝜋 cm2

Answers - Easy

S.A = cm2

S.A = cm2

Tall and thin.

S.A = cm2

S.A = cm2

S.A = cm2

S.A = cm2

Answers - HardS.A = cm2

Because both start at the same place and reach the middle but the slant height must reach a higher point too.

S =

No. (Reasoning is the same as Question 2.)

𝐶=2𝜋 𝑟⇒𝑟=𝐶2𝜋

𝐴=𝜋 𝑟2⇒ 𝐴=𝜋 ∙ ( 𝐶2𝜋 )2

= 𝐂𝟐

𝟒𝝅

S.A = = =

S.A =C

S.A =S.A =