# SAT II Math II : 2-Dimensional Geometry

## Example Questions

### Example Question #1 : Geometry

Note: Figure NOT drawn to scale.

Refer to the above diagram. , and  and  are right angles. What percent of  is colored red?

Explanation:

, 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?

Explanation:

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 #1 : 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?

Explanation:

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 ?

Explanation:

As an interior angle of a regular decagon,  measures

.

.

can be found using the Law of Cosines:

### Example Question #1 : 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?

Explanation:

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 #6 : 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?

Explanation:

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:

### Example Question #7 : 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?

Explanation:

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 #8 : 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?

Explanation:

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 #9 : Geometry

Give the area of  to the nearest whole square unit, where:

Explanation:

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 #1 : Geometry

Give the area of  to the nearest whole square unit, where:

Explanation:

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.