2-Dimensional Geometry - GED Math

is a right triangle with legs ; its area is half the product of its legs, which is

is a right triangle with legs

and

;

its area is half the product of its legs, which is

The shaded region is the former triangle removed from the latter triangle; its area is the difference of the two: .

A regular hexagon can be divided into six triangles, each of which can be easily proved equilateral, as seen in the diagram below:

All segments shown are congruent, and, since the diameter shown in the original diagram is 4, each sidelength is half this, or 2.

Each equilateral triangle has area

.

There are six such triangles, so the total area of the hexagon is six times this, or .

A hexagon has six sides; a regular hexagon with perimeter 36 has sidelength

.

A regular hexagon can be divided into six triangles, each of which can be easily proved equilateral, as seen in the diagram below:

Each equilateral triangle has sidelength 6, so each has area

.

The total area of the hexagon is the area of six such triangles:

Write the area of a square.

Substitute the side into the formula.

The answer is:

Examine the bottom figure, in which the bottom two sides have been connected. Note that the figure is now a rectangle cut out of a rectangle, and, since the opposite sides of a rectangle have the same length, we can fill in some of the side lengths as shown:

The figure is a 60-by-40 rectangle cut from a 100-by-100 square, so, to get the area of the figure, subtract the area of the former from that of the latter. The area of a rectangle is equal to the product of its dimensions, so the areas of the rectangle and the square are, respectively,

and

,

making the area of the figure

.

Since the circle is inscribed in the square, the diameter of the circle is the same length as the length of a square.

Start by finding the area of the square.

For the given square,

Now, because the diameter of the circle is the same as the length of a side of the square, we now also know that the radius of the circle must be . Next recall how to find the area of a circle.

Plug in the found radius to find the area of the circle.

Now, the shaded area is the area left over when the area of the circle is subtracted from the area of the square. Thus, we can write the following equation to find the area of the shaded region.

Start by drawing out the square yard and the circular pool in a way that maximizes the area of the pool.

Notice that the diameter of the pool will be the same length as the side of the square.

Since the question asks about the area that is left over after the pool is built, we can find that area by subtracting the area of the pool from the area of the square.

Start by finding the area of the square.

Next, find the area of the circular pool.

Since the diameter of the pool is , the radius of the pool must be . Recall how to find the area of a circle:

Plug in the radius of the circle.

Subtract the area of the circle from that of the square.

The area of the circle is the sum of the areas of the sectors, which is

.

The degree measure of the arc of the shaded sector is

.

The radius can be found by solving for and substituting in the area formula:

The length of arc is

.

The circumference of the circle is the sum of the lengths of the arcs, which is

.

The white sector has degree measure

.

The radius of the circle can be found using the circumference formula, setting :

The area of the white sector is therefore

.

There are a total of 360 degrees to a complete circle. The shaded sector has degree measure , so the unshaded sector has degree measure

Also, a sector of is of the circle, so, setting , we find that the unshaded sector is

of the circle. This reduces to

,

the correct percentage.

There are a total of in a circle. The unshaded portion of the circle represents a sector, so the shaded portion represents a sector of measure

.

This sector represents

of the circle.

What percentage of a circle is covered by a sector with a central angle of ?

To find the percentage of a circle from the central angle, we need to use the following formula:

Where theta is our central angle.

Plug in our given degree measurement and simplify.

So, our answer is 66.67%

The total number of degrees in a circle is , so the shaded sector represents of the circle. In terms of percent, this is

.

The shaded sector is 40% if the circle, so the unshaded sector is of the circle.

Recall that the length of an arc is a proportion of the circumference, just like how the measure of a central angle is a proportion of the total number of degrees in a circle.

Thus, we can write the following equation to solve for arc length.

Plug in the given central angle and circumference to find the length of the minor arc .

The radius of a circle, given its area , can be found using the formula

Set :

Find by dividing both sides by and then taking the square root of both sides:

The circumference of this circle is found using the formula

:

A arc of the circle is one fourth of the circle, so the length of the arc is

If a sector covers of the area of a given circle, what is the measure of that sector's central angle?

While this problem may seem to not have enough information to solve, we actually have everything we need.

To find the measure of a central angle, we don't need to know the actual area of the sector or the circle. Instead, we just need to know what fraction of the circle we are dealing with. In this case, we are told that the sector represents four fifths of the circle.

All circles have 360 degrees, so if our sector is four fifths of that, we can find the answer via the following.

So our answer is:

The diameter of a circle is twice the radius:

Plug in the radius value:

The radius is half of the diameter, or 11.

The radius is half of the diameter:

From here we plug in our diameter measure of 12 and solve for the radius: