Today's soundtrack is Rhye: Blood.
This morning, I'm working on chapter 16 of Basic Math & Pre-Algebra for Dummies, "Picture This: Basic Geometry."
Geometry, "one of the most useful areas of math" (p. 217), is "the mathematics of figures such as squares, circles, triangles, and lines" (p. 217). I remember from my other readings that geometry was regarded as a thing of beauty by the Greeks; they saw it as the method of understanding the universe.
Plane geometry examines figures in 2d (a plane). A plane continues infinitely in all directions in a flat line. There are four main concepts comprising plane geometry: "points, lines, angles, and shapes" (p. 218).
Points:
A point is a place or location on a plane; it has "no size or shape" (p. 218), and though it is not actually large enough to be visible, we represent them with dots. Intersecting lines share a point; "each corner of a polygon is a point" (p. 218).
Lines:
A line is the shortest distance between two points, and it extends infinitely on a straight path in each direction. Because it has only length but no width, it is a 1d figure. Two points determine the path of a line. As mentioned in the above paragraph, if two lines intersect, they share a point; if they never intersect, they are parallel. There are three kinds of lines: straight lines have arrows on both sides and continue infinitely in both directions. A line segment has a definite start and stop point, both indicated by dots. A ray has a definite start point (indicated by a dot) and continues infinitely in the other direction (indicated by an arrow).
Angles:
Angles are formed when one point is the starting place of two rays. We use angles to tell us how sharp a corner is. We measure angles in degrees; they are measured from 0-360 degrees. Angles that are 90 degrees are right angles; angles less than 90 degrees are acute angles; angles greater than 90 degrees are obtuse angles; angles that are exactly 180 degrees form a straight line and are called straight angles.
Shapes:
Shapes are "any closed geometrical figure that has an inside and an outside" (p. 221). The area of a shape is how much space exists inside the shape. Polygons, however many sides they have, has all straight sides. A triangle, one kind of polygon, has three sides; there are four kinds of triangles: equilateral triangles have three sides the same length with 60 degree angles on all three sides; isosceles triangles have two sides of equal length and two equal angles; a scalene triangle has three different lengths and three different angles; a right triangle has one right angle, and can be either isosceles or scalene. Quadrilaterals, another kind of polygon, have four straight sides. Squares, rectangles, rhombuses, parallelograms, trapezoids, and kites are all kinds of quadrilaterals. There are two other kinds of polygons: regular polygons and irregular polygons. Regular polygons have equal side lengths and equal angles: pentagons and hexagons, for example. Irregular polygons look like random shapes drawn in straight lines. Circles have an inner part called a radius; any line segments that reach both the outer part of the circle and the radius are the same length, so the circle is perfectly round. There are no straight edges in circles; the Greeks "thought that the circle was the most perfect geometric shape" (p. 224).
Solid geometry looks at objects in three dimensions. Every solid shape "has an inside and an outside" (p. 225). A polyhedron is a polygon in three dimensions; thus, it only has straight edges, and is solid. A cube is a polyhedron; it is made up of six flat-faced square-shaped polygons. A tetrahedron is a pyramid; it is made of four flat-faced triangle-shaped polygons. Many solids have one or more curved surfaces; these are not polyhedrons. Examples include spheres, cylinders, and cones.
MEASURING POLYGONS can involve looking at either a shape's perimeter (the length of its sides) or area (how big it is inside). Area is always measured in square units.
When measuring perimeter of a shape, "s" = "side," and "P" = "perimeter"; so when we say that the formula for finding the perimeter of a square is "P = 4 x s", and the sides of the square are 5" each, the perimeter of the square is 20". To find the area of the same square, we say that A (area) = s²; therefore, the area of the square is 16"².
To measure a rectangle, we refer to its length as "l", and its width as "w". The formula for finding the perimeter of a rectangle is P = 2 x (l + w). The formula for finding a rectangle's area is A = l x w.
To measure a rhombus, we use s to represent the length of a side, and we use h to represent its height, which is "the shortest distance from one side to the opposite side" (p. 230). To find the perimeter of a rhombus, we use "P = 4 x s" (the same as a square); however, to find the area of a rhombus, we calculate it as A = s x h, similarly to how we find the area of a rectangle.
To measure a parallelogram, we consider its top and bottom bases (b), its sides (s), and the shortest distance between its bases (h). To find the perimeter of a parallelogram, we calculate P = 2 x (b + s). To calculate the area of a parallelogram, use the formula A = b x h.
To measure a trapezoid, we look at its parallel bases (b1 and b2) and its height (h), which is the shortest distance between the bases. Add the lengths of the trapezoid's four sides to find its perimeter; to find its area, calculate A = 1/2 x (b1 + b2) x h.
Measuring triangles is quite straightforward; just add up the lengths of the sides to find the perimeter, and calculate A = 1/2 x b x h to find its area. The base of a triangle is the length of one side. The height of the triangle sits at a right angle to the base.
To measure a right triangle, we look at its hypotenuse (c) (the long side of the triangle) and the two short sides (a and b), called "legs". We can use the measurements of these legs in the Pythagorean theorem, which is a² + b² = c², through which we can find the length of a right triangle's hypotenuse.
Measuring circles requires us to consider the centre of the circle; the radius (r), which is the distance from the centre of the circle to its outer edge; and the longest distance across the circle, called the diameter (d). Two radii makes the diameter; thus, d = 2 x r. Instead of finding a circle's perimeter, we say that we are finding its circumference (C), which is calculated with this formula: C = π x d. To find the area of a circle, we calculate A = π x r².
MEASURING IN THREE DIMENSIONS involves measuring volume (V) instead of area; volume is the measurement of how much space an object takes up. It is measured in cubic units: for example, 4³, or four cubed.
Measuring a cube requires knowing the length of one of its sides (s); the formula is V = s³.
Measuring a box requires us to consider its length (l), width (w), and height (h). To find the volume of the box, we use the formula V = l x w x h.
Measuring a prism and measuring a cylinder both require knowing only two measurements: the height (h) and the area of the base (Ab). The formula is V = Ab x h.
Measuring a cone and measuring a pyramid both use the formula V = 1/3 x Ab x h.
