Polygons p. 232
Commonly known as shapes, polygons are many-sided figures (poly = many, gon =
side) on a plane, or flat, 2D shapes. These include: triangles (three sides,
three angles), quadrilaterals (four sides), pentagons (five sides), hexagons
(six sides), heptagons (seven sides), octagons (eight sides), nonagons (nine
sides), decagons (ten sides), enneagons (eleven sides), dodecagons (twelve
sides) … etc.
Regular polygons have equal angles and equal sides; regular triangles are also
known as equilateral triangles, while regular quadrilaterals are also known as
squares.
Polygons have invisible diagonals, or lines connecting one vertex or angle to
another vertex or angle through the middle of the shape. A convex polygon has
all diagonals interior to the shape (and each interior angle is less than
180°), while a concave polygon has one or more diagonal exterior to the shape
(and one or more interior angles is greater than 180°).
Triangles are named by their three corners or vertices (e.g.
rABC). The angles always add up to exactly 180°. There are several types of triangles:
Equilateral – triangles having equal sides and equal angles.
Isosceles – triangles having two equal sides and two equal angles.
Scalene – triangles having no equal sides and no equal angles.
Right – triangles having one right angle (90°).
Obtuse – triangles having one angle larger than 90°
but less than 180°.
Acute – triangles having all angles smaller than 90°.
The base of the triangle can be any side of the triangle. The
height or altitude is the perpendicular measure from the base to
the opposite vertex, so it might be interior or exterior to the triangle
depending on the type of triangle.
By contrast, a median goes from the vertex to the midpoint of the
opposite side, and an angle bisector evenly divides an angle. There are
three medians and three angle bisectors to every triangle. Also note the
interesting fact: if the segment is two of these (median, altitude, angle
bisector) it is automatically the third.
If two sides are equal, the opposite angles will also be equal, e.g., if AB =
AC, then
ÐC will equal
ÐB. This is the case for all isosceles triangles.
For equilateral triangles, all sides and angles are equal, so the angles have
to equal 60°.
For other triangles, the longest side will always be opposite the largest
angle, and the other angles will be proportionally smaller and opposite the
proportionally smaller sides. For example, as in a right triangle, where the
longest side is the hypotenuse and is opposite the 90°
angle.
Also note, for every triangle, no matter the type, the sum of any two sides
must be larger than the third side.
An exterior angle will always be equal to the sum of the two opposite angles.
In the example below, x = y + z:
(Alternatively, you can subtract x from 180°.)
The Pythagorean Theorem states that a2 + b2 = c2
for all right triangles. This means that the sum of the squares of the legs of
a right triangle will equal to the square of the hypotenuse.
Pythagorean triples are sets of three numbers that show up as a pattern for
right triangles’ legs and hypotenuses. We have several on our memorization
chart, including those that apply to isosceles right triangles (or 45°-45°-90°
triangles) and special right triangles (or 30°-60°-90°
triangles).
The most common triple is 3, 4, 5 and its multiples (6, 8, 10 and 9, 12, 15,
etc.), followed by the 5, 12, 13. Less common are 7, 24, 25 and 8, 15, 17.
Standardized tests use triples like these frequently.
The isosceles right triangle triple is 1, 1,
Ö2 and its multiples (x, x, xÖ2, for example 2, 2, 2Ö2 or 8, 8, 8Ö2, etc.). Remember that
Ö2
»
1.3, so this side will always be the hypotenuse.
The other special right triangle triple is 1,
Ö3, 2, and its multiples (x, xÖ3, 2x, for example 2, 2Ö3, 4 or 8, 8Ö3, 16, etc). Remember that
Ö3
»
1.7, so it is the measure of the longest leg, while the 2x side will be the
measure of the hypotenuse.
All four-sided figures are quadrilaterals (quadri = 4, lateral = side). The
sum of the interior angles will always equal 360°. The most common types of quadrilaterals are the square, rectangle,
parallelogram, rhombus, and trapezoid, but there are irregular quadrilaterals
that do not fit into these categories.
Square – all equal parallel sides and equal angles, the only truly regular
quadrilateral; the diagonals are equal, bisect each other, bisect the angles,
and are perpendicular at their intersection.
Rectangle – two opposite pairs of equal parallel sides, all equal angles;
diagonals are equal and bisect each other.
Parallelogram – opposite sides are parallel and equal, opposite angles are
equal, consecutive angles are supplementary; diagonals bisect each other but
are not equal unless it is a rectangle.
Rhombus – a parallelogram with equal sides but not necessarily equal angles
though the opposite angles will be equal; the diagonals need not be equal but
they will bisect, be perpendicular, and be angle bisectors.
Trapezoid – will have a pair of parallel sides, but everything else is
undetermined. Note that an isosceles trapezoid has legs of equal measure and
thus diagonals which are equal, and thus also base angles will be equal as
well.
The sum of the interior angles of any polygon can always be determined based
on the number of sides: (n – 2)180°
=
S(Ðint)
This is why triangles have sums of 180°
and quadrilaterals have sums of 360°, etc.
These formulas are included on our memorization chart. The perimeter of any
polygon is always the sum of all sides, but these formulas work as well:
For any square where x is the length of one side, P□ = 4x. The area is the
square of that side, or A□ = x2.
For any parallelogram, including rectangles and rhombuses, where l is the
length and w is the width (alternatively, use b for base and h for altitude or
height), P
= 2l + 2w or 2(l + w), and A
= lw, or length times width (alternatively, use bh or base times height
particularly for non-rectangular parallelograms).
For any triangle where a, b, and c are the sides, Pr
= a + b + c, and where b is the measure of the base and h is the altitude or
height, Ar
= ½bh, or one-half base times height.
Trapezoids are special cases; again, the perimeter is the sum of the sides,
i.e., two bases (b1 and b2) and two legs (x and y), or
P = b1 + b2 + x + y; the area will be
A = ½h(b1 + b2) or one-half the
height times the sum of the bases.
Circles are single-sided figures (or two-sided, if you count them as having an
inside and an outside) that include the set of all points equidistant from a
center point by which it is named.
The parts of the circle are the radius, diameter, chord, and arc. The
perimeter is called the circumference.
We use the measure
p
or pi (pronounced like “pie” for United States English) to calculate the
circumference and the area. Pi is approximately equal to 3.14 (p
»
3.141592658… - pi is a transcendental number and has an infinite,
non-repeating decimal value; as such, it is an irrational real number), and
though our textbook often calculates pi into the answers, most of the time
standardized tests just have
p
as part of the answer.
Radius – the measure from the center to the edge of the circle, represented
with a lowercase r.
Diameter – the measure from one edge to the other passing through the center,
represented with a lowercase d. The diameter equals twice the radius, or d =
2r.
Chord – any line segment whose endpoints are on the edges of the circle. It
need not pass through the center; the longest chord is always the diameter.
Arc – the measure around the edge of the circle from one point to another,
measured in either length or degrees, and named by its endpoints.
The formulas for circles are on our memorization sheet:
The circumference equals twice the radius times pi, or the diameter times pi,
i.e., C = 2pr or
pd.
The area equals the radius squared times pi, or half of the diameter squared
times pi, i.e., A○ =
pr2 or
p(½d)2.
Angles in a circle are used to calculate the measure of arcs. Central angles
are formed by two radii, while inscribed angles have their vertices at the
edge of the circle. The degree measure of a central angle is equal to the
degree measure of the arc, but the degree measure of an inscribed angle is
half the measure of the arc. Contrast the two following examples:
Some other terminology for circles:
Concentric circles are circles with the same central point.
Tangents are lines touching a circle at one point and are always perpendicular
to the radius.
Note that similarity means having proportional measures: all circles are
similar and all squares are similar. However, congruence means having
identical measures.
Volume is the three-dimensional measure of the interior of any solid figure or
prism, while surface area measures the two-dimensional area of each surface of
a solid figure or prism. The formulas are also included on the memorization
sheet. Volume is always given in cubic units, and surface area, like regular
area, is given in square units.
The most important solid figures are the cube, the rectangular solid, and the
cylinder (also known as a right circular cylinder on most exams). I have
included other shapes such as the cone, pyramid, sphere, and other prisms just
in case.
For cubes, volume is equal to one side measure cubed, or
V = x3 where x is the measure of one
side. The surface area is six times that side measure squared, or
SA = 6x2, because there are six sides
for any cube. For rectangular solids, the principles are the same but the formulas are slightly different. Volume will be the product of the three dimensions or V = lwh, for length times width times height, and surface area will be the sum of twice the area of each of the three pairs of similar sides, or A = 2lw + 2lh + 2hw. A right circular cylinder has bases which are circular and a lateral side that is perpendicular to the bases. The volume will be the product of the area of the base and the height, or V = pr2h. The lateral surface area is the measure of the area of the rectangle wrapped around the cylinder, which equals the product of the base’s circumference and the height of the cylinder, or LSA = 2prh, while the total surface area of a cylinder is the sum of the area of the two bases and the lateral surface area, or total SA = 2prh + 2pr2.
The book gives most of these formulas in a chart on p. 272-274.
