Klp 3 English

7/28/2019 Klp 3 English

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Cube

A cube is a region of space formed by six square, all sides of which are the

same in largerst. Three edges join at each corner to form a vertex. The cube canalso be called a regular hexahedron.

figure of ABCD. EFGH cube

Parts of a cubea. Side

A cube has six sides which are all squares, so each side has four equal

edges and all angles are right angle.

b. VertexA point on where three edges meet to each other. A cube has 8 vertices.

ABCD.EFGH cube has 8 vertices, are :

A, B, C, D, E, F, G, H,

(corner symbolized by " ")

c. EdgeA line segment on where two sides meet. A cube has 12 edges. Because all

sides are squares and congruent to each other, all edges of which are the

same in length.

ABCD. EFGH cube has 12 edges of the same length, namely:


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Base edges : AB, BC, CD, AD

Vertical edges : AE, BF, CG, DH

Horizontal edges : EF, FG, GH, EH

d. Side DiagonalsSide diagonals is line segment linking the opposite corners of a side. Sidediagonal divide side into two parts of the same area.

Each side has two, for a total of 12 in the cube. Diagonals length of the

ABCD.EFGH cube are same, AC = BD = EG = HF = AF = BE = CH =

DG = AH = ED = BG = CF

e. SpaceDiagonalsSpace diagonal of a cube is a line segment connecting two opposite corner

point in a cube which are not are the same side. A cube has 4 space

diagonals. Space diagonals intersect in the middle of the cube.

Diagonal length of the space AG = BH = CE = DF


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f. Diagonal PlaneDiagonal Plane of the cube is plane through two opposite edges.

There are six areas of the diagonal, namely: ACGE, BDHF, ABGH,

CDEF, ADGF, BCHE

Diagonal plane ACGE = BDHF = ABGH = CDEF = ADGF = BCHE

Surface Area of a CubeSurface Area of ABCD EFGH cube with side length s units is equal to the

area of side x 6

The area of BCGF = s x s

= s2

Surface Area of ABCD.EFGH cube = 6 x Area of side

= 6 x s2

Surface Area cube with side length s units are 6 x s2 (unit area)

Volume of a cubeVolume of a cube ABCD with side length s units


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The area of base (ABCD) = side x side

= s x s

= s2 (unit area)

Volume of a cube = the area of base x height of cube= s2 x s

= s3

Volume of the cube with side length s = s3 (unit volume)


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Cuboid

A cuboid is a three dimensional object with six sides of a rectangle with

opposite sides are congruent.

figure of ABCD.EFGH cuboid

Parts of a Cuboida. Side

A cuboid has six sides which opposite sides are congruent.

In ABCD. EFGH cuboid, there are :

1. Side of base = ABCD2. Side up = EFGH3. Front side = ABFE4. The back side = CDHG5. The left side = ADHE6. Right side = BCGF

Sides of ABCD = EFGH, the ABFE = CDHG, the ADHE =

BCGF

b. VertexA point on where three edges meet. ABCD.EFGH cuboid has 8vertices, are :

A, B, C, D, E, F, G, H,

(corner symbolized by " ")


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c. EdgeA line segment on where two sides meet. A cuboid has 12 edges.

Because in the cuboid there are two sides are congruent, so there are

4 pieces edges of which the same length.

AB = EF = DC = HG, BC = FG = AD = EH, and EA = FB = HD =

GC

d. Side DiagonalsSide diagonals of a cuboid is a segment connecting two points on a

side opposite angle. There are 12 pieces of side diagonals in a cuboid.

Diagonal length of the AC = BD = EG = HF

Diagonal length of the AF = BE = CH = DG

Diagonal length of the AH = ED = BG = CF

e. Space DiagonalsSpace diagonal of a cuboid is a segment connecting two opposite

corner point in a cube which are not are the same side. Space diagonals

of cuboid intersect in the middle of a cuboid.

There are 4 space diagonals in a cuboid which are the same in length,

AG = BH = CE = DF


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f. Diagonal PlaneDiagonal plane of a cuboid is plane through two opposite edges.

Diagonal plane divide the cuboid into two equal parts.

There are six areas of the diagonal, namely: ACGE, BDHF, ABGH,

CDEF, ADGF, BCHE

Diagonal plane ACGE = BDHF, ABGH = CDEF, and ADGF = BCHE

Surface Area of a CuboidThe surface area of a cuboid is the sum area of all sides.

Area of (ABCD + EFGH + BCFG + ADEH + ABEF + DCHG)

as we know, ABCD = EFGH, BCFG = ADEH, ABEF = DCHG

so the formula can be simplified into,

The surface area of a cuboid = 2( AB x BC) + 2(BC x CG) + 2(AB x CG)

AB is equal to length of cuboid, BC is equal to width of cuboid and CG is

equal to height of cuboid.


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Volume of a CuboidVolume = base area (ABCD) x height of cuboid

as we know area of ABCD = AB x BC

So the volume of a cuboid = length x width x height (unit volume)


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Prism

Definition of PrismPrism is a solid geometry that have two parallel side. This is one of

the example of prism.

Characteristic of PrismThe characteristic of the prism such as :

1. A prism has two bases, which are congruent polygonslying in parallel planes.

2. Base area of the prism can be n-edges3. Prism has a lateral edges thet the lines formed by

connecting the corresponding vertices, which form a

sequence of parallel segments.

4. Prism has lateral sides that the parallelograms formed bythe lateral edges.

Prism has two base there are upper and lower bases.

Triangle ABC is called triangle prism lower base. Triangle DEF is

called triangle prism upper base. AD, BE, and CF is lateral edges

of the prism that perpendicular with the base of the prism.
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Types of prismThere are many types of prism such as :

Right prismA prism whose lateral edges are perpendicular to the bases. In a

right prism, a lateral edge is also an altitude.

Oblique prismA prism whose lateral edges are not perpendicular to the base.

ParallelepipedsThe bases of parallelepiped prisms are parallelograms.

CuboidThe sides of cuboids are rectangular side.


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Net of PrismThere are so many shape of net prism, its depending of the base prism

itself.

Triangular Prism

Number of endpoints : 6 Number of sides : 5 Number of edges : 9

Rectangular prism (cube or solid)


Pentagonal Prism


The conclusion from the three example above are :

Number of endpoints : n x 2 Number of sides : n + 2 Number of edges : n x 3


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Volume and Surface area of prism Volume

Volume of prism is base area multiplied by height or can be

written

V= b x h

Surface areaSurface area of prism is perimeter multiplied by height add

with two times of base or can be written

A : (2 x b) + (c x h)


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Cylinder

Definition of CylinderCylinder is a solid geometry whose base is circle.

Characteristic of CylinderThe characteristic of cylinder such as :

1. Base and top area of the cylinder are same circle with the sameradius.

2. Height of the cylinder is distance of base and top of circlecenter point

3. A cylinder has two parallel sides that are circles and one sidesthat wraps around the two circles at right angles to each.

Types of CylinderThe types of cylinder such as :

Right Circular CylinderCylinder that the base are perpendicular to the height of

cylinder


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Oblique Cylinder

Cylinder that the base and top is displaced each other.

Net of CylinderThis is the net of the cylinder.

Number of sides : 3 The blanket of cylinder like the rectangular shape

Volume and Surface Area Volume

Volume of cylinder is surface base area multiplied by

height or can be written.

V : r2h

Surface AreaThe surface area of cylinder is two times surface base area

and add by circumference circle multiplied by height or can

be written

A : 2r2 + 2rh


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sphere

Sphere is a body, bounded by a spherical surface. It is possible to receive a

ball, rotating a half-circle (or a circle) about its diameter. All plane sections of aball circles (Fig.90). The largest circle is in a section, going through a center of

a ball and is called a large circle. Its radius is equal to a radius of a ball. Any two

large circles are intersected along a diameter of a ball (AB, Fig.91). This

diameter is also a diameter for each of these intersecting circles. Through two

points of a spherical surface, placed on the ends of the same diameter (A and B,

Fig.91 ), it is possible to draw an innumerable set of large circles. For instance,

through the poles of Earth it is possible to draw an infinite number of meridians.

Sphere

A three-dimensional figure with all of its points equidistant from its center.

Radius: r

Diameter: d (2R)


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Sector of a Sphere

The part of a sphere between two right circular cones that have a common

vertex.

At the center of the sphere, and a common axis. (The interior cone may have a

base with zero radius.)

Radius: r

Height: h

Segment and Zone of a Sphere

Segment: the portion of a sphere cut off by two parallel planes.Zone: the curved surface of a spherical segment.

Radius of sphere: r

Radii of bases: r1, r2

Height: h


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Shell of sphere

An object that is bounded by some part of sphere and the shell of cylinders or

truncated cones are draw in a cylinders.

Axis

In general axis can be defined as imaginary line around which something

rotates. A line that passes through a sphere in such a way that the part of the

sphere on one side of the line is a mirror reflection of the part on the other side

of the line.

Arc

Part of the circumference (edge) of a circle, or part of any curve.Given two

points on a circle, the minor arc is the shortest arc linking them. The major arc

is the longest.

A. Surface Area of Sphere1. The curved surface area of segment of sphere

The curve area of segment of sphere is

Where:

t: borderline high

r: radius of sphere

2. The curved surface area of shell of sphereThe curved surface area of shell of sphere is

3. Surface area of a sphere


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The curved surface area & the total surface area of the sphere are equal.

The surface area of a sphere is given by the following formula

Where:

r = radius of sphere

d = 2r

Volume of a Sphere FormulaVolume of a sphere is a measurement of the occupied units of a

sphere. The volume of a sphere is represented by cubic units like cubic

centimeter, cubic millimeter and so on. Volume of a sphere is the

number of units used to fill a sphere.

Formula for volume of a sphere was found by Archimedes.

Archimedes found after several experiments that the volume of a sphere

and also its surface area is exactly of the volume and the surfacearea of a cylinder with the same outer dimensions.

In the above diagram, let rbe the radius of the sphere. Since the

over all dimensions of both the sphere and the cylinder are the same, the

height of the cylinder is 2r.

Volume of a cylinder = Area of the base x Height of the cylinder.
http://en.wikipedia.org/wiki/Surface_areahttp://en.wikipedia.org/wiki/Surface_area


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= x 2r

=

Therefore, as per Archimedes formula the volume of the sphere is,

() ()


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PYRAMID

Definition of Pyramid

Pyramid is a three dimensional figure that has a polygon for its base

and whose faces are triangles having a common vertex. Line t is the height

of the pyramid and T is the common vertex. The name of pyramid is depend

on its base.

The elemen of pyramidElements owned by a pyramid:

1. VertexVertex is a point where two lines meet to form an angle

2. Common VertexCommon vertex is the vertex which is made from more then two

lines. The highest point on the pyramid

3.

SideSide is th plane which bound it to be a three dimensional figure

4. Field side5. Slant length

Slant length, is the distance from the common vertex to the base

through the field side. In this case, the slant length is same with the

height of the field side from the pyramid.

6. Base


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Base is the surface on which figure stands, in pyramid the base is

polygonal figure.

7. Ribs The caracteristics of a pyramid:

1. Fields on the form of a point2. Plane as the base3. The form of field side is a triangle

The nets of Pyramid

d.

b.

c.

a


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Kinds Of Pyramid

There many kind of pyramid, the type of pyramid can be found from

the form of its base such as :

1. Tetrahedron (Triangular Pyramid) :This type of pyramid has :

a. 4 Vertex. They are A,B,C and Tb. 4 field side. There are ABC, ABT, BCT, and ACTc. 6 ribs. They are AB, BC, AC,AT,BT and CT

2. Square Pyramid:This type of pyramid has:

a. 5 vertex. They are A,B,C,D and Tb. 5 Side :

1 base : ABCD

4 feild side : ABT, ADT, BCT, and CDT

c. 8 ribs.4 Ribs in the base : AB, BC, CD and AD

4 Ribs in feild side : AT, BT, CT, and DT

3.

Pentagonal PyramidThis type of pyramid has:

a. 6 vertex. They are A,B,C,D and Tb. 6 Side :

1 base : ABCD

5 feild side : ABT, ADT, BCT, and CDT

c. 10 ribs.5 Ribs in the base : AB, BC, CD, DE and EA


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5 Ribs in feild side : AT, BT, CT, DT, and ET

Pyramid has volume same as the other three dimensional figure. Becausepyramid is three dimensional figure so we can find the area of its base and

also the area of its field side.

- The volume of a pyramid is a measure of how much it would taketo fill the shape. For a pyramid, the formula is:

Bxh3

1

Where B is the area of the base figure, and h is the height from

the base to the vertex. The volume is expressed in measurement

units, cubed, like cubic inches. See if you can imagine little cubes

filling up the interior space of the shape.

This formula is true for pyramids of any shape base.

- The area of its base is depend on the shape of the base- The area of the field side is


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CONE

Definition of ConeA right circular cone is similar to a regular pyramid except that its

base is a circle. The vocabulary and equations pertaining to the right circular

cone are similar to those for the regular pyramid.

The elemen of ConeElements owned by a Cone:

1. VertexVertex of cone is the angle on the top of the cone

2. SideThe side of the cone is not a triangle or rectangle like the other

three dimensional figure.

3. Slant lengthSlant length is the distance from the common vertex of the cone

to the base through the side of hte cone.

4. BaseBase is the place where the side stands, the shape of the cone is a

circle.

5. HeightHeight is the measurement from the base to the top of a cone

6. RadiusRadius is a stright line from the centre of a circle to any point on

its outer edge.


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The caracteristics of a cone:1. Fields on the form of a point2. Circle as the base3. It has a flat base4. It has one curved side5. Because it has a curved surface it is not a polyhedron

Cone is a pyramid which had base a circle, as same as the other threedimensional figure, cone also has volume, area of its base and the area of its

side.

1. Volume of a coneGiven the radius and h, the volume of a cone can be found by using the

formula:

Where B is the area of the base of the cone and H is the height of the

cone. Since the base is a circle, area of the base = . Thus, the formula

is Vcone =

2. The formula to find the area of its base is , where r is the radius of its base.

3. The area of its side are- -


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Where r is the radius of its base, h is the height of the cone and s is the

slant length.

Conic surfaceThe conic surface consist of two parts, one is the curve side, the other is the

base of cone. If we slice and open the curve and then we spread out at the flat

field, we will get one of circle nets. Given the radius of the circle is r, the

curve side is a circle nets that had a radius and the length of arc is

circumference of cone base, that is 2r. From this, we get the formula of conicsurface is the sum of the area from the base of cone and also the side of cone.

The formula of the area of conic surface is (r+s).

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