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MATHEMATICS PYRAMID

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

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

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

• Right Pyramid vs Oblique Pyramid• Regular Pyramid vs Irregular Pyramid• Convex Pyramid vs Concave Pyramid• Types of Pyramids by their base

TYPES OF PYRAMID

RIGHT PYRAMIDThe apex lies directly

above the midpoint of the base

It has isosceles triangle as its faces

Its base is a regular polygon

OBLIQUE PYRAMIDThe 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

REGULAR PYRAMIDThe base of this

pyramid is a regular polygon

IRREGULAR PYRAMIDThe base is an

irregular polygon

REGULAR PYRAMID AND IRREGULAR PYRAMID

CONVEX PYRAMIDIt has convex

polygon as its base - Convex means extending outward

CONCAVE PYRAMIDIt has concave

polygon as its base - Concave means having an outline that goes inward

CONVEX PYRAMID AND CONCAVE PYRAMID

TRIANGULAR PYRAMID

The base is a tr iangle.

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

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 pLwhere p represents the perimeter of the base and l the slant height.

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.

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)=24inchesl= 5L.S.A.=1/2(24)(5)=60inches^2

LATERAL SURFACE AREA

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 alt itude is 15 inches.

T.S.A = 1/2pl + BThe 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

Definition: The number of cubic units that will exactly fill a pyramid.

VOLUME

FORMULA OF VOLUME OF PYRAMID =

1 / 3  × BASE AREA ×

PERPENDICULAR HEIGHT

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 threeDifferent 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

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.

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

UNFOLD OF A PYRAMIDAL FRUSTUM

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

AREA AND VOLUME OF PYRAMIDAL FRUSTUM

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)

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

SOLUTION

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.htmlhttp://www.ditutor.com/solid_gometry/

types_pyramids.htmlhttp://study.com/academy/lesson/pyramid-in-math-

definition-lesson-practice-problems.html

REFERENCES