NOTES: A quick overview of 2-D geometry
What is 2-D geometry?
 Also called plane geometry, it’s the geometry that deals with two dimensional
shapes—flat things that have length and width, such as a piece of paper.
 You should be familiar with the following 2-d shapes:
 2 sets of congruent, parallel sides
 4 right angles
rectangle
 actually just a special kind of parallelogram—
one that has 4 right angles
 2 sets of parallel sides
 4 congruent sides & 4 right angles
square
 actually just a special kind of rectangle—one
that has equal length & width
 2 sets of congruent, parallel sides
 height (h) is straight down from the
highest point to form a right angle
with the base; never the slanty side
 half a parallelogram/rectangle/square
 height (h) is straight down from the
highest point to form a right angle
with the base; never the slanty side
parallelogram
h
triangle
h
h
h
b
b
b
right
triangle
equilateral
triangle
isosceles
triangle
scalene
triangle
h
b
trapezoid
b1
h
b2
pentagon,
hexagon,
octagon
circle
d
r
 Triangles are classified by sides:
equilateral (all sides equal), isosceles (2
sides equal), and scalene (no sides equal).
 They’re also classified by angle:
right (one 90), obtuse (one >90),
acute (all <90)
 has 2 bases: b1 & b2
 height (h) is straight down from b1 to
b2; never the slanty side
 half a parallelogram with base b1 + b2
 5 sides, 6 sides, 8 sides
 diameter (d) = distance from 1 side to
the other through the center
 radius (r) = ½ the diameter; distance
from the center to the edge
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What are some useful bits of vocabulary to know?
 polygon: a closed 2-d shape made with straight line segments
 regular polygon: a polygon with all sides the same length
regular hexagon
hexagon that isn’t regular
 base (lowercase “b”): the side of certain 2-d shapes at right angles to
(perpendicular to) the height; usually but not always the “bottom”
 congruent: identical in shape and size; the word we use for “equal” when we’re
talking about shapes instead of numbers; symbolized by  ( ABC   DEF)
D
A
and hash marks on diagrams
B
C
F
E
 parallel: always the same distance apart; symbolized by || (L1 || L2) L1
L2
 perpendicular: at right angles (90); symbolized by  (L3  L4)
and on diagrams with
L3
L4
What is perimeter, and how do you calculate it?
 Perimeter is the distance around a shape.
 Think of counting your steps with a pedometer as you walk around.
 Perimeter is always measured in regular units: in., ft., cm, m. It’s a distance.
EXAMPLE #1: What’s the perimeter of the rectangle below?
2.1 m
6.3 m
You could use the formula from the formula sheet:
P = 2L + 2w
= 2(6.3) + 2(2.1)
= 16.8 m
Or you could just add up all the sides.
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EXAMPLE #2: What’s the perimeter of a regular octagon with a side length of 7 in?
There’s no formula here. Since an octagon has 8 sides, the perimeter is
8(7) = 56 in.
EXAMPLE #3: What’s the perimeter of the room shown below?
Remember to find the unmarked sides!
Then add them up:
P = 13 + 24 + 13 + 4 + 5 + 10 + 5 + 10
= 84 ft.
EXAMPLE #4: If the perimeter of a rectangle is 12.6 m and the length is 4.2 m,
what’s the width?
4.2 m
LEVEL 2
?
?
4.2 m
Without algebra:
4.2 + 4.2 = 8.4 Since P = 12.6, there’s 12.6 – 8.4 = 4.2
left to be divided between each side.
The width is 2.1 m
With algebra:
P = 2L + 2w
[from the formula sheet]
12.6 = 2(4.2) + 2w
12.6 = 8.4 + 2w
4.2 = 2w
w = 2.1 m
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What is area, and how do you calculate it?
 Area is the space inside a 2-d shape.
 Think of covering something with little 1 ft  1 ft area rugs.
 Area is always measured in square units: in2, ft2, cm2, m2.
 Don’t confuse perimeter and area!
 See the formula sheet.
EXAMPLE #1: What’s the area of the rectangle below?
2.1 m
6.3 m
A = Lw
= (6.3)(2.1)
= 13.23 m2
EXAMPLE #2: What’s the area of a square with side length 11 in?
You could use the formula sheet: A = s2
= 112
= 121 in2
Or you could recognize that a square is just a special kind of rectangle,
namely, one with length the same as width. A = Lw
= (11)(11)
= 121 in2
EXAMPLE #3: What’s the area of the parallelogram below?
6.3 m
2.1 m
First, notice that if you slide the triangular piece
to the right, you get the rectangle of EXAMPLE #1
so the area should be the same as it was there.
To get the area here, you multiply (6.3)(2.1) = 13.23 m2
Instead of “length” and “width” we just have “base” and “height.”
Base and height always form a right angle. Height is never the slanty side.
Note that a rectangle is actually just a special kind of parallelogram, namely,
one in which there are 4 right angles. You can think of the area of a rectangle
as really A = bh, too.
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4m
EXAMPLE #4: What’s the area of the triangle on the right?
First, notice that a triangle is really half a parallelogram.
3m
4m
3m
3m
3m
3m
3m
Since the area of a parallelogram is A = bh,
the area of a triangle should be half that,
and it is: A = ½ bh
Just as for parallelograms, height on triangles
is always perpendicular to the base. It’s never
the slanty side.
For the original triangle, the base is 6 m (3m + 3 m) and the height is 4 m.
A = ½ bh
= ½ (6)(4) = 12 m2
Remember that taking ½ of something (multiplying it by ½) is the same as
dividing by 2. So you could just multiply base by height and divide the
result by 2 instead.
8 ft
6 ft
EXAMPLE #5: What’s the area of the trapezoid on the right?
We can’t just do A = bh with the bottom base
16 ft
since that would be the area of the big dotted rectangle
below, and that’s too big. But if we used the other base
(perpendicular to the height), we’d get a rectangle that’s too small.
16 ft
6 ft
6 ft
8 ft
8 ft
16 ft
What we need is the average of the big base and the little base.
Well, to find the average of 2 numbers, you add them and divide by 2.
Another way to say that is to add them and take ½ of the result.
From this we can make sense of the formula for the area of a trapezoid:
A = ½ (b1 + b2) h
[I’ve moved the h to make the formula more intuitive.]
= ½ (16 + 8)(6)
= 72 m2
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This page has harder “backwards” examples.
EXAMPLE #6: If the area of a rectangle is 16.32 cm2 and the width is 3.4 cm,
what’s the length?
LEVEL 2
A = Lw
16.32 = 3.4L
L = 4.8 m
EXAMPLE #7: If the area of a square is
LEVEL 2
4
25
mi, what’s the length of 1 side?
A = s2
4
25
= s2
s=√
4
25
=
2
5
mi.
0.2 m
EXAMPLE #8: If the area of the parallelogram shown is 0.08 m2, what’s the base?
A = bh
LEVEL 2
0.08 = 0.2 b
b = 0.4 m
EXAMPLE #8: If a right triangle has one leg length of 12 inches and an area of
30 in2, what’s the length of the other leg?
LEVEL 2
Note that since it’s a right triangle,
one leg is the base and the other is
the height.
12 in
m
A = ½ bh
30 = ½ (12)h
h = 5 in
EXAMPLE #8: If the area of a trapezoid is 36 yds2, its height is 4 yds,
and one base is 10 yds, what’s the other base?
LEVEL 3
A = ½ (b1 + b2) h
36 = ½ (10 + b2) (4)
36 = 2 (10 + b2)
36 = 20 + 2b2
16 = 2b2
b2 = 8 yds.
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How do you find the circumference & area of circles?
 Calculations with circles all involve the irrational number , a decimal that goes
on FOREVER without repeating. The GED test formula sheet and many books
suggest using 3.14 as a good enough approximation of .
 Circumference for circles is like perimeter for polygons.
Imagine putting a “fence” around a circle and finding its length.
 Circumference is a length, so it’s in regular units: inches, feet, meters, etc.
 There are 2 formulas on your sheet for circumference: 2r and d
Since the diameter (d) is twice the radius (r), these are equivalent.
Use whichever is more convenient.
EXAMPLES:
1) What’s the circumference of a circle
with a radius of 5 m?
C = 2r
= 2(3.14)(5) = 31.4 m
2) Rounded to the nearest foot,
what’s the circumference of a circle
with a diameter of 3 ½ ft?
C = d
= (3.14)(3 ½) = 10.99  11 ft
3) If a circle has a circumference of 35.168 cm, what’s the radius?
C = 2r
35.168 = 2(3.14)(r)
r = 5.6 cm
LEVEL 2
 The circle area formula on your sheet (A = r2) is only in terms of radius.
If you’ve got the diameter, remember to divide this by 2 to get the radius.
Remember that area is ALWAYS in square units.
EXAMPLES:
1) To the nearest square kilometer,
what’s the area of a circle with a
diameter of 5.5 km?
A = r2
= (3.14)(2.75)2
= 23.74625  24 km2
2) What’s the diameter of a circle with
an area of 50.24 cm2?
LEVEL 2
A = r2
50.24 = (3.14)r2
r2 = 50.24  3.14 = 16
r = √16 = 4 cm
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How do you find the area of composite shapes?
 Composite (or compound) shapes are simply several shapes put together.
 To find the area of these, just add or subtract pieces—whichever is
appropriate or most intuitive for you.
EXAMPLE: Find the area of the room shown below.
Addition method
Subtraction method
5 ft
10 ft
area room = area big + area little
= (24)(13) + (10)(5)
= 362 ft2
10 ft
area = area whole – area left – area right
room
= (24)(18) – (10)(5) – (5)(4)
= 362 ft2
How do you use the Pythagorean Theorem?
 The Pythagorean Theorem states a relationship among the side lengths
of a right triangle. It’s on your formula sheet.
a2 + b2 = c2
a & b are the lengths of the legs of a right triangle
[It doesn’t matter which one you call a and which you call b.]
c
a
b
c is the length of the hypotenuse (the longest side,
which is opposite the right angle)
[It does matter that you call the longest side c.]
 Geometrically, the theorem says that the area of the square on one
leg plus the area of the square on the other leg add up to the area
of the square on the hypotenuse. Mostly, though, you’ll be thinking
of squares algebraically.
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EXAMPLE #1: What’s the missing side length in the figure below?
[Find the hypotenuse.]
?
3 in.
a2 + b2 = c 2
32 + 42 = c2
4 in.
9 + 16 = c2
25 = c2
c = √25 = 5 in.
 Most Pythagorean problems don’t work out so nicely with all whole
numbers. The ones that do have a special name—Pythagorean triples.
It’s worth memorizing the most common one: 3-4-5 because lots of
questions use these numbers or multiples of them (like doubling everything for 68-10 or tripling everything for 9-12-15). If you can spot a Pythagorean Triple, you
can save yourself time calculating.
o
other Pythagorean Triples: 5-12-13; 9-15-17; 7-24-25
EXAMPLE #2: What’s the missing side length in the figure below? [Find the leg.]
13 in.
5 in.
a2 + b2 = c 2
52 + b2 = 132
?
25 + b2 = 169
b2 = 144
b = √144 = 12 in.
 Finding a leg is harder than finding the hypotenuse since you
need to solve a one-step equation to isolate b2.
EXAMPLE #3: What’s the hypotenuse of a right triangle with a height of 4.2 cm
and a base of 6.8 cm? Round your answer to the nearest tenth.
a2 + b2 = c 2
(4.2)2 + (6.4)2 = c2
58.6 = c2
c = √58.6 = 7.655 etc.  7.6 cm.
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