Geometry Flashcards
Area of a Triangle
area = ½ base × height
Angles in a Triangle
All angles in a triangle
sum to 180
x + y + z = 180
Pythagorean Theorem
In a right triangle, a2 + b2 = c2,
where a and b are the legs
(shorter sides) and c is the
hypotenuse.
Pythagorean Triples
The most common triples are 3–4–5,
5–12–13, 7–24–25, and 8–15–17.
Similar Triangles
Similar triangles have all the
same angles, and sides that
are proportionate.
Third Side Rule
The third side of a triangle is greater than
the difference of the other two sides and
less than their sum.
(a + b) > c > (b − a)
Angle/side Relationship
Side length corresponds with
angle size. Angle z and side c are
the largest. Angle x and side a
are the smallest.
Special Right Triangles
Isosceles and Equilateral
equilateral = all sides are the same length
isosceles = two sides are the same length
In an equilateral triangle, all angles are 60°.
Area of a Rectangle
area of a rectangle = length × width
area of a square = side2
Area of a Parallelogram
area of a parallelogram = base × height
Sum of Angles in a Polygon
sum of angles in a polygon = 180(n − 2),
where n = number of sides
Volume and Surface Area of a Box
volume = l × w × h
surface area = 2lw + 2wh + 2lh
Volume and Surface Area
of a Cylinder
volume = πr2h
surface area = 2πr^2 + 2πrh
Lines and Angles
All angles in a line add up to 180°
x + y = 180
Parallel Lines
When a line intersects with two parallel angles, the resulting intersections are identical.
Intersecting Lines
When two lines intersect, opposite angles are equal.
Distance and Midpoint on a Coordinate Plane
If you have endpoints (x1, y1) and
(x2, y2) of a line on a coordinate plane:
Sector
Sector area and arc length are proportional to angle size. Here, the angle is 60°.
Since there are 360° in a circle, the sector (grey area) takes up 1/6 of the circle. Thus,
its area is 1/6 of the total area, and its arc length is 1/6 of the circumference.
Slope, x-, y-intercept
y = mx + b, where m = slope, b = y intercept
slope = rise/run or (y2-y1)/(x2-x1)
The y-intercept is the y-coordinate on
a line at which x = 0.
The x-intercept is the x-coordinate on
a line at which y = 0.
Circles Facts
diameter = 2r
circumference = πd or 2πr
area = πr2
Central Angle (circles)
A central angle has a vertex that
lies at the center point of the
circle. An inscribed angle has its
vertex on the circle itself rather
than on the center of the circle.
An inscribed angle is equal to
half of the arc it intercepts.
Triangles in Circles
If one of the sides of an
inscribed triangle is a diameter
of the circle, then the triangle
must be a right triangle.
Coordinate Plane
- Slope: (change in y / change in x) (rise / run)
- Slope-intercept: y = mx + b
- “Line contains point” means you could plug a coordinate pair, f.e. (5, d)
into a line, f.e. y = 3x + 4, by replacing x = 5 and y = d … d = 3(5) + 4 - parallel lines (do not intersect … same slope)
- perpendicular lines (slope is opposite reciprocal .. the product of their
slope is -1) - finding distance between two points (make a right. triangle where the
distance is the hypotenuse)
Lines/Angles
- angles on a line add up to 180º
- when two lines are parallel, all the small angles are the same, and all the
big angles are the same - if we’re told lines are parallel, find the clone angles
Circles/Cylinders
- d = 2r Circ = π*d Area = π * r^2
- Arclength = (angleº / 360º) *circumfer.
- Area of sector = (angleº / 360º) * Area
- Capacity of cylinder: Vol = π r^2 * h
- Volume of liquid: Vol = π r^2 * h (of liquid)
- Rectangle inscribed in circle? Diagonal = Diam
- Circle inscribed in square? Diameter = Side
- Triangle inscribed in semi-circle? Right triangle.
- Two of triangles sides are radii? Isosceles
Polygons
- Triangles have 180º. Every time you add a side, you add another 180º.
n – 2 * 180 = total degrees (4 sides: 360º, 5 sides: 540º, 6 sides: 720º ..) - Area of Trapezoid = (Avg. of Bases) * Height
- Area of Rhombus = ½ (product of diagonals)
- Volume of a Rectangular Solid: Vol = lwh
- Diagonal of Rectangular Solid: a^2+b^2+c^2 = d2
- Volume of Cube: Vol = s^3
Triangles/Diagnoles
- 4 Special Right Triangles:
3:4:5 / 5:12:13 / 45-45-90 (x : x : x√2) / 30-60-90 (x : x√3 : 2x)
√2 = approx. 1.4 (2/14)
√3 = approx. 1.7 (3/17)
- Diagonal of square = hypotenuse of 45-45-90
- Two equal sides <—> Two equal angles
- Area of equilateral triangle = (s^2 * √3) / 4 (or break into two 30-60-90 and solve that way)
- Difference < 3rd side of a triangle < Sum
- Exterior angle rule (if an angle is supplemental to one of a triangle’s three
angles, it’s equal to the sum of the triangle’s other two angles.) - SIMILAR TRIANGLES (triangles with the same set of three angles will
have proportionate sides) .. this usually occurs when there are mini triangles
within bigger triangle
