Pyramid and Frustum

class activity

1. Pyramid & Frustum

2. Introduction The  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.

3. *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?

4. 2/4/2016 Unfold (2D)

5. 2/4/2016 Used in 3D games The 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.

6. Pyramid and Frustum What is Pyramid? Types of Pyramid What is the different between Right P yramid & Oblique P yramid? Total Surface A rea Volume Frustum of Pyramid

7. 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 Trilateral Quadrilateral P olygon shape A pyramid has at least four outer triangular surfaces including the base

8. Types of Pyramid Pyramid 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

9. 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

10. Pentagonal Pyramid The base is a pentagon Hexagonal Pyramid The base is a hexagon

11. 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

12. Total Surface A rea 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

13. 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

14. 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

15. 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) = 600cm ᵌ

16. 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 cm ᵌ

17. Volume of hexagonal pyramid 1 st need to find the area of the hexagon

18. 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 = 144 cm ᵌ 3 5

19. 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:  

20. A L = Area of every side of pyramid = x AP A r = Total surface area of frustum pyramid = x AP + A + A’   Area of pyramidal frustum P = 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

21. 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

22. A r = x AP + A + A ’ = 912 + 576 +196 = 1584 cm²   A L = x AP = x 12 = 912cm²   = h = 12cm  

23. 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.

24. Volume of a frustum pyramid Main formula that is used to calculate the volume of a frustum pyramid : Height : h Area of bases : B 1 & B 2

25. 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

26. 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 cm 2 , 12cm 2 and height is 9cm. B 1 : 10cm 2 B 2 : 12cm 2 H : 9cm

27. How is it used? Scenario 2 : Angle instead of height is given. Example 1 : Find the volume of the frustum pyramid.

28. Find them one by one B 1 (area of top square) : 3ft x 3ft : 9ft 2 B 2 (area of bottom square) : 7ft x 7ft : 49ft 2 Height : tan 62 o 30 = : h = (2)(tan 62 o 30) : h = 3.842ft

29. B 1 = 9ft 2 B 2 = 49ft 2 H = 3.842ft