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Unit 1.4 Surface area right pyramids & cones Name: Date: Shape definitions to know: memorize Tetrahedron 4 triangles joined Square pyramid Square bottom, 4 attached triangle faces meet at point at top. Rectangular pyramid Rectangular bottom, 4 triangle faces

Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

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Page 1: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Unit 1.4 Surface area right pyramids & cones

Name: Date:

Shape definitions to know: memorize

Tetrahedron 4 triangles joined

Square pyramid

Square bottom, 4 attached triangle faces meet at point at top.

Rectangular pyramid

Rectangular bottom, 4 triangle faces

Page 2: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Pentagonal pyramid

Pentagon bottom [5 sides], 5 triangle faces

Hexagonal pyramid

Hexagon bottom [6 sides], 6 triangle faces

Triangular pyramid

Triangle bottom, 3 triangle faces … same as tetrahedron

Apex The point

where the faces meet.

Page 3: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Height how tall the pyramid is [centre of floor to apex]. INSIDE height … or CENTRE height.

Slant height on an angle, how tall is the triangle outside. … distance on outside – up the middle of the outside face.

RIGHT pyramid

The faces are ALL triangles … base is any polygon shape.

Page 4: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Net The ‘exploded’ form of a 3D shape.

Or

Also, recall … and memorize the formulae for area of

shapes. “area” means FLAT object

Square 𝑨 = 𝒍 × 𝒘 𝒐𝒓 𝑨 = 𝒔𝟐

Rectangle 𝑨 = 𝒍 × 𝒘

Triangle 𝑨 =

𝟏

𝟐𝒃𝒉 𝒐𝒓 𝑨 =

𝒃𝒉

𝟐

Circle 𝑨 = 𝝅𝒓𝟐

Remember: r = radius half way across circle

d = diameter all the way across

Page 5: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Surface area – means the outside area of a 3D object [NOT

flat]. Must use the ‘net’.

UNITS for area and surface area are ‘squared’: cm2, ft2, …

So, to find the surface area of a right square pyramid, you

will USE ONLY the base length … and slant height.

So … to calc SURFACE AREA of the pyramid, see the net

and see that you need area of the middle square … and

area of all 4 triangles.

Of course … you

won’t always be

given these … you

might have to

calculate them!

Page 6: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

So if you are given …

Then surface area = area [square] + 4[area of triangle]

Page 7: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

𝑺𝑨 = [𝟖𝒄𝒎 × 𝟖𝒄𝒎] + 𝟒 [𝟖𝒄𝒎 × 𝟏𝟎𝒄𝒎

𝟐]

= 𝟔𝟒𝒄𝒎𝟐 + 𝟒 [𝟖𝟎𝒄𝒎𝟐

𝟐]

= 𝟔𝟒𝒄𝒎𝟐 + 𝟒[𝟒𝟎𝒄𝒎𝟐]

= 𝟔𝟒𝒄𝒎𝟐 + 𝟏𝟔𝟎𝒄𝒎𝟐

= 𝟐𝟐𝟒𝒄𝒎𝟐

e.g. 1) page 28: Determine the area of this regular

tetrahedron … given its slant height.

Always write

in your units.

ALWAYS pay

attention to

them!!

Page 8: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Solution: (1) Think of the net … what shapes are here?

(2) Calc area of one triangle

* “Regular” tetrahedron means

EVERY side is identical.

(3) Multiply that one triangle by 4, since there

are 4 triangles in a tetrahedron.

So: 𝑨∆ =𝒃×𝒉

𝟐 =

𝟗.𝟎𝒄𝒎×𝟕.𝟖𝒄𝒎

𝟐 =

𝟕𝟎.𝟐𝒄𝒎𝟐

𝟐 = 𝟑𝟓. 𝟏 𝒄𝒎𝟐

CHECK!! Are you done?

Don’t forget: 𝑺𝑨 = 𝟒 × 𝟑𝟓. 𝟏 𝒄𝒎𝟐 = 𝟏𝟒𝟎. 𝟒 𝒄𝒎𝟐

e.g. 2 page 29) A right rectangular pyramid has base

dimensions 8 ft by 10 ft, and height of 16 ft. Calculate the

SA of the pyramid to the nearest square foot.

Solution:

[1] Sketch and label the diagram.

[2] Sketch and label the NET!!

* If you don’t do this … you’ll miss an important

fact watch

Page 9: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

NET: Gives you area of the square.

To get area of each triangle, you need height of each

TRIANGLE … not height of the actual pyramid. BUT you

use height of the pyramid … and base lengths … to

calculate triangle heights [via PYTHAGORUS].

Notice that

you are not

told slant

height!

Page 10: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐

(𝟏𝟔𝒇𝒕)𝟐 + (𝟒𝒇𝒕)𝟐 = 𝒄𝟐

𝟐𝟓𝟔𝒇𝒕𝟐 + 𝟏𝟔𝒇𝒕𝟐 = 𝒄𝟐

𝟐𝟕𝟐𝒇𝒕𝟐 = 𝒄𝟐

Thus 𝒄 = √𝟐𝟕𝟐𝒇𝒕𝟐 = 𝟏𝟔. 𝟒𝟗 𝒇𝒕 [square root of ft2 = ft]

… and that is only the SIDE of that triangle … and its

partner.

You NEED the red-

dashed line – or you

can’t calc area of that

triangle. You KNOW

the 16 ft height … and

the red line [4ft]. Use

pythag to calc.

Page 11: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Now look again …

𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐

(𝟏𝟔𝒇𝒕)𝟐 + (𝟓𝒇𝒕)𝟐 = 𝒄𝟐

You also NEED the

green-dashed line – or

you can’t calc area of

that triangle. You

KNOW the 16 ft height

… and the green line

[5ft]. Use pythag to

calc.

Page 12: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

𝟐𝟓𝟔𝒇𝒕𝟐 + 𝟐𝟓𝒇𝒕𝟐 = 𝒄𝟐

𝟐𝟖𝟏𝒇𝒕𝟐 = 𝒄𝟐

Thus 𝒄 = √𝟐𝟖𝟏𝒇𝒕𝟐 = 𝟏𝟔. 𝟕𝟔 𝒇𝒕 [square root of ft2 = ft]

… and that is FINALLY the SIDE of that triangle … and its

partner.

Now you can actually answer the question. The surface

area is:

𝑺𝑨 = 𝒂𝒓𝒆𝒂 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒍𝒆 + 𝒂𝒓𝒆𝒂 𝟐 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 + 𝒂𝒓𝒆𝒂 𝒐𝒕𝒉𝒆𝒓 𝟐

Page 13: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

= (𝟏𝟎𝒇𝒕 × 𝟖𝒇𝒕) + 𝟐 × (𝟏𝟎𝒇𝒕 × 𝟏𝟔. 𝟒𝟗𝒇𝒕

𝟐) + 𝟐 × (

𝟖𝒇𝒕 × 𝟏𝟔. 𝟕𝟔𝒇𝒕

𝟐)

= 𝟖𝟎𝒇𝒕𝟐 + 𝟏𝟔𝟒. 𝟗𝒇𝒕𝟐 + 𝟏𝟑𝟒. 𝟎𝟖𝒇𝒕𝟐

= 𝟑𝟕𝟖. 𝟗𝟖 𝒇𝒕𝟐

** Note that your units are correct for SA … so you can be

quite sure that you did it right!

Done part 1 of section 1.4

ASSIGNMENT:

CYU 1 page 28

then pages 34-35: q’s 4, 5, 9, 13a

** p.s. LATERAL AREA means sides & top only … not

bottom.

Next do CYU 2 page 29

Page 14: Unit 1.4 Surface area right pyramids & cones€¦ · form of a 3D shape. Or Also, recall ... So, to find the surface area of a right square pyramid, you will USE ONLY the base length

Then do pages 34-35: q’s 8a, 10, 13b, & 18.

THIS is the order I TAUGHT this stuff. It’s likely easiest to

do the work in this order.