BASIC MATH 12 -- GEOMETRY
VOLUME OF CYLINDERS
V = Bh so V = pi r squared x h (area of the circular base x the height)
VOLUME OF CUBES & RECTANGULAR PRISMS
V = LXWXH (basically V = Bh (cubes in base x number of layers in height) LENGTH X WIDTH X HEIGHT (note: HEIGHTH IS INCORRECT)
ANGLES FORMED BY INTERSECTING LINES:
VERTICAL ANGLES SUPPLEMENTARY ANGLES
TANGENTS
a trig ratio (opposite/adjacent of right triangle) but also illustrated in context of a circle: literally a tangent line or POINT OF TANGENCY
GRAPH EQUATION FOR A CIRCLE (using Pythagorean Theorem)
(very good, by Khan Academy) very perspicuous!:) Ludwig Wittgenstein Quote: "A mathematical proof must be perspicuous."
EXTERIOR ANGLES OF A TRIANGLE***
***
SUMS OF INTERIOR ANGLES OF POLYGONS
***SEE PIC, esp bottom
PYTHAGOREAN THEOREM
*there are at least 350 proofs for the theorem! NOTE: Pythagorus: Greek philosopher and mathematician from 6th century BC supposedly, however, the theorem was intuitively understood as sseen on Babylonian tablets 1800 BC
Rhombus has special features in addition to those of a parallelogram
1 its diagonals bisect the angles they connect 2 its diagonals are perpendicular 3 it is a parallelogram
45 45 90 Triangle
1. both legs are the same length 2. the hypotenuse is square root of 2 times as long as either leg
Parallelogram is special in two ways
1. its opposite angles are congruent 2. its diagonals bisect each other
RIGHT PRISM
1. the congruent and parallel faces of this prism are its bases 2 the lateral faces are rectangles 3 the altitude of this prism is one of its edges, thus making it a right prism can be other polygon based figures like in pic: square prism
30-60-90 Triangle
1. the hypotenuse is twice as long as the short er leg 2. the longer leg is square root of 3 times as long as the shorter leg
AREA OF A CIRCLE
1. using radius to cut the circle into slices, arrange then the slices to fit, one beside the other to resemble a parallelogram. 2 notice that the smaller the slice, the more slices and the less deviation in the lines of the figure 3 ultimately, as the slices become thinner, the shape becomes more like a simple rectangle. the area of a rectangle is b x h 4. what is the circle's height? the radius (the height of every slice) (as the slices become thinner and thinner, and a rectangle is rendered) 5. what is the length of the rectangle? outlay the circumference: 1/2 of circumference (for half slices on top, half slices on bottom) 6. C = pi x diameter 7. so 1/2 (pi x diameter) x radius 1/2 (pi x 2r) x r pi x r x r pi r squared = area of circle
VOLUME OF PYRAMIDS
1/3 Bh (in other words 1/3 the volume of a rectangular prism with the same base) Oblique Pyramid: same: V = 1/3 Bh fascinating!: see Volume of a Pyramid, Deriving the Formula by MathematicsOnline - you tube
TRIG RATIOS (_?__)
6 Note: cosecant, secant, and cotangent are simply the inverses (or reciprocals) of sin, cosine, and tangent
SHAPE NET FOR AN OBLIQUE CYCLINDER
?WHAT ABOUT CYLINDER WHOSE PARALLEL FACES ARE ELLIPSES RATHER THAN CIRCLES? (THAT'S WHAT THIS SHAPE NET REPRESENTS) SAME WAY TO FIND VOLUME? notice how the falling/laying of the bases (in reforming the oblique cylinder) is governed/determined by the concave/convex contour of the lateral face
TRIANGLE AREA
A = 1/2 BH (a triangle is half of a parallelogram/rectangle)
AREA OF PARALLELOGRAM
A = bh
AREA FOR CYCLINDER
A=2πrh+2πr2 !!!!!!!!!!!!!!!! (rectangle represents area of the rectangular portion formed by the unfurling of the lateral face) 2 (pi x radius squared - area of each circle base) + pi x diameter (circumference of circle which equates to the length of the rectangle rendered upon unrolling of the cylinder's lateral face) x h (height of the cylinder is the same as the width of the rectangle) question: oblique cylinders....? how to?
A PYRAMID IS NAMED ACCORDING TO THE SHAPE OF ITS ______________
BASE triangular, square, hexagonal, octagonal, etc (all faces are triangles)
ANGLES FORMED BY LINES CUT BY A TRANSVERSAL
CORRESPONDING ANGLES ALTERNATE EXTERIOR ANGLES ALTERNATE INTERIOR ANGELS ***if the two lines are parallel, then the formed angles have special relationships
pi
For all circles, the ratio of the circumference to the diameter (C/d) is lwyas the same: approximately 3.14, or 22/7 (called pi) ITERATIVE; ITERATION: (iterate, from iterum:again) make repeated use of a mathematical or computational procedure, applying it each time to the result of the previous application, typically as a means of obtaining successively closer approximations to the solution of a problem
INTERESTING!!!
GOD HAS CREATED THE WORLD/UNIVERSE IN SUCH A WAY, AND MAN IN SUCH A WAY -- INCLUDING IN HIS IMAGE -- SO THAT MAN MIGHT RULE OVER AND SUBDUE HIS CREATION UNTO GOD'S GLORY ALONE. GOD HAS ESTABLISHED PRINCIPLES, PERHAPS GREAT ONES FROM SMALL ONES; AND REASON, AND SO FORTH. HE HAS IMBUED MAN WITH THE REASON, DISCERNMENT, AND MORE. IN THIS, GOD HAS GIVEN MAN THE GIFT OF LEARNING AND CREATING AND SUBDUING/USING FOR GOOD,NOT FOR EVIL, TO GOD'S GLORY ALONE. YET IT IS ALSO WRITTEN, "THE WRATH OF MAN SHALL PRAISE HIM(GOD)."
CONGRUENT
HAVING THE SAME SHAPE AND SIZE; angles - degrees sides - length
AREA OF RECTANGLE
L X W example: 8 and 4: 8 squares, four times
ANY TRIANGLE IS HALF OF A ____________________--
PARALLELOGRAM
CIRCUMFERENCE
PI x Diameter
SOLID FIGURES
PRISMS CUBES & RECTANGULAR PRISMS PYRAMIDS CYLINDERS CONES SPHERES
PYTHAGOREAN THEOREM (PROOF 2) Proof by __________________-
Rearrangement
SURFACE AREA OF PRISMS
SA = 2B + LA (surface area = area of both bases, + area of the lateral faces - LA) trapezoid prism: make a SHAPE NET for, then figure out 2B and LA
SURFACE AREA OF CONE proof
SA = B + LA SA = pi r-squared + pi x r x s (slant height)
SURFACE AREA OF PYRAMID
SA = B + LA (base + lateral area) B + 4(1/2bh)
POLYHEDRON & PRISM
a 3-dimensional closed figure with faces that are polygons. Prism: polyhedron with at least two faces that are congruent and parallel
QUADRILATERAL
a closed 2-dimensional figure with four sides that are line segments
POLYGON
a closed figure whose sides are all line segments
TESSELLATIONS
a covering of a plane without overlaps or gaps using combinations of congruent figures rectangles tessellate equilateral triangles tessellate rectangles with equilateral triangles tessellate regular pentagons don't tessellate
SCALE DRAWING
a drawing that is the same shape but not the same size as the object it shows/represents ex: blueprints and maps
LINE SYMMETRY
a figure has line symmetry if it can be folded such it has two parts matching exactly. the fold line is called the LINE OF SYMMETRY
TURN SYMMETRY
a figure that rotates onto itself before turning 360 degrees has turn symmetry
PLANE
a flat surface that extends indefinitely in all directions; a collection of points that that forms a flat surface infinitely wide and infinitely long
DIAGONAL of a polygon
a line segment whose endpoints are two non-adjacent vertices notice formula...
TRANSFORMATIONs
a process by which one figure, expression, or function is converted into another that is equivalent in some important respect but is differently expressed or represented. ex: TRANSLATION (sliding) ROTATION (turning) REFLECTION (flipping) DILATION (changing the size of a figure) *label the corresponding points on the transformation image with the same letters, but also with a prime sign (X to 1, Y to 1, etc, instead of XYetc)
CIRCLE
a set of points, all of which are the same distance from a given point that point is the center of the circle circle A = circle whose center is point A VERY INTERESTING (referencing proof for volume of circle): what is a point? no dimension. God's eternal attributes in things seen. + God created things seen with things unseen. -- the proof for V of circle had to rely on "approaching infinity") interesting
LINE
a straight path of points that has no endpoints. it has length, but no width an infinite set of points forming a straight path extending in both directions infinitely
TRIANGLE
a three-sided polygon plus height: perpendicular from vertex to base
TRIANGLE types
acute equiangular obtuse right isosceles scalene (no congruent sides) obtuse
ANGLE NAMES BY DEGREES
acute: less than 90 right: 90 obtuse: more than 90, less than 180 straight: 180 REFLEX: more than 180, less than 360
SOME FIGURES THAT ARE ALWAYS SIMILAR
all equilateral triangles all 45 45 90 triangles all 30 60 90 triangles all squares all circles ***any two REGULAR POLYGONS that have the same number of sides (reg polygon: closed figure with all sides and angles equal)
VOLUME OF PRISMS
amount of space inside the prism; Volume is measured in cubic units V = Bh V is volume of a prism: (and for oblique prism) B is number of cubic units needed to cover the base h is number of layers
POINT
an exact position in space; no dimension
ANGLES
angles are formed by two rays with a common endpoint called the VERTEX see pic: may name angle: angle ABC angle CBA angle B angle 1
CENTRAL ANGLES
angles formed by intersecting diameters and radii in a circle
INTERIOR ANGLES
angles formed by sides of the polygon, inside the figure
CONGRUENT ANGLES
angles with the same measure
VOLUME OF TRIANGULAR PRISM
basically, the volume of its rectangular prism counterpart: V = L x W x H (except you divide it by 2 since a triangular prism is only half the size/volume of its rectangular counterpart)
TRANSLATIONS
called a slide (every point in the figure slides the same distance in the same direction. Use a slide arrow to show the movement.
COMPLEMENTARY
combining in such a way as to enhance or emphasize the qualities of each other or another. harmonious, compatible
COORDINATE GEOMETRY uses a ____________________ grid, called the _____ __________ _________. It's formed by a vertical y axis that intersects at 90 degrees a horizontal x axis; their point of intersection is called the ___________. any point consists of 2 numbers, called _____________.
coordinate; Cartesian Coordinate System, origin, coordinates
OPEN CURVE vs CLOSED CURVE
curve: A BENT LINE OR nameLINE SEGMENT
AREA OF TRAPEZOID PROOF 2*** real cool (most direct for comprehending the formula
cut in half, with line parallel to the other two parallel lines rotate top to fit along right side; this renders a parallelogram! then take the top length, (a) which has been rotated to rest along the bottom base line, and add to the original base (b), (they form base 1 plus a (base 2) then multiply their sum by 1/2 h (height)
AREA OF TRAPEZOID formula and proof 1
essentially: area of a rectangle + area of a triangle base1xh + 1/2(base2-base1)h
OVERLAPPING CLASSIFICATIONS OF TRIANGLES
ex: isosceles may be acute, right, obtuse right may be isosceles, scalene, etc
TERMS FOR SOLID FIGURE face edge vertices
face: flat surface of a solid figure edge: line segment at which two faces of a solid figure meet vertices: corner points of a solid figure
SIMILAR FIGURES
figures having the same shape, but not necessarily the same size
CONCAVE
having a surface or outline that curves inward like the interior of a circle or sphere LATIN con - together cavus - hollow
CONVEX
having an outline or surface curved like the exterior of a circle or sphere
CONGRUENT
having exactly the same size and shape
ARCS OF A CIRCLE
major and minor; central angles: angles formed from the center
TOPOLOGICALLY EQUIVALENT
if you can make a figure into another by stretching, shrinking, or bending it AND without connecting or disconnecting points, then the two figures are topologically equivalent. Ex: topologically equivalent to an S: Z 1 2 3 5 7 not topo eq to an S: A X 0 4 6 8 9 Note: all polygons are topologically equivalent to each other***:)
CONVEX vs CONCAVE POLYGON
in a convex polygon all the diagonals (endpoints = two non-adjacent vertices) are inside the closed figure; in a concave polygon, one or more of the diagonals are outside/partly outside the closed figure Note: convex: vaulted, curved; having outside like exterior of circle or sphere concave: having outline like interior of a circle or sphere
CONE
in geometry, a cone is a 3-dimensional figure with one circular base; a curved surface connects the base to the vertex.
PYRAMID
in geometry, a pyramid is a : polyhedron (3-D closed figure with multiple sides that are polygons) with 1 a single base that is a polygon, and 2 faces that are triangles*:) apothem = slant height apex = vertex
ROTATIONS
in geometry, means turning the figure around a point called the TURN CENTER or POINT OF ROTATION
PERPENDICULAR BISECTOR
line, line segment, ray, or plane that forms a right angle with a line segment and divides that segment into two congruent parts
PARALLEL LINES
lines in the same plane that never cross because they are always the same distance apart
SKEW LINES
lines that are neither parallel nor intersecting; there is no one plane that can contain them
INTERSECTING LINES
lines that do cross - they either have one point or all points in common - if two lines intersect, there is one plane that can contain them
PERPENDICULAR LINES
lines that form right angles where they intersect
coordinate grid: quadrants:
moving counter-clockwise from upper-right: first (+++), second (-+), third (- - ), fourth (+-)
RAY
part of a line that has one endpoint and extends indefinitely in the other direction
LINE SEGMENT
part of a line that has two endpoints
archimedes
pi
PLANE FIGURES
polygons triangles special characteristics of right triangles quadrilaterals circles similarity congruence transformations tessellations topology
REGULAR PYRAMID
pyramid whose vertex is directly above (perpendicular to) the center of its base which is a regular polygon
ANGLE BISECTOR
ray that separates an angle into two congruent angles
REFLECTIONS
reverse image of what you're looking at has been flipped so to speak
You can also use the Pythagorean theorem to tell whether you can make a ______ _______ from three given line segments.
right triangle
SPHERES
sa = 4 pi r squared (the area of the circle that defines/renders/engenders the sphere would have to be multiplied 4 xs to cover the entire area of the sphere
RIGHT TRIANGLES: SPECIAL CHARACTERISTICS
sides of: hypotenuse and leg(s) if one angle is 30 degrees, then one leg will be 1/2 as long as the hypotenuse! such relationships are called TRIGONOMETRIC RATIOS surveyors, astronomers, and engineers use trig ratios to find distances or angles that are impossible or impractical to measure directly
SHORTCUT FOR FINDING AREA OF RECTANGULAR PRISM
since rectangular prism has three pairs of congruent sides, simply: SA (surface area) = 2lw + 2lh + 2wh(widthxheight)
TRIANGLE INEQUALITY THEOREM
stated two ways: 1. the sum of the lengths of any two sides of triangle is greater than the length of the third side (visualize motion) 2.in other words: the third side is always less than the sum of the other two sides
AREA
the area of a figure is the number of square units inside the figure
OBLIQUE PRISM
the bases of this prism are hexagons and the faces are parallelograms because the altitude of this prism is not one of its edges, this is an oblique prism
PERIMETER
the distance around a polygon LATIN peri around metron measure
VOLUME OF SPHERES
the volume of a sphere is 2/3 that of a cylinder! V = 4/3 PI R CUBED (tentative explanation of proof is that sphere is comprised of square pyramids, the apexes of which meet at the center. Each square pyramid is 1/3 volume of its counterpart cylinder, or 1/3 Bh. Height is the radius of the sphere, essentially. Base, when added to the bases of all the other square pyramids which comprise the sphere, represents the total surface area of the sphere, which is already understood to be 4pi r squared. So 4pi r squared x 1/3 x base (already covered in first part) x height (radius) = volume of sphere. So: 4 pi r squared x 1/3 x r (height) = 4/3 pi r cubed [however, my question: the bigger the square pyramids, the less adaptable to the concept above; the smaller the square pyramids -- approaching infinity -- the less adaptable? are do they just infinitely get smaller and smaller? one might say, but there must be a point of finality. yet is that how God has created creation? It appears finite, but smaller becomes smaller, and larger larger. God's majestic signature in his works? ***Romans 1:20 20 For since the creation of the world His invisible attributes are clearly seen, being understood by the things that are made, even His eternal power and Godhead, so that they are without excuse,
classifying QUADRILATERALS
trapezoid: exactly one pair of parallel sides parallelogram: opposite sides the same length and parallel rectangle: parallelogram with four right angles rhombus: parallelogram with four congruent sides square: rhombus with four right angles; a special rectangle***:)
SUPPLEMENTARY ANGLES
two angles whose sum is 180 degrees ***supplementary (basic definition): completing or enhancing something
COMPLEMENTARY ANGLES
two angles whose sum is 90 degrees
OBLIQUE PYRAMID
vertex is not directly above the center of its base.
PYTHAGOREAN THEOREM (PROOF 1) Euclid's Proof (and Einstein when 12?)
very eloquent proof***:)
VOLUMES OF CONES
volume of a cone is 1/3 the volume of its cylinder counterpart (same base area B, and same height h) so, V = 1/3 x pi r squared x h (height) (proof seems to involve math I've not learned yet)
