Geometry and Graphs - GED Math

Card 0 of 4270

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

A circular swimming pool at an apartment complex has diameter 18 meters and depth 2.5 meters throughout.

The apartment manager needs to get the interior of the swimming pool painted. The paint she wants to use covers 40 square meters per can. How many cans of paint will she need to purchase?

You may use 3.14 for $\pi$ .

Answer

The pool can be seen as a cylinder with depth (or height) 2.5 meters and a base with diameter 18 meters - and radius half this, or 9 meters.

The bottom of the pool - the base of the cylinder - is a circle with radius 9 meters, so its area is

$B = \pi r ^{2} = 3.14 \cdot 9 \cdot 9 = 254.34$ square meters.

Its side - the lateral face of the cylinder - has area

$LA = 2 \pi r h = 2 \cdot 3.14 \cdot 18 \cdot 2.5 = 141.3$ square meters.

Their sum - the total area to be painted - is square feet. Since one can of paint covers 40 square meters, divide:

$395.64 \div 40 \approx 9.891$

Nine cans of paint and part of a tenth will be required, so the correct response is ten.

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Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

A given regular icosahedron has edges of length two inches. Give the total surface area of the icosahedron.

Answer

The area of an equilateral triangle is given by the formula

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

Since there are twenty equilateral triangles that comprise the surface of the icosahedron, the total surface area is

$SA = 20 A = 20 \cdot \frac{s^{2}\sqrt{3}}{4} = 5 s^{2}\sqrt{3}$ .

Substitute :

$SA = 5 \cdot 2^{2}\sqrt{3} = 5 \cdot 4 \sqrt{3} = 20 \sqrt{3}$ square inches.

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Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length three inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

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

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$ .

Substitute :

$SA =2 \cdot 3^{2}\sqrt{3} = 18 \sqrt{3}$ square inches.

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Question

A regular tetrahedron has four congruent faces, each of which is an equilateral triangle.

A given tetrahedron has edges of length five inches. Give the total surface area of the tetrahedron.

Answer

The area of an equilateral triangle is given by the formula

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

Since there are four equilateral triangles that comprise the surface of the tetrahedron, the total surface area is

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

Substitute :

$SA = 5^{2}\sqrt{3} = 25 \sqrt{3}$ square inches.

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Question

A water tank takes the shape of a sphere whose exterior has radius 24 feet; the tank is six inches thick throughout. To the nearest hundred, give the surface area of the interior of the tank in square feet.

Use 3.14 for $\pi$ .

Answer

Six inches is equal to 0.5 feet, so the radius of the interior of the tank is

feet.

The surface area of the interior of the tank can be calculated using the formula

$SA = 4 \pi r^{2}$

$SA \approx 4 \cdot 3.14 \cdot 23.5 ^{2} \approx 6,936$ ,

which rounds to 6,900 square feet.

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Question

A water tank takes the shape of a closed cylinder whose exterior has a height of 40 feet and a base with radius 15 feet; the tank is three inches thick throughout. To the nearest hundred, give the surface area of the interior of the tank in square feet.

Use 3.14 for $\pi$ .

Answer

Three inches is equal to 0.25 feet, so the height of the interior of the tank is

$40 - 2 \times 0.25 = 39.5$ feet.

The radius of the interior of the tank is

feet.

The surface area of the interior of the tank can be determined by using this formula:

$SA = 2 \pi r (r + h)$

$SA = 2 \cdot 3.14 \cdot 14.75 (14.75 + 39.5)$

$SA \approx 2 \cdot 3.14 \cdot 14.75 \cdot 54.25\approx 5,025$ ,

which rounds to 5,000 square feet.

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Question

Give the total surface area of the above cone to the nearest square meter.

Answer

The base is a circle with radius , and its area can be calculated using the area formula for a circle:

$B= \pi r^{2} = \pi \times 4^{2} \approx 3.14 \times 16 = 50.24$ square meters.

To find the lateral area, we need the slant height of the cone. This can be found by way of the Pythagorean Theorem. Treating the height and the radius as the legs and slant height as the hypotenuse, calculate:

$l^{2} = h^{2} + r^{2}$

$l^{2} = 10^{2} + 4^{2} = 100 + 16 = 116$

$l = \sqrt{116} \approx 10.77$ meters.

The formula for the lateral area can be applied now:

$LA = \pi r l \approx 3.14 \cdot 4 \cdot 10.77 \approx 135.34$

Add the base and the lateral area to obtain the total surface area:

$SA = B + LA \approx 50.24 + 135.34 \approx 185.58$ .

This rounds to 186 square meters.

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Question

Above is a diagram of a conic tank that holds a city's water supply.

The city wishes to completely repaint the exterior of the tank - sides and base. The paint it wants to use covers 40 square meters per gallon. Also, to save money, the city buys the paint in multiples of 25 gallons.

How many gallons will the city purchase in order to paint the tower?

Answer

The surface area of a cone with radius and slant height is calculated using the formula $S = \pi r^{2} + \pi r l$ .

Substitute 35 for and 100 for to find the surface area in square meters:

$S = \pi \cdot 35^{2} + \pi \cdot 35 \cdot 100$

$\approx 3.14 \cdot 1,225 + 3.14\cdot 35 \cdot 100$

$\approx 3,848 + 10,996$

$\approx 14,844$ square meters.

The paint covers 40 square meters per gallon, so the city needs

$14,844 \div 40 =371.1$ gallons of paint.

Since the city buys the paint in multiples of 25 gallons, it will need to buy the next-highest multiple of 25, or 375 gallons.

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Question

A cube has a height of 9cm. Find the surface area.

Answer

To find the surface area of a cube, we will use the following formula:

$SA = 6 \cdot l \cdot w$

where l is the length, and w is the width of the cube.

Now, we know the height of the cube is 9cm. Because it is a cube, all lengths, widths, and heights are the same. Therefore, the length and the width are also 9cm.

Knowing this, we can substitute into the formula. We get

$SA = 6 \cdot 9\text{cm} \cdot 9\text{cm}$

$SA = 6 \cdot 81\text{cm}^2$

$SA = 486\text{cm}^2$

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Question

A sphere has a radius of 7in. Find the surface area.

Answer

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 7in.

So, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (7\text{in})^2$

$SA = 4 \cdot \pi \cdot 49\text{in}^2$

$SA = 196\text{in}^2 \cdot \pi$

$SA = 196\pi \text{ in}^2$

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Question

Find the surface area of a cube with a length of 12in.

Answer

To find the surface area of a cube, we will use the following formula:

$SA = 6 \cdot l \cdot w$

where l is the length, and w is the width of the cube.

Now, we know the length of the cube is 12in. Because it is a cube, all sides are equal. Therefore, the width is also 12in. So, we can substitute. We get

$SA = 6 \cdot 12\text{in} \cdot 12\text{in}$

$SA = 6 \cdot 144\text{in}^2$

$SA = 864\text{in}^2$

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Question

A cube has a height of 8cm. Find the surface area.

Answer

To find the surface area of a cube, we will use the following formula:

$SA = 6 \cdot l \cdot w$

where l is the length, and w is the width of the cube.

Now, we know the height of the cube is 8cm. Because it is a cube, all lengths, widths, and heights are the same. Therefore, the length and the width are also 8cm.

Knowing this, we can substitute into the formula. We get

$SA = 6 \cdot 8\text{cm} \cdot 8\text{cm}$

$SA = 6 \cdot 64\text{cm}^2$

$SA = 384\text{cm}^2$

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