All SAT II Math II Resources
Example Questions
Example Question #1 : Area
Note: Figure NOT drawn to scale.
Refer to the above diagram. , , and and are right angles. What percent of is colored red?
, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:
.
The area of , the shaded region, is half the products of its legs:
The area of is half the product of its hypoteuse, which we can see as the base, and the length of corresponding altitude :
comprises
of .
Example Question #1 : Geometry
Note: Figure NOT drawn to scale
Refer to the above figure, which shows a square garden (in green) surrounded by a dirt path (in orange). The dirt path is seven feet wide throughout. What is the area of the dirt path in square feet?
The area of the dirt path is the area of the outer square minus that of the inner square.
The outer square has sidelength 75 feet and therefore has area
square feet.
The inner square has sidelength feet and therefore has area
square feet.
Subtract to get the area of the dirt path:
square feet.
Example Question #3 : Geometry
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange). The dirt path is six feet wide throughout. Which of the following polynomials gives the area of the garden in square feet?
The length of the garden is feet less than that of the entire lot, or
;
The width of the garden is less than that of the entire lot, or
;
The area of the garden is their product:
Example Question #1 : Geometry
The above figure is a regular decagon. If , then to the nearest whole number, what is ?
As an interior angle of a regular decagon, measures
.
.
can be found using the Law of Cosines:
Example Question #2 : Geometry
The circle in the above diagram has its center at the origin. To the nearest tenth, what is the area of the pink region?
First, it is necessary to determine the radius of the circle. This is the distance between and , so we apply the distance formula:
Subsequently, the area of the circle is
Now, we need to find the central angle of the shaded sector. This is found using the relationship
Using a calculator, we find that ; since we want a degree measure between and , we adjust by adding , so
The area of the sector is calculated as follows:
Example Question #1 : 2 Dimensional Geometry
You own a mug with a circular bottom. If the distance around the outside of the mug's base is what is the area of the base?
You own a mug with a circular bottom. If the distance around the outside of the mug's base is what is the area of the base?
Begin by solving for the radius:
Next, plug the radius back into the area formula and solve:
So our answer is:
Example Question #1 : Geometry
You have a right triangle with a hypotenuse of 13 inches and a leg of 5 inches, what is the area of the triangle?
You have a right triangle with a hypotenuse of 13 inches and a leg of 5 inches, what is the area of the triangle?
So find the area of a triangle, we need the following formula:
However, we only know one leg, so we only know b or h.
To find the other leg, we can either use Pythagorean Theorem, or recognize that this is a 5-12-13 triangle. Meaning, our final leg is 12 inches long.
To prove this:
Now, we know both legs, let's just plug in and solve for area:
Example Question #1 : Geometry
You have a rectangular-shaped rug which you want to put in your living room. If the rug is 12.5 feet long and 18 inches wide, what is the area of the rug?
You have a rectangular-shaped rug which you want to put in your living room. If the rug is 12.5 feet long and 18 inches wide, what is the area of the rug?
To begin, we need to realize two things.
1) Our given measurements are not in equivalent units, so we need to convert one of them before doing any solving.
2) The area of a rectangle is given by:
Now, let's convert 18 inches to feet, because it seems easier than 12.5 feet to inches:
Now, using what we know from 2) we can find our answer
Example Question #5 : 2 Dimensional Geometry
Give the area of to the nearest whole square unit, where:
The area of a triangle with two sides of lengths and and included angle of measure can be calculated using the formula
.
Setting and evaluating :
Example Question #311 : Sat Subject Test In Math Ii
Give the area of to the nearest whole square unit, where:
The area of a triangle, given its three sidelengths, can be calculated using Heron's formula:
,
where , , and are the lengths of the sides, and .
Setting , , and , evaluate :
and, substituting in Heron's formula:
To the nearest whole, this is 260.