Solid Geometry[1]

1+

PRINCIPLES IN



=1+

 1!

+

   1  2!

+⋯

Polyhedrons Polyhedrons are solids whose faces are plane polygons.

Regular polyhedrons are those which have identical faces. There are only five known regular polyhedrons, namely tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. These solids are known as Platonic solids.

Let m

= number of polygons meeting at a vertex, n =

number of vertices of each polygon, f

=number of faces of the polyhedron, e =number

of edges of the polyhedron, and v =number

of vertices of the polyhedron. a =

length of the edge

For any polyhedron    =    =

 2  

 Solids for which Volume = (Area of base) x (Altitude) Solids in which V = Ab × h This is a group of solids in which the volume is equal to the product of the area of the base and the altitude. Two types of solids belong to this group namely, prisms and cylinders A solid is said to be prismatic or cylindrical if every cutting plane parallel to base are the same in both shape and size. If the base is a closed polygon the solid is a prism, whereas, the solid is a cylinder if the base is a closed loop of a curve line. The name of the prism and cylinder is according to the shape of its base. A prism with a pentagon base is called a pentagonal prism and a cylinder of circular base is called circular cylinder. The volume of these two solids is given by the formula

 =  ℎ where V = volume, Ab = area of the base, and h = altitude.

prism Prisms are polyhedron whose bases are equal polygons in parallel planes and whose sides are parallelograms. Prisms are classified according to their bases. Thus, a hexagonal prism is one -whose base is a hexagon, and a regular hexagonal prism has a base of a regular hexagon. The axis of a prism is the line joining the centroids of the bases. A right prism is one whose axis is perpendicular to the base. The altitude "h" of a prism is the distance between the bases.

,  =   ℎ =    ,  =    Ar  = Area of right section L = Lateral edge  Ab = Area of base Pr  = Perimeter of right section

Cylinder  A cylinder is the surface generated by a straight line intersecting and moving along a closed plane curve, the directrix, while remaining parallel to a fixed straight line (called the axis) that is not on or  parallel to the plane of the directrix. Like prisms, cylinders classified according to bases.

are their 

,  =  ℎ =    ,  =  

 Ar  = Area of right section L = Lateral edge  Ab = Area of base Pr  = Perimeter of right section

Right Circular  Cylinder 

,  =  h  =   ℎ  ,   = 2ℎ



 Solids for which Volume = (Area of base) x (Altitude) 

This is a group of solids in which the volume is equal to one-third of the product of base area and altitude. There are two solids that belong to this group; the pyramid and the cone.

Like cylinder and prism, the name of pyramid and cone is according to the shape of the base. If the base is pentagon, the pyramid is called pentagonal pyramid, and if the base is circle, the cone is called circular cone. The formula for the volume of these solids is

1  =   ℎ 3 Where Ab = area of the base, and h = altitude.

cone  The surface generated by a moving straight line (generator) which always passes through a fixed point (vertex) and always intersects a fixed plane curve (directrix) is called conical surface. Cone is a solid bounded by a conical surface whose directrix is a closed curve, and a plane which cuts all the elements. The conical surface is the lateral area of the cone and the plane which cuts all the elements is the base of the cone.

1  =   ℎ 3

 

Where Ab = area of the base, and h = altitude.

 The Right Circular Cone  Any cone with circular right section is a circular cone. Right circular cone is a circular  cone whose axis is perpendicular to its base. Properties of Right Circular Cone The   slant height  of a right circular cone is the length of an element. Both the slant height and the element are denoted by L. The   altitude   of a right circular is the perpendicular drop from vertex to the center of the base. It coincides with the   axis  of the right circular cone and it is denoted by h. If a right triangle is being revolved about one of its legs (taking one leg as the axis of revolution), the solid thus formed is a right circular cone. The surface generated by the hypotenuse of the triangle is the lateral area of the right circular  cone and the area of the base of the cone is the surface generated by the leg which is not the axis of rotation. All elements of a right circular cone are equal. Any section parallel to the base is a circle whose center is on the axis of the cone. A section of a right circular cone which contains the vertex and two points of the base is an isosceles triangle. •

•

•

•

,  =

1 3

  ℎ

 ,   = 

•

•

Slant height, L  =

  + ℎ

pyramids A pyramid is a polyhedron with a polygon base of any shape, and all other  faces are triangles which have common vertex.





Properties of a Pyramid  The lateral faces are all triangles meeting at the vertex of the pyramid  The   altitude   of the pyramid is shortest distance between the vertex and the base. It is the drop distance from the vertex perpendicular to the base.  If a cutting plane parallel to the base will pass through the pyramid, the smaller  pyramid thus formed is similar to the original pyramid. By similar solids

   

=

ℎ 

If two pyramids have equal base area and equal altitude, any section made by a cutting plane parallel to the base are equal. From the figure, if  =  then  =  The pyramid is said to be a right pyramid if the vertex is directly above the centroid of the base, otherwise it is an oblique pyramid

Regular pyramids A regular pyramid is one whose base is a regular polygon whose center coincides with the foot of the perpendicular dropped from the vertex to the base PROPERTIES OF A REGULAR PYRAMID  The edges of a regular pyramid are equal; it is denoted by e.  The lateral faces of a regular pyramid are congruent isosceles triangles (see figure).  The altitudes of the lateral faces of a regular  pyramid are equal. It is the slant height of the regular pyramid and is denoted by L.  The altitude of the regular pyramid is perpendicular to the base. It is equal to length of the axis and is denoted by h.  The vertex of regular pyramid is directly above the center of its base when the pyramid is oriented as shown in the figure.  If a cutting plane is passed parallel to the base of regular pyramid, the pyramid cut off is a regular pyramid similar to the original pyramid.

Formula for Regular pyramids Area of one lateral face

  = Lateral Area, AL

1 2



  =     =  2    = 2

Length of lateral edge, e

=

 + ℎ 

Slant height, L

= =

  + ℎ   + /2



Volume, V

1     =   ℎ = ℎ =  ℎ 3 6 6

Where Ab = area of the base (regular polygon) A1 = area of one lateral face AL = lateral area AT = total area  x = length of side of the base h = altitude of pyramid (this is the length of axis of the pyramid) L = slant height of pyramid (this is the altitude of triangular face) P = perimeter of the base e = length of lateral edge For x, R, r, n, and θ, see The Regular Polygon

 Truncated Prism the part of a prism between the base and a nonparallel plane that cuts the prism.

,  = 

,  =  

ℎ 

ℎ + ℎ + ⋯ ℎ−  + ℎ 

Where: AR = area of the right section n = number of sides.

frustum Frustum of a pyramid (cone) is a portion of pyramid (cone) included between the base and the section parallel to the base not passing through the vertex Formula for Volume of a Frustum The volume of a frustum is equal to onethird the product of the altitude and the sum of the upper base, the lower base, and the mean proportional between the bases. In

=

1 3

  +   +

  ℎ

Lateral Area, AL   (Right Frustum) The lateral area of frustum of regular pyramid is equal to one-half the sum of the perimeters of the bases multiplied by the slant height.

 =

1 2

  +  

 The Sphere 

Sphere is a solid bounded by closed surface every point of which is equidistant from a fixed point called the center.

Properties of a Sphere  Every section in the sphere made by a cutting plane is a circle. If the cutting plane passes through the center of the sphere, the section made is a   great circle; otherwise the section is a  small circle.  For a particular circle of a sphere, the axis is the diameter of the sphere perpendicular to the plane of the circle.  The ends of the axis of the circle of a sphere are called poles.  The nearer the circle to the center of the sphere, the greater is its area.  The largest circle in the sphere is the great circle.  The radius (diameter) of the great circle is the radius (diameter) of the sphere.  All great circles of a sphere are equal.  Every great circle divides the sphere into two equal parts called hemispheres.  The intersection of two spherical surfaces is a circle whose plane is perpendicular to the line joining the centers of the spheres and whose center is on that line. (See figure to the right.)  A plane perpendicular to a radius at its extremity is tangent to the sphere.

FORMULAS FOR A SPHERE  Surface Area, A The surface area of a sphere is equal to the area of four great circles.

  = 4    =  

Volume, V

= =

4 3 1 6

   

 Spherical Zone  A zone is that portion of the surface of the sphere included between two parallel planes.

Area of the Zone The area of any zone (one base or two bases) is equal to the product of its altitude h and the circumference of the great circle of the sphere.

A = circumference of the great circle x altitude

  = 2ℎ Properties of Spherical Zone  The   bases   of the zone are the circumference of the sections made by the two parallel planes.  The   altitude   of the zone is the perpendicular distance between these two parallel planes.  If one of the bounding parallel planes is tangent to the sphere, the surface bounded is a zone of one base.

Note that when h = 2R, the area of the zone will equal to the total surface area of the sphere which is 4πR2.

 Spherical Sector  A spherical sector is a solid generated by revolving a sector of a circle about an axis which passes through the center of the circle but which contains no point inside the sector. If the axis of revolution is one of the radial sides, the sector thus formed is spherical cone; otherwise, it is open spherical sector.

Formulas for Spherical Sector Total surface area, A The total surface area of a spherical sector is equal to the area of the zone plus the sum of the lateral areas of the bounding cones.

  = 2ℎ +  +    = (2ℎ +  + ) Note that for spherical cone, b = 0 and the equation will reduce to

  = (2ℎ + )

Properties of Spherical Sector  Spherical sector is bounded by a zone and one or two conical surfaces.  The spherical sector having only one conical surface is called a spherical cone, otherwise it is called open spherical sector .  The base of spherical sector is its zone.

The volume of spherical sector, either open spherical sector or spherical cone, is equal to one-third of the product of the area of the zone and the radius of the sphere. This is similar to the volume of a cone which is Vcone   = 1/3 A bh. In spherical sector, replace Ab with Azone and h with R.

=

2 3

  ℎ

 Spherical Segment  Spherical segment is a solid bounded by two parallel planes through a sphere. In terms of spherical zone, spherical segment is a solid bounded by a zone and the planes of a zone's bases.

Properties of Spherical Segment •The   bases   of a spherical segment are the sections made by the parallel planes. The radii of the lower and upper sections are denoted by a and b, respectively. If either a or b is zero, the segment is of one base. If both a and b are zero, the solid is the whole sphere. •If one of the parallel planes is tangent to the sphere, the solid thus formed is a   spherical  segment of one base. •The spherical segment of one base is also called spherical cap and the two bases is also called spherical frustum. •The  altitude   of the spherical segment is the perpendicular distance between the bases. It is denoted by h.

 Spherical Segment formulas Total Area, A The total area of segment of a sphere is equal to area of the zone plus the sum of the areas of the bases.

  = 2ℎ +   +     = (2ℎ +  +  ) Volume, V The volume of spherical segment of two bases is given by

=

1 6

ℎ(3  + 3  + ℎ )

The volume of   spherical segment of  one base is given by

=

1 3

ℎ  (3  ℎ)

 Spherical Wedge and Spherical Lune  A spherical wedge is a solid formed by revolving a semi-circle about its diameter by less than 360°. Spherical Lune is the curve surface of the wedge, it is a surface formed by revolving a semi-circular arc about its diameter by less than 360.

Volume of wedge, Vwedge Volume of wedge / central angle = Volume of sphere / 1 revolution

 

=

    0

=

   70

Area of Lune,Alune Area of Lune / central angle = Area of sphere / 1 revolution

 

=

  0

=

   90

 Similar Figures Two surfaces or solids are similar if any two corresponding sides or planes are proportional.

The areas of similar surfaces A1 and A2 have the same ratio as the squares of any two corresponding lines x1 and x2

 

 

=









The volumes of similar solids V 1 and V2 have the same ratio as the cube of any two corresponding lines x1 and x2 In similar figures of any kind, pairs of corresponding line segments such as   ,   and  ,  have the same ratio.

 

=

 

 

=







