2-Dimensional Geometry - GED Math

Card 0 of 2760

Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. . $\angle C$ is a right angle. What percent of $\Delta ACE$ has been shaded in?

Answer

$\Delta BCD$ is a right triangle with legs ; its area is half the product of its legs, which is

$Area= \frac{1}{2} \times 3 \times 4 = 6$

$\Delta ACE$ is a right triangle with legs

and

;

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

$Area= \frac{1}{2} \times 5 \times 12 = 30$

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

This region is therefore

$\frac{24}{30 } \times 100 % = 80 %$ of $\Delta ACE$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. . $\angle C$ is a right angle. What is the area of the shaded region?

Answer

$\Delta BCD$ is a right triangle with legs ; its area is half the product of its legs, which is

$Area= \frac{1}{2} \times 8 \times 8 = 32$

$\Delta ACE$ is a right triangle with legs

and

;

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

$Area= \frac{1}{2} \times 12 \times 24= 144$

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

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Question

The above hexagon is regular. Give its area.

Answer

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

$A = \frac{s^{2} \sqrt{3}}{4} = \frac{2^{2} \sqrt{3}}{4} = \frac{4\sqrt{3}}{4} = \sqrt{3}$ .

There are six such triangles, so the total area of the hexagon is six times this, or $6\sqrt{3}$ .

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Question

Give the area of a regular hexagon with perimeter 36.

Answer

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

$36 \div 6 = 6$ .

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

$A = \frac{s^{2} \sqrt{3}}{4} = \frac{6^{2} \sqrt{3}}{4} = \frac{36 \sqrt{3}}{4} = 9 \sqrt{3}$ .

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

$6 \cdot 9 \sqrt{3} = 54 \sqrt{3}$

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Question

Determine the area of a square with a side length of .

Answer

Write the area of a square.

Substitute the side into the formula.

$A=(2x^5)^2 = (2x^5)(2x^5) = 4x^{10}$

The answer is: $4x^{10}$

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Question

Figure NOT drawn to scale.

Refer to the above figure. Every angle shown is a right angle.

Give its area.

Answer

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,

$40 \times 60 = 2,400$

and

$100 \times 100 = 10,000$ ,

making the area of the figure

.

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Question

A circle is inscribed in square that has a side length of , as shown by the figure below.

Find the area of the shaded region. Use $\pi=3.14$ .

Answer

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.

$\text{Area of Square}=\text{side}^2$

For the given square,

$\text{Area}=144$

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.

$\text{Area of Circle}=\pi r^2$

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

$\text{Area of Circle}=\pi(6)^2=36\pi=113.04$

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.

$\text{Area of Shaded Region}=144-113.04=30.96$

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Question

Josh wants to build a circular pool in his square yard that measures feet on each side. He wants to build the pool as big as possible, then pave the rest of his yard in tile. In square feet, what is the area of the yard that will be tiled? Round your answer to the nearest tenths place.

Answer

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.

$\text{Area of Square}=50^2=2500$

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:

$\text{Area}=\pi r^2$

Plug in the radius of the circle.

$\text{Area}=\pi (25)^2=625\pi$

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

$\text{Tiled Area}=2500-625\pi=536.5$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

The area of the shaded sector is $24 \pi$ . The area of the white sector is $40 \pi$ .

What is the length of arc $\widehat{MN}$ ?

Answer

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

$A = 24 \pi + 40 \pi = 64 \pi$ .

The degree measure of the arc of the shaded sector is

$\frac{24 \pi}{64 \pi} \times 360 ^{\circ } = 135 ^{\circ }$ .

The radius can be found by solving for and substituting $A = 64 \pi$ in the area formula:

$A = \pi r^{2}$

$64 \pi = \pi r^{2}$

$r^{2} = 64$

The length of arc $\widehat{MN}$ is

$\frac{135}{360} \cdot 2 \pi r = \frac{135}{360} \cdot 2 \cdot 8 \pi = 6\pi$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

The length of arc $\widehat{MN}$ is $12 \pi$ .

The length of arc $\widehat{MYN}$ is $18 \pi$ .

What is the area of the white sector?

Answer

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

$C = 12 \pi + 18 \pi = 30 \pi$ .

The white sector has degree measure

$\frac{ 18 \pi }{ 30 \pi} \times 360 ^{\circ} = 216 ^{\circ}$ .

The radius of the circle can be found using the circumference formula, setting $C = 30 \pi$ :

$C = 2 \pi r$

$30 \pi = 2 \pi r$

$r = 30 \pi \div 2 \pi$

The area of the white sector is therefore

$A = \frac{216}{360 } \cdot \pi r^{2} = 0.60 \cdot \pi \cdot 15^{2} = 225 \cdot 0.60 \cdot \pi = 135 \pi$ .

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Question

What percent of the above circle has not been shaded in?

Answer

There are a total of 360 degrees to a complete circle. The shaded sector has degree measure $80^{\circ }$ , so the unshaded sector has degree measure

$360 ^{\circ }- 80^{\circ }= 2 80^{\circ }$

Also, a sector of $N^{\circ }$ is $\frac{N}{360} \times 100 %$ of the circle, so, setting , we find that the unshaded sector is

$\frac{280}{360} \times 100 %$

$= \frac{28,000}{360} %$

of the circle. This reduces to

$\frac{28,000\div 40 }{360 \div 40 } %$

$=\frac{700 }{9 } %$

$=77\frac{7}{9 } %$ ,

the correct percentage.

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Question

What percent of the above circle has been shaded?

Answer

There are a total of $360^{\circ }$ in a circle. The unshaded portion of the circle represents a $140^{\circ }$ sector, so the shaded portion represents a sector of measure

$360^{\circ } - 140^{\circ } = 220 ^{\circ }$ .

This sector represents

$\frac{ 220 ^{\circ }}{360^{\circ } } \times 100 %$

$= \frac{ 22000}{360 } %$

$= \frac{ 22000 \div 40 }{360 \div 40 } %$

$= \frac{ 550 }{9 } %$

$= 61\frac{1}{9} %$

of the circle.

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Question

What percentage of a circle is covered by a sector with a central angle of $240^{\circ}$ ?

Answer

What percentage of a circle is covered by a sector with a central angle of $240^{\circ}$ ?

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

$%=\frac{\theta}{360^{\circ}}100$

Where theta is our central angle.

Plug in our given degree measurement and simplify.

$%=\frac{240^{\circ}}{360^{\circ}}100=66.67 %$

So, our answer is 66.67%

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Question

Figure is not drawn to scale.

What percent of the circle has not been shaded?

Answer

The total number of degrees in a circle is $360^{\circ }$ , so the shaded sector represents $\frac{144^{\circ }}{360^{\circ }}$ of the circle. In terms of percent, this is

$\frac{144^{\circ }}{360^{\circ }} \times 100 % = \frac{14,400^{\circ }}{360^{\circ }} %= 40%$ .

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

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Question

Find the length of the minor arc if the circle has a circumference of $81\pi$ .

Answer

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.

$\text{Arc length}=\frac{\text{central angle}}{360}(\text{circumference})$

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

$\text{Arc length}=\frac{40}{360}(81\pi)=9\pi$

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Question

A circle has area $81 \pi$ . Give the length of a $90 ^{\circ }$ arc of the circle.

Answer

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

$A = \pi r^{2}$

Set $A= 81 \pi$ :

$\pi r^{2} = 81 \pi$

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

$\pi r^{2} \div \pi = 81 \pi \div \pi$

$r^{2} = 81$

$r = \sqrt{81} = 9$

The circumference of this circle is found using the formula

$C = 2 \pi r$ :

$C = 2 \pi (9)$

$C= 18 \pi$

A $90 ^{\circ }$ arc of the circle is one fourth of the circle, so the length of the arc is

$L = \frac{1}{4}C$

$= \frac{1}{4} \cdot 18 \pi$

$= \frac{9}{2} \pi$

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Question

If a sector covers $\frac{4}{5}$ of the area of a given circle, what is the measure of that sector's central angle?

Answer

If a sector covers $\frac{4}{5}$ 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.

$\measuredangle =\frac{4}{5}(360^{\circ})=288^{\circ}$

So our answer is:

$288^{\circ}$

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Question

What is the diameter of a circle with a radius of 9?

Answer

The diameter of a circle is twice the radius:

Plug in the radius value:

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Question

What is the radius of a circle given the diameter is 22?

Answer

The radius is half of the diameter, or 11.

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Question

What is the radius, in inches, of a circle with a diameter of 12 inches?

Answer

The radius is half of the diameter:

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

$\frac{12}{2}=r$

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