Math PYRAMIDS
- 2. BRIEF HISTORY OF PYRAMIDS The first precision measurements of the pyramid were made by Egyptologist- Sir Flindere Petrie in 1880-82 and published as The Pyramids and Temples of Gizeh. The great pyramids of Gizeh are the most magnificent man made structures in history Egyptian mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry. They were only concerned with practical application . WHAT IS PYRAMID
- 3. A pyramid is a three dimensional shape whose base is a polygon. Each corner of a polygon is a singular apex, which gives the pyramid its distinctive shape. each base edge and apex form a triangle WHAT IS PYRAMID
- 4. The faces of a pyramid are all triangles. If he base is a regular polygon, the triangles are all congruent(that is same shape and size), and isosceles (two sides the same length) if the apex is directly over the centre of a regular base as it is above,, it's called a right pyramid. if the apex is not the centre of the base, it is called an online pyramid and the faces are not congruent. WHAT IS PYRAMID
- 5. • Right Pyramid vs Oblique Pyramid • Regular Pyramid vs Irregular Pyramid • Convex Pyramid vs Concave Pyramid • Types of Pyramids by their base TYPES OF PYRAMID
- 6. RIGHT PYRAMID The apex lies directly above the midpoint of the base It has isosceles triangle as its faces Its base is a regular polygon OBLIQUE PYRAMID The apex is not directly above the center of its base The faces are not isosceles triangle It has a square base RIGHT PYRAMID AND OBLIQUE PYRAMID
- 7. REGULAR PYRAMID The base of this pyramid is a regular polygon IRREGULAR PYRAMID The base is an irregular polygon REGULAR PYRAMID AND IRREGULAR PYRAMID
- 8. CONVEX PYRAMID It has convex polygon as its base - Convex means extending outward CONCAVE PYRAMID It has concave polygon as its base - Concave means having an outline that goes inward CONVEX PYRAMID AND CONCAVE PYRAMID
- 9. TRIANGULAR PYRAMID The base is a triangle. PENTAGONAL PYRAMID The base is a pentagon. SQUARE PYRAMID The base is a pentagon. HEXAGONAL PYRAMID The base is a pentagon. TYPES OF PYRAMID BY THEIR BASE
- 10. SURFACE AREA The lateral surface area of a regular pyramid is the sum of the areas of its lateral faces. The total surface area of a regular pyramid is the sum of the areas of its lateral faces and its base. The general formula for the lateral surface area of a regular pyramid is L.S.A.=1/2 pL where p represents the perimeter of the base and l the slant height.
- 11. SURFACE AREA The general formula for the total surface area of a regular pyramid is T.S.A.=1/2 pl + B Where p represents the perimeter of the base, l the slant height and B the area of the base.
- 12. Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 88 inches and the slant height is 55 inches. The perimeter of the base is the sum of the sides. L.S.A = 1/2pl p=3(8)=24inches l= 5 L.S.A.=1/2(24)(5)=60inches^2 LATERAL SURFACE AREA
- 13. Find the total surface area of a regular pyramid with a square base if each edge of the base measures 16 inches, the slant height of a side is 17 inches and the altitude is 15 inches. T.S.A = 1/2pl + B The perimeter of the base is 4 x s since it is a square. p=4(16)=64inches The area of the base is s^2 B=16^2 = 256 inches T.S.A.=1/2(64)(17)+256 =544+256 =800inches^2 TOTAL SURFACE AREA
- 14. Definition: The number of cubic units that will exactly fill a pyramid. VOLUME FORMULA OF VOLUME OF PYRAMID = 1/3 × BASE AREA × PERPENDICULAR HEIGHT
- 15. Base as B and Height as H B is the area of the base of the pyramid H is its height. The height must be measured as the vertical distance from the apex down to the base. VOLUME Prism can be cut into three Different pyramid that do not Overlap . It can be shown that these pyramid have the same volume . Within the prism whose volume its base multiplied by its height is BH . If the three pyramid are equal volume , then the volume of each pyramid is Bh/3
- 16. VOLUME This pyramid has base ABC and vertex E. This pyramid has base ACF and vertex E. This pyramid has base ACF and vertex E. Every pyramid, is EXACTLY one third the volume of a triangular prism with the same base and height . So the volume of ANY Pyramid is (Area of its Base x height) divided by three.
- 17. It is a 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 The apothem is the height of any of its slides FRUSTUM OF PYRAMID
- 18. UNFOLD OF A PYRAMIDAL FRUSTUM
- 19. APOTHEM OF PYRAMIDAL FRUSTUM To calculate, we have to have: the height, the apothem of the biggest base and the apothem of the minor base. Then apply the Pythagorean theorem to determine the length of the hypotenuse of the shaded triangle to obtain the apothem Hypotonuse2 = (base2+ altitude2) square root of that Nd the 2 is square So hypotenuse square is equal to square root of base square plus altitude square
- 20. AREA AND VOLUME OF PYRAMIDAL FRUSTUM
- 21. Area is equal to perimeter of the large base plus perimeter of the small base divided by two multiplied by the apothem of the truncated pyramid Volume is equal to height divide by 3 multiplied by (area of large base times small base plus square root of area of large times small)
- 22. Calculated the lateral area, surface area and volume of the truncated square pyramid who larger base edge is 24, smaller base edge is 14 cm and who lateral edge is 13 cm. EXAMPLE
- 23. SOLUTION
- 24. http://hotmath.com/hotmath_help/topics/surface-area-of-a- pyramid.html http://www.ditutor.com/solid_gometry/frustum_pyramid.html http://www.mathsisfun.com/geometry/pyramids.html http://www.ditutor.com/solid_gometry/types_pyramids.html http://study.com/academy/lesson/pyramid-in-math- definition-lesson-practice-problems.html REFERENCES