Geometry : General Formulas

Mid segment of a triangle

a mid segment connects the midpoint of 2 sides of a triangle and is equal to ½ the side not containing the 2 midpoints.

Pythagorean theorem

a squared plus b squared equals c squared. the hypotenuse is always c.

Polygons

a triangle has 3 sides. a quadrilateral has 4 sides. a pentagon has 5 sides. a hexagon has 6 sides. a octagon has 8 sides. a decagon has 10 sides.

Reflection over the x axis

change the sign of the y coordinate.

Area of a trapezoid

½ times the height times the sum of the 2 parallel sides.

Supplementary angles

2 angles when added together that equal 180 degrees.They do not have to be adjacent angles.

Complementary angles

2 angles when added together that equal 90 degrees. They do not have to be adjacent angles.

Surface area of a cylinder

2 times pi times radius times the sum of the height and radius.

Lateral surface area of a cylinder

2 times pi times the radius times the height.

Surface area of a rectangular prism

2 times the sum of the length times the width, the length times the height. and the width times the height.

Surface area of a sphere

4 times pi times the radius squared.

Area of a rectangle

Base times height or length times width.

Area of a square

Base times height or side squared.

Area of a parallelogram

Base times the height

Area of a rhombus

Base times the height

Area of a triangle

Base times the height divided by 2.

Transformations

Dilations are a capital D. Reflections are a lower case r. Rotations are a capital R. Translations are a capital T.

Dilation

Multiply both coordinates by the scale factor.

Trapezoid

One set of parallel sides.

Isosceles trapezoid

One set of parallel sides. Non-parallel sides are congruent. Base angles are congruent. Diagonals are congruent.

Rhombus

Opposite sides are parallel. All sides are congruent. Diagonals are congruent and Perpendicular(make right angles). Diagonals bisect each other and the Angles of the rhombus.

Median of a trapezoid

The top parallel side plus the other parallel side divided by 2.

Angles

acute angles are less than 90 degrees. Right angles are 90 degrees. obtuse angles are between 90 and 180 degrees. Straight angles are 180 degrees and reflex angles are greater than 180 degrees.

Triangles classified by angles

acute triangles have 3 acute angles. right triangles have a 90 degree and 2 acute angles. obtuse triangles have an obtuse and 2 acute angles.

Alternate interior angles

alternate interior angles are congruent.

Similar triangles

angles in similar (∼) triangles are congruent. sides are in proportion. angles are in a proportion of one to one. (1:1)

Median

bisects the opposite side into 2 congruent line segments. they meet in a triangle at a point called the centroid. median segments are in a ratio of 2 to 1.

Proving triangles congruent

can not be angle angle side (A.S.S.) or side side angle(S.S.A.).

Reflection over the y axis

change the sign of the x coordinate.

Corresponding angles

corresponding angles are congruent.

Perimeter

distance around the outside of a polygon.

Intersection of a tangent and a radius

form right angles when they intersect.

Volume of a sphere

four thirds times pi times the radius to the thrid power.

Slope

from left to right. up the hill is positive slope. down the hill is negative slope. a horizontal line has 0 slope and a verical line as an undefined slope.

Rotation of 180 degrees

graph the coordinate. rotate the graph 180 degrees counter clockwise and read the point.

Rotation of 270 degrees

graph the coordinate. rotate the graph 270 degrees counter clockwise and read the point.

Rotation of 90 degrees

graph the coordinate. rotate the graph 90 degrees counter clockwise and read the new point.

Slopes of parallel lines

have the same slope and a different y intercept.

Volume of a rectangular prism

length times width times height.

Proving triangles similar

need only 2 angles to be congruent to probe 2 triangles similar.

Volume of a cone

one third times pi times the radius squared times the height.

Volume of a pyramid

one third times the area of the base times the height. the base could be a triangle, rectangle, or square

Square

opposite sides are parallel. all sides are congruent and meet at right angles. diagonals are congruent and perpendicular(make right angles). diagonals bisect each other and the angles of the square.

Parallelograms

opposite sides are parallel. opposite sides and angles are congruent. consecutive angles are supplemental (add to 180). diagonals bisect each other.

Rectangle

opposite sides are parallel. opposite sides are congruent. contains 4 right angles. diagonals are congruent and bisect each other.

Volume of a cylinder

pi times radius squared times the height.

Lateral surface area of a cone

pi times the radius times the slant height.

Triangles classified by sides

scalene triangles have no equal sides. isosceles triangles have at least 2 equal sides. equilateral triangles have 3 equal sides.

Adjacent angles

share a common vertex, a common side, but not common interior points.

Isosceles triangles

sides opposite congruent angles are congruent. angles opposite congruent sides are congruent.

Reflection over the line y=x

switch the x and y coordinates.

Sum of the angles in a triangle

the 3 angles of a triangle add up to 180 degrees.

Exterior angle of a triangle

the exterior angle of a triangle equals the sum of the 2 opposite interior angles.

Find the endpoint of a line given the midpoint and the other endpoint

the integers added to the coordinates of C to get the coordinates of M are added to the coordinates of M to get the coordinates of D.

Distance formula

the square root of x two minus x one squared plus y two minus y one squared.

Triangle inequality theorem

the sum of 2 sides of a triangle must be greater than the 3rd side.

Formula for angles of polygons

the sum of the interior angles equals the number of sides minus 2 times 180 degrees. the sum of the exterior angles equal 360 degrees.

Midpoint formula

x one plus x two divided by two is the x coordinate. y one plus y two divided by two is the y coordinate.

Slope intercept form of a line

y equals slope (m) times x plus the y intercept (b)

Point slope form of a line

y minus y one equals slope (m) times x minus x one.

Finding the slope and y intercept of a line

y must be positive and isolated on one side of the equation.

Slope formula

y two minus y one divided by x two minus x one. rise over run.