SOLID GEOMETRY Flashcards
REFERS TO ‘MANY’ ‘FACES’ . CONSIST OF PLANES POLYGONS AS ITS FACE,
POLYHEDRA / POLYHEDRON
TETRA/HEXA/OCTA/DODECA/ICOSA HEDRONS HAS HOW MANY FACES?
4/6/8/12/20
FOR TETRA/HEXA/OCTA/DODECA/ICOSA HEDRONS, WHAT IS THE POLYGON OF EACH OF ITS FACE
TRIANGLE CUBE TRIANGLE PENTAGON TRIANGLE
FOR TETRA/HEXA/OCTA/DODECA/ICOSA HEDRONS, HOW MANY VERTICES DOES EACH OF THEM HAVE?
4/8/6/20/12
HOW DO YOU GET THE NUMBER OF EDGES IN A POLYHEDRONS
F-E+V=2
#OF FACES - #OF EDGES + #OF VERTICES = 2
WHAT IS THE FORMULA FOR THE VOLUME OF A TETRA/HEXA/OCTA/DODECA/ICOSA HEDRONS
TETRA: since this a three faced triangle,
V= √2/(12) s^3
HEXA:since this is a cube,
V= s^3
OCTA: since this is a back to back pyramid,
V= √2 /(3) s^3
DODECA: idk,
V= 7.66 s^3
ICOSA: V= 2.18 s^3
s= length of one side
WHAT IS THE FORMULA FOR THE surface area OF A TETRA/HEXA/OCTA/DODECA/ICOSA HEDRON
TETRA: since this a three faced triangle,
SA: √3 s^2
HEXA:since this is a cube,
V= 6s^2
OCTA: since this is a back to back pyramid,
V= 2√3 s^2
DODECA: idk,
V= 20.65 s^2
ICOSA: V= 5√3 s^2
WHAT IS THE FORMULA FOR THE radius of an inscribed sphere witin A TETRA/HEXA/OCTA/DODECA/ICOSA HEDRO.
r of sphere = 3Vol / SA
describe a prism
- a polygon with added length
- stacked up polygon
- a polyhedron with 2 faces parallel and congruent and whose remaining faces are parallelogram
a triangular prism is also called a ? and its volume can be computed as?
wedge
-mukang tent
v = 1/2bhl
two main parts of a PRISM
THE LATERAL FACES AND THE BASE
`
BASE = POLYGON SHAPES
LATERAL FACES= PARALLELOGRAM THAT CONNECTS THE TWO PARALLEL POLYGON SHAPE
A TRIANGULAR PRISM’S LATERAL SURFACE AREA and total surface area IS COMPUTED AS
SA= l(a+b+c)
SA(total) = l(a+b+c) + bh
a square/rectangular prism can be called as ______ or ______
cuboid or rectangular parallelipiped
a cuboid’s volume,lateral surface area & total surface area can be computed as
V= (a b c) SA(lateral)= 2(ab+ac) SA(total)= 2(ab+ac+bc)
three Classification/TYPE of a prism
RIGTH AND OBLIQUE PRISM
RIGTH IS WHERE THE LATERAL FACE IS PERPENDICULAR TO THE BASE AND OBLIQUE IS WHERE IT’S NOT.
TRUNCATED PRISM IS WHERE THE LATERAL EDGES HAVE DIFFERENT LENGTH!!!
lateral SURFACE AREA OF RIGTH PRISM & oblique prism FORMULA
rigth SA(lateral)= (Pb)(h)
Pb= perimeter of base h= height
oblique SA(lateral= (Pk) (L)
Pk= perimeter of a polygon 'formed' that is perpendicular to the length of the lateral face. L= length, since height and length is different as it is not parallel to the base.
total SURFACE AREA OF RIGTH PRISM & oblique prism FORMULA
rigth SA= (Pb)(h) + 2 (Ab)
Ab= area of base/polygon Pb= perimeter of base h= height
oblique SA= (Pk) (L) +2(Ab)
volume OF RIGTH PRISM & oblique prism FORMULA
rigth V= (Ab) (h)
oblique V= (Ab) (h) = (Ak) (L)
VOLUME OF A TRUNCATED PRISM
V= Ab(h1 + h2 +h3+ ….hn)/5
V=Ab(average height)
T or F cyclinder is a prism, why?
False, since circle is not a polygon, therefore a cylinder is not a prism.
RIGHT VS OBLIQUE cylinder
right is where the bases is placed directly above each other, oblique is like the leaning tower of pisa where it lean and the sides are not perpendicular to the bases.
formula for lateral surface area of a RIGHT AND OBLIQYE CYLINDER
SA= 2πr h RIGHT!!!! TANGINA PAGOD NA KO HAHAHA
SA= 2πr L OBLIQUE
you can also use the h- height
where h= L sinθ (from the right angle form in the inclination)
formula for total surface area of a RIGHT AND OBLIQYE CYLINDER
total right SA= 2πrh+ 2Ab = 2πr+2πr^2
Ab= base area = πr^2
total oblique SA= 2πrL +2Ab
formula for VOLUME of a RIGHT AND OBLIQYE CYLINDER
V = Ab (h) = πr^2h right or oblique!!!
Ab= πr^2 =area of base h= vertical height = Lsinθ
is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles.
prismatoid OR PRISMOID
VOLUME FOR PRIMOID
V= h/6 (A1+4A2+A3)
A1 &A3 = end areas
A2= area at mid section
h=distance betw end areas…
prismoidal formula!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
7 common prismoid famillies
PYRAMID- has 1 plane & 1 vertex PRISM PARALELLOPIPED FRUSTUM COPULA WEDGE ANTIPRISM
parts of a pyramid
apex = the tip base= the polygon base height- vertical length perpendicular the the base slant height- length between base to tip element- edge of triangle
Pyramid’s lateral surface area, and total surfacearea & volume formula.
Al=1/2 P L
At= 1/2 P L + Ab
V=1/3 Ab h
P= perimeter of base L= slant height Ab= Area of base
a shape where it has a circular base and an apex
cone!!! ays kreem
cone’s lateral surface area, and total surfacearea & volume formula.
Al= πr√(h^2 + r^2) At= πr( r + √(h^2 + r^2)) V= 1/3 πr^2 h
s^2=h^2 + r^2 = pythagorean
h= vertical height s= slant height r= radius
a pyramid or cone that has its top part cut off, like cupcake or buckeet.
frustum
frustum’s lateral surface area, and total surfacearea & volume formula.
Al= (P1 + P2)/2 (l) At= (P1 + P2)/2 (l) + B1 + B2 V= h/3 (B1B2) (√(B1B2)
l= slant height of the frustum
P1 & P2= perimeter of the bases
B1 & B2= area of the bases
h= height of figure
a polyhedron defined by two trangles & 3 trapezoid faces
WEDGES
wedge’s lateral surface area, and total surfacearea & volume formula.
Al= bh + 2ls As= bh + 2ls + bl V= 1/2 bhl
s= slant height of the wedge l = length of the base b= width of the base h= height of the wedge
a polyhedron having two faces that are identical polygons in parallel planes but rotated from each other while the other faces are a series of alternately oriented triangles
antiprism`
a prismatoid formed by joining two polygon
CUPOLA
WHAT IS THE SHAPE OFTHE EARTH
SPHEREOID SINCE IT SPINS ( OVAL LIKE)
SURFACE AREA OF A SPHERE
volime OF A SPHERE
SA=4πr^2
V=4/3 πr^3
different figures you can slice from a sphere
ex: orange
Orange sphere as SOLID slices: spherical; *segment *sector *wedge *pyramid
Orange sphere surface: (skin) spherical; *zone *lune *polygon
the half of a sphere is called a ?
hemishpere
a segment of a sphere that is less than a hemisphere is called? give the v formula.
spherical CAP
V=(πh^2)/3 (3R-h)
R= radius from the center of the main SPHERE
r= radius from the segment center
two types of SPHERICAL SEGMENT
ONE BASE AND TWO BASE SPEHRICAL SEGMENT
- ONE BASE IS SLICING IT ONCE TO OBTIAN A SEGMENT
- 2 BASE MEANS SLICING IT TWICE.
OTHER TERM FOR TWO BASE SPHERICAL SEGMENT. GIVE V FORMULA.
SPHERICAL FRUSTUM.
V=(πh)/6 (3a^2+ 3b^2 + h^2
a= radius 1
b= radius 2
h= height between the two circle base.
what do you call the surface area of a spherical segment. also give the laterAL SURFACE AREA FORMULA.
spherical zone:
LSAsegment= Z = 2πRH R= is radius from the main sphere r= radius of the segment.
a slice from a sphere that is shaped in a combination of a cone and a spherical cap. (like a snowcone! yum!)
Spherical sector
volume formula for spherical sector
V= 2/3 πR^2 h
volume formula for wedge (ex: a Piece of orange pulp group)
V= (πR^3) (θdeg)/270
the surface area of a SPHERICAL WEDGE
SPHERICAL LUNE
DERIVE THE FORMULA FOR SPHERIAL LUNE (OTHER VOLUME FORMULAS IN SPHERICAL SLICES CAN ALSO BE DERIVED THIS WAY)
SA(lune) = kθ
[since the surface area is directly proportional to the angle of the lune we can subtitute it with a proportionality constant k]
SA(lune)/θ = k = SA(total)/θ(total)
[k can also be expressed as the total of each measurement where θ of the sphere = 360, SAtotal of the sphere = 4πr^2]
so,
SA(lune)= (4πR^2 (θ) / 360)
simplifying will give us;
SA(lune)= (πr^2 (θ) /90)
what do you call the largest circle in a sphere, largest circle that can be form (greatest diameter)
within the hemishpere,
they are called the great circles`
the surface area of a polygon in a sphere formed by different circles around the sphere.
SPHERICAL POLYGON\
SA=πR^2 (E) / 180
E= angles - (n-2) (180) (in degrees) E= largest spherical excess is 720
the solid formed by a spherical polygon when vertices is all connected to the center of the sphere
SPHERICAL PYRAMID
Vpyramid = kE
so,
V= πR^3 (E) / 540
a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
ELLIPSE`
equation of an ellipse
1= ((x-h)^2 / a^2) + ((y-k)^2 / b^2)
a= semi major= larger= length of axis b= semi minor= smaller h= location of vertex= x coordinate k= y coordinate
ellipsoid
a solid formed when an ellipse if revolved arond an axis. aka A SPHEROID, LOOKS LIKE A MUSHED BALL, earth, foot ball.
two types of SPHEROID
PROLATE : ROTATED ABOUT THE MINOR AXIS
AND
OBLATE : ROTATED ABOUT THE MAJOR AXIS
VOLUME OF PROLATE AND OBLATE
Vgeneral = INTEGRATING THE ELLIPSE
Vprolate=4/3π (a)(b^2)
Voblate = 4/3π (a^2)( b)
b=length of semi-munir axis
a= length of semi-major axis
s a plane curve which is mirror-symmetrical and is approximately U-shaped. HALF OF AN ELLIPSE
PARABOLA
PARABOLA EQUATION
(x-h)^2 = 4a(y-k)^2
h= x coordiante OF THE VERTEX k= y coordinate O THE VERTEX a= distance from the vertex to the FOCUS
A PARABOLA REVOLVED AROUND THE AXIS OF SYMMETRY (LOOKING LIKE A SATELLITE DISH)
PARABOLOID
V OF A PARABOLOID
imagine the paraboloid is inside a cylinder. like a double glass mug
V = 1/2 Vcylinder = 1/2πr^2 h r= radius of circumscribing cylinder h= height of circumscribing cyliner
a solid od revolution obtained by rotating a circle about a line not intersecting with it. (DONUT!!!)
TORUS
v AND Lateral area of a torus
V= 2(π^2) (R) (r^2) A= 4π^2Rr
r= radius of the CIRCLE being rotated (katawan ng donut) R= distance between the ceter of the TORUS TO THE RADIUS OF THE CIRCLE. ( gitna ng donut papunta sa gitna ng katawan)
