Dimensional shapes. Vertex

height Edge

depth

width

4.0 Definiton of 3-Dimensional shapes

As human,we live in a three-dimensional world.We can see or touch the object that has

three dimensions that can be measured such as length,width,and height.

So many three-dimensional objects in our surrounding.The room you are sitting in can be

described by these three dimensions.

In the world around us,there are many three-dimensional geometric shapes.

3-dimensional(3D) is a shapes that takes up space which is not flat and called a solid.3D

shapes can be solid or hollow.They have width,height and length.Every solid has a fixed

number of edges,vertices and surfaces.

There are two main types of solids that are polyhedral and non-polyhedra.Polyhedra is a

solid whose faces are all flat.Each face must be polygon(they must have straight

sides).Polyhedrons or polyhedral are named according to the number of faces they have.This

terms comes from the Greek words poly,which means “many”,and hedron,which means

“face”.So quite literally,a polyhedron is a three-dimensional object with many faces.The other

parts of polyhedron are its edges,the line segments along which two faces intersect,and its

vertices,the points at which three or more faces meet.The examples of polyhedra are

cube,cuboid,pyramid and prisms.

Some common space figures that are non-polyhedra.These figures have some things in

common with polyhedra,but they all have some curved surfaces,while the surfaces of a

polyhedra are always flat.The examples of non-polyhedra are cylinders,cones and spheres.

(Refer to references A)

4.1 The meaning of 3-dimensional shapes

Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There

are two competing incompatible definitions of a cuboid in the mathematical literature. In the more

general definition of a cuboid, the only additional requirement is that these six faces each be a

quadrilateral, and that the undirected graph formed by the vertices and edges of the polyhedron

should be isomorphic to the graph of a cube.[1] Alternatively, the word “cuboid” is sometimes used to

refer to a shape of this type in which each of the faces is a rectangle, and in which each pair of

adjacent faces meets in a right angle; this more restrictive type of cuboid is also known as a right

cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular

parallelepiped.

Prism

In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated

copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All

cross-sections parallel to the base faces are the same.

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or

sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is

one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and

of trigonal trapezohedron. The cube is dual to the octahedron.

A cube is the three-dimensional case of the more general concept of a hypercube.

It has 11 nets. If one were to colour the cube so that no two adjacent faces had the same colour,

one would need 3 colours.

Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular

base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane

base and the surface (called the lateral surface) formed by the locus of all straight line segments

joining the apex to the perimeter of the base. The term "cone" sometimes refers just to the surface

of this solid figure, or just to the lateral surface.

The axis of a cone is the straight line (if any), passing through the apex, about which the lateral

surface has a rotational symmetry.

In common usage in elementary geometry, cones are assumed to be right circular, where right

means that the axis passes through the centre of the base (suitably defined) at right angles to its

plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in

which the axis does not pass perpendicularly through the centre of the base. In general, however,

the base may be any shape, and the apex may lie anywhere (though it is often assumed that the

base is bounded and has nonzero area, and that the apex lies outside the plane of the base). For

example, a pyramid is technically a cone with a polygonal base.

Pyramid

A pyramid is a building where the outer surfaces are triangular and converge at a point. The base

of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at

least three outer surfaces (at least four faces including the base). The square pyramid, with square

base and four triangular outer surfaces, is a common version.

A pyramid's design, with the majority of the weight closer to the ground, means that less material

higher up on the pyramid will be pushing down from above: this distribution of weight allowed early

civilizations to create stable monumental structures.

Cylinder

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points

at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this

surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and

the volume of a cylinder have been known since deep antiquity.

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a

round ball. Like a circle in three dimensions, a perfect sphere is completely symmetrical around its

center, with all points on the surface lying the same distance r from the center point. This distance r

is known as the radius of the sphere. The maximum straight distance through the sphere is known

as the diameter of the sphere. It passes through the center and is thus twice the radius

(Refer to references A)

4.2 Classify of 3D shapes.

Solid that are 3D shapes are cube,pyramid,cylinder,prisms,cone and spheres.

Cylinder Cone Sphere

Prism Cube Pyramid

4.2.1 There are some ways to classify the 3- Dimensional shapes.There are:

Types of figures

Types of solids

Types of figure

Types of solids

Types of surfaces for solids

Result of movement by solids

Using number faces of the solids

Using corners of the solids

Using base of the solids

Using parallel faces

Plan figures-Polygon,many angle

Solid figures-Polyhedron,many faces

Polyhedra-cube,cuboid,prism,pyramid

Non-polyhedra-Cone,Cylinder,sphere

Types of surfaces for solids

Results of movement by solids

Using number faces of the solids

Flat surface solid-Cube,cuboid,pyramid,prism

Curved surface solid-Sphere

Flat and curved surface and solid-Cylinder,cone

Can be stacked-Cube,cuboid,prism,pyramid

Can be roll and side-Cone,cylinder,sphere

One faces-Sphere

Two or more faces-Cube,cuboid,prism,pyramid,cylinder,cone

Using corner of the solids

Using base of the solids

Using base of solid

Using parallel faces

No corner-Cone,cylinder,sphere

One or more corner-Cube,cuboid,prism,pyramid

No base-Sphere

One or more bases-Cube,cuboid,prism,pyramid,cylinder,cone

No parallel faces-Cone,sphere

Have parallel faces-Cube,cuboid,cylinder,prism,pyramid

4.3 Characteristics of 3-Dimensional shapes

CYLINDER

CONE

SPHERE

CUBE

PRISMS

Prisms can be divided into regular and irregular prisms.

A prisms has two faces that are parallel and congruent.The prisms are name after the shape of the two parallel and congruent faces.For instance,a rectangular prism has bases that are rectangles and a pentagonal prism has bases that are pentagons.

Rectangular Prisms

Pentagonal Prisms

Hexagonal Prism

-6 rectangular faces

-12 edges

-8 vertices

-2 pentagonal faces

-5 rectangular faces

-15 edges

-10 vertices

-2 hexagonal faces

-6 rectangular faces

-18 edges

-12 vertices

PYRAMID

4.3.1 3D Shape Properties in tables

This 3D shape has no flat faces and no straight

edges. It has just one curved face.

It is a cube.

This 3D shape has one curved face and one flat

face. The flat face is a circle.

It is a cylinder.

This 3D shape has 6 flat square faces, 12 straight

edges and 8 corners.

It is a cuboid.

This 3D shape has one curved face and 2 flat

circular faces.

It is a cone.

This 3D shape has 6 flat faces; 2 are squares and 4

are rectangles. It has 12 straight edges and 8

corners.

It is a sphere

4.4 Differentiation between the types of 3-D shapes

CUBOID

Key FeatureSix faces which are all rectangles

Faces 6Corners 8Edges 12

CUBE

Key FeatureSix faces which are all squares

Faces 6Corners 8Edges 12

TRIANGULAR PRISM

Key FeatureA prism with a triangular cross-section

Faces 5Corners 6Edges 9

HEXAGONAL PRISM

Key FeatureA prism with a hexagonal cross-section

Faces 8Corners 12Edges 18

CYLINDER

Faces,Corners and EdgesThe normal definitions of faces,corners and edges are not appropriate for a cylinder.

CONE

Key Feature

The point of the cone is directly above the centre of

the circular base.

Faces,Corners, and edges

The normal definitions of faces,corners and edges are

not appropriate for a con.

SPHERE

(Refer to references B)

4.5 Derivation formula of 3-Dimensional shapes

Shapes Vertices Edges

Corners Formula

8 12 6 Face Area  =  Area of the face = units 2

Surface Area =  Total area of all the faces  =     

units 2

Volume of Cuboid = Length x Breadth x

Height    =  units 3

6 9 Volume of a prism = Area x Length

Example: What is the volume of a prism whose ends are 25 in2 and which is 12 in long:

Answer: Volume = 25 in2 × 12 in = 300 in3

8 12 6 Area of Square = length x breadth = l2

Volume of cube = Ah = l2x h = l3 units3

The volume of a cube

Volume of a cube = a × a × a = a³where a is the length of each side of the cube.

Triangular prism

6 9 5Area of triangle = x base x perpendicular

height = bh

Volume of triangular prism = Ah= bhl units3

( where l is the length of the prism)

0 2 3 Area of circle = π x radius x radius = π r2

so Volume of cylinder = Ah = π r2 units3

V = [Area of circle] x [Height of cylinder]

V = (π x R x R) x H

10 15 7 Area of pentagonal = SA = 2B + Ph

SA = 2(1/2ans) + nsh

SA = 2(1/2a)(5)s + 5sh

SA = 5as + 5sh

Volume =V = Bh

V = 1/2ansh

V = 1/2a(5)sh

V = 5/2ash

12 18 8 Area of hexagonal =Surface Area of Prism =

6as + 6sh = 6as + Ph

Volume of Prism = 3ash = Ah

where

        a = apothem length, s = side, h = height

4 6 Area of Pyramid : SA = B + n(1/2sl)

SA = s2 + (4)(1/2sl)

SA = s2 + 2sl

Volume = V = 1/3Bh

V = 1/3(b2)h

= 1/3b2h

1 1 1 Area of cone = π r2 + πrl

Volume = 1/3 π r2hUsing this knowledge, you can find the volume

of a cone by using the almost same formula of

the cube, but since it takes 3 cones to make 1

cube, we can multiple by 1/3 (or divide by 3; it's

the same thing afteral.

Volume of a Cone vs CylinderThe volume formulas for cones and cylinders are very similar:

The volume of a cylinder is: π × r2 × h

The volume of a cone is: π × r2 × (h/3)

So, the only difference is that a cone's volume is one third (1/3) of a cylinder's

Since the base area of a cone is a circle,again we can substitute the area fomula for a circle into the volume formula,in place of the base area.

Volume of cylinder vs prism

The cylinder is somewhat like a prism.It hass parallel congruent bases,but its bases are circles rather than polygons.You find the volume of a cylinder in the same way that you find the volume of prism:it is the product of the base area times the height of the cylinder:

V = Bh

Since the base of a cylinder is always circle,we can substitute the formula for the area of circle into the formula for the volume,like this:

V =r2 h

Comparison of the volume of 3D shapes

Volume (Vco) of the cone is give by:

Vco = (1 / 3)*area of base * height

= (1 /3) * π r 2 * r = (1 /3) * π r 3

Volume (Vcy) of the cylinder is give by:

Vcy = area of base * height

= π r 2 * r = π r 3

Volume (Vhe) of the hemisphere is give by:

Vhe = (1 / 2) volume of a sphere

= (1/2) (4/3) π r 3 = (2 / 3) π r 3

The volume of the cylinder is the largest. The volume of the cone is one third of the volume of the cylinder and it is the smallest. The volume of the hemisphere is twice the volume of the cone or two thirds the volume of the cylinder.

Volume of square pyramid

1.Identify the length and width of the base.

2.Calculate the area of the base.

3.Multiply the area of the base by the height..

4.Multiply the previous answer by one third, or divide by 3.

Volume of sphere

An easier way to obtain the answer:

Imagine a sphere which is divided into an infinite amount of prisms with a common vertex at the

centre of the sphere. By calculating the volume of all these prisms, one can obtain the volume of the

sphere.

The formula for the volume of a prism is (1/3)bh. If we apply this formula to the infinite number of

pyramids, the total area of the bases (b) would be the SA of the sphere, (4πr2), the height (h), would

be the distance from the surface area to the centre, which is the radius (r).

This means the formula for finding the Volume of a sphere would then be (1/3)bh which is (1/3)(r)

(4πr2) which could then be simplified to (4/3)πr3

(Refer to references C)

4.3.2 Platonic Solid

There are five Platonic Solids.

Each one is a polyhedron with every face being a regular polygon of the same size and shape.

They are also convex (no "dents" or indentations in them).

Tetrahedron

4 Faces 4 Vertices 6 Edges

Cube

6 Faces 8 Vertices 12 Edges

Octahedron

8 Faces 6 Vertices 12 Edges

Dodecahedron

12 Faces 20 Vertices 30 Edges

4.3.3 Some examples of cross-section

Icosahedron

20 Faces 12 Vertices 30 Edges

Example:

The cross section of a rectangular pyramid is a rectangle

Task B

Procedures on how to produce these 3-D shape models

Triangular prism

Example:

The cross section of a circular cylinder is a circle

Example:

The cross section of as quare prism is square

The triangular prism has five faces. three faces are rectangles and two are triangles.

A triangular prism has 6 vertices and 9 edges.Draw three rectangle with the measure is

11.5 cm.Make sure the measure are accurate.Then draw two same size of triangles with

the measure is 4.5 cm.

Score the net along ALL fold lines before attempting to assemble

Use a ruler to aid scoring

Use sticky tape on the tabs to assemble this shape rather than glue. It is much easier.

Hexagonal prism

The hexagonal prism is a prism composed of two hexagonal bases and six rectangular sides. It is

an octahedron. The regular right hexagonal prism of edge length has surface area and volume

Firstly,draw two same size hexagonal bases with the measure is 3.5 cm each side.

Then,draw six same size of rectangular sides which is the measure is 5.52 cm each side.

Score the net along ALL fold lines before attempting to assemble

Use a ruler to aid scoring

Use glue on the tabs to assemble this shape rather than sticky tape. It is much easier.