Pyramid &Frustum

IntroductionThe frustum of a pyramid or truncated pyramid is the result of cutting a pyramid by a plane parallel to the base and separating the part containing the apex.

*The lateral faces of a pyramidal frustum are trapezoids.

*The height of the pyramidal frustum is the perpendicular distance between the bases.

*The apothem is the height of any of its sides.

In computer graphics the screen is called the viewport. Everything within the frustrum will get projected onto the viewport to create an perspective image on your screen.

How does an UNFOLD frustum pyramid look like?

01/05/2023

Unfold (2D)

01/05/2023

Used in 3D gamesThe pyramid is constructed so that it fits neatly within the viewing screen and extends far enough to include all the model.

The part of the pyramid from the screen to the extreme left is called a frustum.

This is a pyramid with its top cut off. In computer graphics the screen is called the viewport. Everything within the frustrum will get projected onto the viewport to create an perspective image on your screen.

Pyramid and Frustum• What is Pyramid?

• Types of Pyramid

• What is the different between Right Pyramid & Oblique Pyramid?

• Total Surface Area

• Volume

• Frustum of Pyramid

What is Pyramid?• A pyramid is a structure whose outer surfaces are triangular and converge to a single vertex

• The base of a pyramid can be

I. Trilateral

II. Quadrilateral

III. Polygon shape

• A pyramid has at least four outer triangular surfaces including the base

Types of PyramidPyramid Base Description

Regular Pyramid The base of a regular pyramid is a regular polygon and its faces are equally sized triangles

Irregular Pyramid The base of an irregular pyramid is an irregular polygon, and as a result, its faces are not equally sized

Right Pyramid A right pyramid has isosceles triangles as its faces and its apex lies directly above the midpoint of the base

Triangular Pyramid The base is a triangle

Oblique Pyramid An oblique pyramid does not have all isosceles triangles as its lateral sides

Pentagonal Pyramid The base is a pentagon

Hexagonal Pyramid The base is a hexagon

Right pyramid VS Oblique pyramid• If the apex is directly above the center of the base, then it is a Right Pyramid.

• If it is not directly above the center of the base, then it is a Oblique Pyramid.

Right Pyramid Oblique Pyramid

Total Surface Area• Total surface area of pyramid = area of base + area of each of the

……………………………………………… lateral faces

Calculate the surface area of the following pyramid. Total surface area = Area of base + Area of four lateral faces = (6×6) + (1/2 × 6 × 12 ×4) = 36 + 144 = 180 cm2

Calculate the surface area of the following pyramid.

Total surface area = Area of base + Area of four lateral faces

= (10×10) + (1/2 × 10 × 13 ×4) = 100 + 260

= 360 cm2

Volume

• Total volume of pyramid = 1/3 (base area) x perpendicular height of pyramid• this formula applies to all pyramids even if they have different base

Volume of square base pyramid

• Total volume of pyramid = 1/3 (base area) x perpendicular height of pyramid

Total volume of pyramid = 1/3 (10x10)(18) = 600cmᵌ

Volume of triangular pyramid

• Total volume of pyramid = ⅓ (base area) x perpendicular height of pyramid

Total volume = ⅓ x {½ x (14 x 8)} x (17) =317 ⅓ cmᵌ

Volume of hexagonal pyramid

• 1st need to find the area of the hexagon

Volume of hexagonal pyramid

• Total volume of pyramid = ⅓ (base area) x perpendicular height of pyramid

Total volume = ⅓ x area of hexagon x perpendicular height = ⅓ x 6(½ x 4 x 6} x (6)

= ⅓ x 72 x 6 = 144cmᵌ

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Apothem of pyramidal frustum

• To calculate the apothem of a pyramidal frustum, the height, the apothem of the biggest base and

the apothem of the minor base must be known.

• Apply the Pythagorean theorem to determine the length of the hypotenuse of the shaded triangle to

obtain the apothem:

a c

b

Pythagoras theorem:

AL = Area of every side of pyramid = x APAr = Total surface area of frustum pyramid = x AP + A +A’

Area of pyramidal frustumP = Perimeter of the larger base

P’ = Perimeter of smaller base

A = Area of the larger base

A’ = Area of the smaller base

AP = Apothem of the truncated pyramid

Example:Calculate the lateral area, surface area and volume of the truncated square pyramid whose larger base

edge is 24, smaller base edge is 14 cm and whose lateral edge is 13 cm.

P = 24 x 4

= 96cm

P’ = 14 x 4

= 56cm

A = 24 x 24

= 576cm²

A’ = 14 x 14

= 196cm²

Ar = x AP + A +A’ = 912 + 576 +196 = 1584cm²

AL = x AP = x 12 = 912cm²

=

h = 12cm

Volume of a Frustum Pyramid

To calculate the volume of a frustum pyramid, 3 main factors must be known; the height, the area of the top and bottom parts of the frustum pyramid.

Without these factors, it is impossible to identify the volume of the frustum pyramid without including external factors and formulas into the mix.

Volume of a frustum pyramidMain formula that is used to calculate the volume of a frustum pyramid :

Height : hArea of bases : B1 & B2

Due to the fact that a frustum pyramid is another form of pyramid with its top cut off, the formula for said frustum pyramid has many similarities to the pyramid’s formula in calculating its volume:

The length and width is removed and replaced with the area’s of the top and bottom parts of the frustum pyramid

How is it used?Scenario 1 : Every information is given.

Example 1 :

Find the volume of the frustum pyramid whose area of bases are 10 cm2, 12cm2 and height is 9cm.

B1: 10cm2

B2: 12cm2

H : 9cm

How is it used?Scenario 2 : Angle instead of height is given.

Example 1 :

Find the volume of the frustum pyramid.

Find them one by oneB1(area of top square) : 3ft x 3ft

: 9ft2

B2(area of bottom square) : 7ft x 7ft: 49ft2

Height : tan 62o30 =

: h = (2)(tan 62o30): h = 3.842ft

B1 = 9ft2

B2 = 49ft2

H = 3.842ft