Geometry

     Quadrilaterals

• Quadrilaterals
  - Parallelogram - Opposite sides and opposite angles are equal
  - Rectangle - All angles are 90°, and opposite sides are equal
  - Rhombus - All sides are equal. Opposite angles are equal
  - Square - All angles are 90°. All sides are equal
  - Trapezoid - One pair of opposite sides is parallel
• Sum of Interior Angles of a Polygon = (n - 2) × 180, where n is the number of sides 
• Perimeter is the sum of the lengths of all sides
• Area
  - Triangle = (Base × Height) / 2
  The base refers to the bottom side of the triangle. The height always refer to a line that is perpendicular (at a 90° angle) to the base
  - Rectangle = Length × Width
  - Parallelogram = Length x Height
  - Cut more complex shapes into rectangles and right triangles, then find the areas of these individual shapes.
  - See p16 in MGMAT Geometry for less common area formulas
• Surface Area = The SUM of the areas of ALL of the faces
• Volume = Length × Width x Height
  - Remember, when you are fitting 3-dimensional objects into other 3-dimensional objects, knowing the respective volumes is not enough. You must know the specific dimensions (l,w,h) of each object to determine whether the objects can fit without leaving gaps.
  
  Triangles & Diagonals
• The sum of the three angles of a triangle equals 180°
• Angels correspond to their opposite sides. The largest angle is opposite the longest side, the smallest angle is opposite the shortest side. If two sides are equal, their opposite angles are also equal.
• Length of sides: (x-y)<(x+y)
• Use Pythagorean Theorem to find the hypotenuse (the side opposite the right angle) of a right triangle: a² + b² = c²
• Common right triangles
  3-4-5                6-8-10
  3² + 4² = 5²      9-12-15
  (9 + 16 = 25)    12-16-20
  5-12-13            10-24-26
  5² + 12² = 13²
  25 + 144 = 169
  8-15-17
  8² + 15² = 17²
  64 + 225 + 289
• An isosceles triangle is one with two equal sides, the two angles opposite these two sides are also equal.
  45° 45° 90°
  leg leg hypotenuse
   x :  x  :  x√2
• An equilateral triangle is one with all three sides (and all three angles) equal.
  Two 30-60-90 triangles make up an equilateral triangle
  30°    60°  90°
  short long hypotenuse
     x  :  x√3 : 2x
• Diagonal of a square: d = s√2, where s is a side of a square
• Main diagonal of a cube: d = s√3, where s is an edge of the cube
• To find the diagonal of a rectangle, you must know either the length and the width or one dimension and the proportion of one to the other
• To find the diagonal of a rectangular solid, you can use the Pythagorean theorem twice or the Deluxe Pythagorean Theorem: d² = x² + y² + z²,where x,y and z are the sides of the rectangular solid and d is the main diagonal.
• Triangles are defined as similar if all their corresponding angles are equal and their corresponding sides are in proportion. Once you find that two triangles have two pairs of equal angles, you know that the triangles are similar, furthermore, if two right triangles have one other angle in common, they are similar triangles.  
• If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a²:b² 
• Be able to see any side of a triangle as the base, not just the side that happens to be drawn horizontally, also be able to draw the height from that base. 
• The area of an equilateral triangle with a side length of S is equal to (S²√3) / 4
• Right triangle/Rectangle DS tips, knowing any two of these will allow to solve for the rest:
  1. Side length 1
  2. Side length 2
  3. Diagonal/Hypotenuse
  4. Perimeter
  5. Area
  
  
• Circles & Cylinders
• A radius is any line segment that connects the center point to a point on the circle
• A chord is any line segment that connects two points on the circle. Any chord that passes through the centre of the circle is called a diameter
• The distance around the circle is termed the circumference: C = 2πr
  A full revolution or a turn of a spinning wheel is equivalent to a wheel going around once, a point on the edge of the wheel travels one circumference in one revolution
• A portion of a distance on a circle is termed an arc. Use the central angle to determine what fraction an arc is of the entire circle (out of a total of 360°)
• The boundaries of a sector are formed by the arc and two radii. Slice of pizza.
• Area of a circle: A = πr² 
• You can find the area of a sector by determining the fraction of the entire area the sector occupies, you can do this by looking at the central angle that defines the sector
• An inscribed angle has its vertex on the circle itself. An inscribed angle is equal to half of the arc it intercepts
• If one of the sides of an inscribed triangle is the diameter of the circle, then the triangle must be a right triangle. Conversely, any right triangle inscribed in a circle must have the diameter of the circle as one of its sides.
• Surface area of a cylinder = 2 circles + rectangle = 2(πr²) + 2πrh
• Volume of a cylinder: πr²h
• If you know the circumference, the radius, the diameter, or the area of a circle, you can use one to find any of the other measurements.
• Sphere
  Surface area: 4πr²
  Volume: 4/3πr³
  
  
• Lines and Angles
• Parallel lines are lines that lie in a plane and that never intersect
• Perpendicular lines are lines that intersect at a 90° angle
• Intersecting lines:
  - the interior angles form a circle, so the sum is 360°
  - angles that combine to form a line sum to 180°
  - angles found opposite each other where two lines intersect are equal. These are called vertical angles.
• An exterior angle of a triangle is equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle.
• Parallel lines cut by a transversal:
  - All acute angles (less than 90°) are equal
  - All obtuse angles (more than 90° but less than 180°) are equal
  - Any acute angle is supplementary to any obtuse angle (they sum to 180°) 
• Complementary angles sum to 90°
• Supplementary angles sum to 180°