Up Your Score (43 page)

Surface area
, or the amount of area on the surface of a solid, is often seen as lots of little areas added up.

Rectangular solid:
surface area = (2 ×
l
×
w
) (
2
×
h
×
w
) (
2
×
h
×
l
). Why? Because there are three different pairs of sides.

Cylinder:
surface area = (2 × π ×
r
²
) + (2 × π ×
r
×
h
). The 2 × π ×
r
²
are the two ends (circles), and the 2 × π ×
r
×
h
is the rectangle you get if you unroll the side.

Important note:
Most of the geometrical formulas are actually printed in the SAT exam instructions. So if you draw a blank, scan the instructions for the formula you need.

Now, solve these sample problems.

1. Jennifer wants to build a fence (God knows why people are always building fences in questions like these) to enclose a circle with an area of 144 × π. How much fencing will she need? (Draw a picture. It usually helps.)

Amount of fence = circumference
Circumference = 2 × π ×
r

Well, we can’t solve it without knowing the radius. So find the radius from the known area (144 × π).

144 × π = π ×
r
²

r
²
= 144

r
= 12

Answer: Jennifer needs 2 × π ×
r
= 2 × π × 12 = 24 × π fence. She also needs a psychoanalyst.

2. Which angles are equal in this picture? Which are supplementary?

Answer: (
a, d
) and (
c, b
) are equal, because in a parallelogram, opposite angles are equal. (
a, c
), (
b, d
), (
a, b
), and (
c, d
) are supplementary, because consecutive angles are supplementary.

But don’t prove it each time. Just know it by looking.

3. What is the measure of angle
x
?

Notice that angle
y
and 150° form a straight line, so they are supplementary. Since
y
+ 150° = 180°,
y
= 30°. Now use the sum of the angles of the triangle to solve for
x
.

Answer: 90°

4. What is the measure of angle
x
if
l
and
m
are parallel?

Remember those parallel lines cut by another line? The ones that are full of congruent and supplementary angles? Draw a third parallel line right through the middle of angle
x
and fill in the angles you know.

Voilà! The answer: 30 + 55° = 85° =
x
.

5. What is the measure of angle
x
?

The 40° angle is
inscribed,
so the arc shown is 80°, as is, therefore, central angle
x
.

Answer: 80°

6. What is the measure of angle
x
?

Break down the figure into the following components to find
x
.

First:

The angles in the following figure are supplementary, therefore:
y
= 100°.

Second:

z
+ 45° =
y
= 100°

z
= 55°

Finally:

x
+ 55° + 90° = 180°

x
= 35°