Maths: Geometry

Congruence

Similar shapes:similar shapes are identical is shape, but not in size, two similar shapes will have equal angle distributions, you can work out if two shapes are mathematically similar by checking if their side lengths are at the same ratio.scale factors:to work out the scale factor of similar shapes, you take a length from one shape, and divide it by the corresponding length in another, this is the scale factor, you can now multiply other sides by this number to convert them between shapesarea and volume scale factors:to calculate area and volume scale factors, you work out the scale factor and then square it for area or cube it for volume, now use this as the new scale factorcongruence:congruent shapes are identical in shape and size, a shape is still congruent if it has been reflected or rotated

Parts:a chord of a circle is a straight line going from one point on the circumference to another,an arc is a section of the circumference between two radii, it looks like a slice of pizza, a segment is the area within a chord, there will be a minor segment and a major segment, a sector is the area within an arc,Area of a sector:too find the area of a sector you just do: angle of the sector / 360 x pi x r^2There is a rule that the perpendicular from the center to the chord bisects the chord, meaning that a line going from the center of the circle straight down to the chord will bisect it as a perpendicular line, this is useful as it means you create two congruent right-angled triangles on either side of the line, this is how you prove that its a perpendicular bisector

Circles

Polygons

A polygon is a 2D closed shape with straight sides,Interior angles:The sum of the interior angles in a polygon are equal to 180 x (n - 2), where n = the number of sides,In triangles, the exterior angle is equal to the sum of the other 2 interior angles, so in a triangle ABC, the exterior angle of C is the same as the sum of A and B, just thought you should know.....

3-D Shapes

The volume of a pyramid is 1/3 x base area x perpendicular heightThe volume of a cone is 1/3 x pi x r^2 x hThe curved surface area of a cone is pi X rl, where l = slant heightvolume of a sphere is 4/3 x pi x r^3surface area of a sphere is 4 x pi x r^2Prisms:a prism is a shape with a regular cross section throughout, to calculate the volume of a prism, multiply the area of the cross section by the length

In graphs, the origin is the point where the x and y axis cross, this point is often the basis for translations of shapes, Bearings are written in three figures, you can preface them with 0's if the angle is lower than 100, bearings always start at 0 degrees facing north, and they move in a clockwise direction,A locus is a path formed by a point which moves according to a rule, for instance, the locus of a clock hand would be a circle around the hand because that is the path the hand takes, too draw a locus around a polygon, you must use a rounded edge for the corners,Constructions are accurate diagrams drawn using a pair of compasses and a ruler, you can use this for constructing things like the perpendicular distance which is the shortest distance from a point to a line,to construct a perpendicular bisector you must open a compass slightly wider than half the length of the line,place your compass on one end of the line segment and draw an arc above and below the line,maintaining the same compass width, do the same on the other side, place a ruler where the arcs cross and draw a line between them,to construct an angle bisector,you place the compass on the point where the lines meet and draw an arc between the lines about half way up the lines, then you move the compass up to where you drew the arc and draw another arc in the middle of the angle, then repeat for the other side, then where those arcs cross you draw a line through them from the intersection of the original linesto construct a perpendicular from a point on a line, you put la compass on the point, open it so it touches near to the edge of the line, then draw a dot on either side of the line, then put the compass on the first dot and draw an arc below the line, same for the second dot, then connect the arc intersect with the point,

Coordinates and bearings

Sine rule

The sine rule is a/sin(A) = b/sin(B) = c/sin(C) or sin(A)/a = sin(B)/b = sin(C)/c, this basically means that if you know "a" and sin("A"), which is the length of a side and that sides opposite angle, and you know either the length or angle of another part of the triangle (sine rule works for any triangle) you can put the information into this equation and get an answer.To do this you must rearrange the equation to make the missing angle or side the subject,so if you want to find the length of a side, you get a/sin(A) = b/sin(B) and multiply both sides by  sin(A)  to get a = ( b / sin(B) ) * sin(A), and if you want the angle you would turn the equation upside-down, i.e. sin(A)/a = sin(B)/b and then multiply both sides by a which gets you sin(A) = (sin(B) / b) * a, bear in mind this gets you sin(A) not "A" so to get the angle you then have to inverse sin the answer.

Cosine rule

The cosine rule is: a2 = b2 + c2 - 2bc cos A, this can be used to find a missing side on any triangle, however it requires you to already know two sides and an angle. bear in mind that this equation returns a squared, not a, so to get the correct value you must square root the answer,The cosine rule can also be rearranged to get: cos(a) = (b2 + c2 - a2) / 2bc, which can be used to find a missing angle in a triangle where you already know all three sides, once again you should bear in mind that this returns cos(a) not a so to get the correct value you gotta inverse cos the answer.

3-D Trigonometry and Pythagoras

to do 3-D trigonometry or Pythagoras, you basically just construct 2-D triangles in 3-D shapes, then use trigonometry or Pythagoras as you normally would to find the correct value