Sat math Essentials

1
2
4
3
m
∠
1 + m
∠
2 + m
∠
3 + m
∠
4 = 360
°
B
C
F
A
D
E
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Practice Question
What is the sum of the interior angles in the figure above?
a.
360°
b.
540°
c.
900°
d.
1,080°
e.
1,260°
Answer
d.
To find the sum of the interior angles of a polygon, use the formula 
S
180(
x
2), with 
x
being the
number of sides in the polygon. The polygon above has eight sides, therefore 
x
8.
S
180(
x
2) 
180(8 
2) 
180(6) 
1,080°
Exterior Angles
The sum of the exterior angles of
any
polygon (triangles, quadrilaterals, pentagons, hexagons, etc.) is 360°.
Similar Polygons
If two polygons are similar, their corresponding angles are equal, and the ratio of the corresponding sides is in
proportion.
Example
These two polygons are similar because their angles are equal and the ratio of the corresponding sides is in
proportion:
2
1
0
0
2
1
1
9
8
2
1
8
4
2
1
3
1
0
5
2
1
18
30
20
135°
75°
135°
75°
60°
9
15
10
60°
8
4
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Practice Question
If the two polygons above are similar, what is the value of
d
?
a.
2
b.
5
c.
7
d.
12
e.
23
Answer
a.
The two polygons are similar, which means the ratio of the corresponding sides are in proportion.
Therefore, if the ratio of one side is 30:5, then the ration of the other side, 12:
d,
must be the same.
Solve for 
d
using proportions:
3
5
0
1
d
2
Find cross products.
30
d
(5)(12)
30
d
60
d
6
3
0
0
d
2
Parallelograms
A 
parallelogram
is a quadrilateral with two pairs of parallel sides.
In the figure above,
A
B
|| 
D
C
and 
A
D
|| 
B
C
.
Parallelograms have the following attributes:
■
opposite sides that are equal 
A
D
B
C
A
B
D
C
■
opposite angles that are equal 
m
∠
A
m
∠
C
m
∠
B
m
∠
D
■
consecutive angles that are supplementary 
m
∠
A
m
∠
B
180°
m
∠
B
m
∠
C
180°
m
∠
C
m
∠
D
180°
m
∠
D
m
∠
A
180°
A
B
D
C
30
12
5
d
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Special Types of Parallelograms
■
A 
rectangle
is a parallelogram with four right angles.
■
A 
rhombus
is a parallelogram with four equal sides.
■
A 
square
is a parallelogram with four equal sides and four right angles.
Diagonals
■
A diagonal cuts a parallelogram into two equal halves.
B
A
C
D
ABC = 
ADC
B
A
C
D
AB = BC = DC = AD
m
∠
A
 = m
∠
B
 = m
∠
C
 = m
∠
D
 = 90
B
A
C
D
AB = BC = DC = AD
A
D
AD = BC
AB = DC
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■
In all parallelograms, diagonals cut each other into two equal halves.
■
In a rectangle, diagonals are the same length.
■
In a rhombus, diagonals intersect at right angles.
■
In a square, diagonals are the same length and intersect at right angles.
B
A
C
D
AC = DB
AC DB
B
A
C
D
AC DB
B
A
C
D
AC = DB
B
A
E
C
D
AE = CE
DE = BE
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Practice Question
Which of the following must be true about the square above?
I
.
a
b
II.
A
C
B
D
III.
b
c
a.
I only
b.
II only
c.
I and II only
d.
II and III only
e.
I, II, and III
Answer
e.
A
C
and 
B
D
are diagonals. Diagonals cut parallelograms into two equal halves. Therefore, the diagonals
divide the square into two 45-45-90 right triangles. Therefore,
a
,
b
, and 
c
each equal 45°.
Now we can evaluate the three statements:
I:
a
b
is TRUE because 
a
45 and 
b
45.
II:
A
C
B
D
is TRUE because diagonals are equal in a square.
III:
b
c
is TRUE because 
b
45 and 
c
45.
Therefore I, II, and III are ALL TRUE.
S o l i d F i g u r e s , P e r i m e t e r, a n d A r e a
There are five kinds of measurement that you must understand for the SAT:
1.
The 
perimeter
of an object is the sum of all of its sides.
Perimeter 
5 
13 
5 
13 
36
5
13
13
5
D
A
a
c
b
C
B
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2. Area
is the number of square units that can fit inside a shape. Square units can be square inches (in
2
),
square feet (ft
2
), square meters (m
2
), etc.
The area of the rectangle above is 21 square units. 21 square units fit inside the rectangle.
3. Volume
is the number of cubic units that fit inside solid. Cubic units can be cubic inches (in
3
), cubic feet
(ft
2
), cubic meters (m
3
), etc.
The volume of the solid above is 36 cubic units. 36 cubic units fit inside the solid.
4.
The 
surface area
of a solid is the sum of the areas of all its faces.
To find the surface area of this solid . . .
. . . add the areas of the four rectangles and the two squares that make up the surfaces of the solid.
 1 cubic unit
 1 square unit
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5. Circumference
is the distance around a circle.
If you uncurled this circle . . .
. . . you would have this line segment:
The circumference of the circle is the length of this line segment.
Formulas
The following formulas are provided on the SAT. You therefore do not need to memorize these formulas, but you
do need to understand when and how to use them.
Circle
Rectangle
Triangle
r
l
w
h
b
A = lw
C = 
2
π
r
A = 
π
r
2
Cylinder
Rectangle
S
olid
h
l
V = 
π
r
2
h
w
r
h
V = lwh
C = 
Circumference
A =
Area
r =
Radiu
s
l =
Length
w =
Width
h =
Height
V =
Volume
b =
Ba
s
e
A
=
1
2
bh
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Practice Question
A rectangle has a perimeter of 42 and two sides of length 10. What is the length of the other two sides?
a.
10
b.
11
c.
22
d.
32
e.
52
Answer
b.
You know that the rectangle has two sides of length 10. You also know that the other two sides of the
rectangle are equal because rectangles have two sets of equal sides. Draw a picture to help you better
understand:
Based on the figure, you know that the perimeter is 10 
10 
x
x.
So set up an equation and solve
for 
x
:
10 
10 
x
x
42
20 
2
x
42
20 
2
x
20 
42 
20
2
x
22
2
2
x
2
2
2
x
11
Therefore, we know that the length of the other two sides of the rectangle is 11.
Practice Question
The height of a triangular fence is 3 meters less than its base. The base of the fence is 7 meters. What is the
area of the fence in square meters?
a.
4
b.
10
c.
14
d.
21
e.
28
Answer
c.
Draw a picture to help you better understand the problem. The triangle has a base of 7 meters. The
height is three meters less than the base (7 
3 
4), so the height is 4 meters:
4
7
10
10
x
x
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The formula for the area of a triangle is 
1
2
(base)(height):
A
1
2
bh
A
1
2
(7)(4)
A
1
2
(28)
A
14
The area of the triangular wall is 14 square meters.
Practice Question
A circular cylinder has a radius of 3 and a height of 5. Ms. Stewart wants to build a rectangular solid with a
volume as close as possible to the cylinder. Which of the following rectangular solids has dimension closest
to that of the circular cylinder?
a.
3 
3 
5
b.
3 
5 
5
c.
2 
5 
9
d.
3 
5 
9
e.
5 
5 
9
Answer
d.
First determine the approximate volume of the cylinder. The formula for the volume of a cylinder is 
V
π
r
2
h.
(Because the question requires only an approximation, use 
π ≈
3 to simplify your calculation.)
V
π
r
2
h
V
≈
(3)(3
2
)(5)
V
≈
(3)(9)(5)
V
≈
(27)(5)
V
≈
135
Now determine the answer choice with dimensions that produce a volume closest to 135:
Answer choice