Solid Geometry - Intermediate Geometry

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Question

Find the diagonal of a cube with a side length of .

Answer

The diagonal of a cube is simply given by:

$diagonal=x\cdot \sqrt{3}$

Where is the side length of the cube.

So since our

$diagonal =1\cdot\sqrt{3}$

$diagonal=\sqrt{3}$

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Question

Find the length of a diagonal of a cube with volume of $\small 64, in.^3$

Answer

There is a formula for the length of a cube's diagonal given the side length. However, we might not remember that formula as it is less common. However, we can also find the length using the Pythagorean Theorem.

But first, we need to find the side length. We know the volume is 64. Our formula for volume is

$\small V=s^3$

Substituting gives

$\small 64=s^3$

Taking the cube root gives us a side length of 4. Now let's look at our cube.

We need to begin by finding the length of the diagonal of the bottom face of our cube (the green segment). This can be done either by using the Pythagorean Theorem or by realizing that the right triangle is in fact a 45-45-90 triangle. Either way, we realize that our diagonal (the hypotenuse) is $\small 4\sqrt{2}$ .

We now seek to find the diagonal of the cube (the blue segment). We do this by looking at the right triangle formed by it, the left vertical edge, and the face diagonal we just found. This time our only recourse is to do the Pythagorean Theorem.

$\small 4^2+(4\sqrt{2})^2=d^2$

$\small 16+32=d^2$

$\small 48=d^2$

$\small d=4\sqrt{3}$

In general, the formula for the diagonal of a cube with side length $\small s$ is

$\small d=s\sqrt{3}$

The length of our diagonal is $\small 4\sqrt{3},in.$

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Question

Suppose the volume of a cube is . What is the length of the diagonal?

Answer

Write the equation for the volume of a cube. Substitute the volume to find the side length, .

Write the equation for finding diagonals given an edge length for a cube.

$d=s\sqrt3$

Substitute the side length to find the diagonal length.

The answer is $2\sqrt3$ .

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Question

If the volume of a cube was one eighth, what is the diagonal of the cube?

Answer

Write the volume of a cube and substitute the given volume to find a side length.

$\frac{1}{8}=s^3$

$s=\frac{1}{2}$

Write the diagonal formula for a cube and substitute the side length.

$d=s\sqrt3 = \frac{\sqrt3}{2}$

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Question

If the volume of a cube was , what is the length of the diagonal?

Answer

Write the equation for finding the volume of a cube, and substitute in the volume.

Write the diagonal equation for cubes and substitute in the given length.

$d=s\sqrt3= 10\sqrt3$

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Question

If the surface area of a cube was , what is the length of the diagonal?

Answer

Write the surface area formula for cubes and substitute the given area.

Write the diagonal formula for cubes and substitute the side length.

$d=s\sqrt3=2\sqrt3$

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Question

Find the distance from point A to point B in the cube below (leave answer in simplest radical form):

Answer

When calculating the diagonal of the cube, point A to point B.

We must first find the diagonal of the base of the cube.

The base of the cube is a square where all sides are 8.

The diagonal of this square is found either by the pythagorean theorem or by what we know about 45-45-90 triangles to get the diagonal of the base below:

$Diagonal\ of\ base=8\sqrt2$

The diagonal of the base would be from point A to point C in the drawing.

We can see that the diagonal of the base and side BC of the cube form the two legs of a right triangle that will allow us to find the 3D diagonal of the whole cube.

Use the pythagorean theorem with BC and the Diagonal of the Base.

$(8\sqrt2)^{2}+8^{2}=AB^{2}$

$128+64=AB^{2}$

$192=AB^{2}$

Take the square root of both sides.

$\sqrt192=AB$

After simplifying everything we get the final answer for the Diagonal of the Cube (AB).

$AB=8\sqrt3$

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Question

A cube has a side length of 6 meters. What is the length of its diagonal across one of the faces?

Answer

Since all sides of a cube are equal and all sides form right angles, we use pythagorean theorem to find the length of the diagonal.

$c=\sqrt{72}=\mathbf{3\sqrt{8}}$ meters

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Question

What is the length of the diagonal of a rectangular box with the dimensions of $3; cm\times 4; cm\times 12 ; cm$ ?

Answer

To solve this problem we need an extension of the Pythagorean Theorem:

$d^{2}=x^{2}+y^{2}+z^{2}$

So the equation to solve becomes $d^{2}=3^{2}+4^{2}+12^{2}=9+16+144=169$

So the distance of the diagonal is .

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Question

Find the diagonal of the prism. The diagonal is represented by the dashed line.

Answer

The length of the diagonal is from the bottom left hand corner closest to us to the top right hand corner that's farthest away from us.

This kind of a problem may seem to be a little more complicated than it really is.

In order to solve for the diagonal length, all that's required is the Pythagorean Theorem. This equation will be used twice to solve for the dashed line.

For the first step of this problem, it's helpful to imagine a triangle "slice" that's being taken inside the prism.

, where the diagonal of interest is D2, and D1 is the diagonal that cuts from corner to corner of the bottom face of the prism. Of this triangle that's outlined in pink dashed lines, the given information (the dimensions of the prism) provides a length for one of the legs (16).

We can already "map out" that D2 (the hypotenuse of the dashed triangle) can be solved by using the Pythagorean Theorem if we can obtain the length of the other leg (D1).

The next step of this problem is to solve for D1. This will be the first use of the Pythagorean theorem. D1 is the diagonal of the base and is limited to a 2D face. This can be represented as:

The hypotenuse of the base, or the mystery length leg of the dashed triangle, can be solved by using the Pythagorean Theorem:

$\sqrt{720}=\sqrt{c^2}$

$\sqrt{720}=c$

${c=12\sqrt{}5}{= D1}$

Now that we calculated the length of D1, D2 can be solved for by using the Pythagorean Theorem a second time:

$16^2+(12\sqrt{5})^2=c^2$

$\sqrt{976}=\sqrt{c^2 }$

$\sqrt{976}=c$

${4\sqrt{61}=c \approx 31.2=D2}$

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Question

A right rectangular prism has a width of cm, a length of cm, and a height of cm. Find the diagonal distance of the prism.

Answer

To find the diagonal distance of a prism, you can use the formula:

$d = \sqrt{w^{2} + l^{2} + h^{2}}$ , where = height; = width, and = length.

So, in this problem

$d = \sqrt{2^{2} + 8^{2} + 10^{2} } = \sqrt{4 + 64 + 100} = \sqrt{168} = \sqrt{4 * 42} = 2\sqrt{42}$ .

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Question

A right rectangular prism has a height of ft, a width of feet and a length that is twice its width. The volume of the prism is cubic feet. What is the diagonal of the prism?

Answer

First, given the volume, you need to find the width and length. The volume of a right, rectangular prism can be found using

, so $36 = 2x \cdot x \cdot 2$ , where represents the length and represents the width.

Solving for , you get

$36 = 2x^{2} \cdot 2, 18 = 2x^{2}, 9 = x^{2}, 3 = x.$

So, the width of the prism is 3 feet.

Remember that the length is twice the width, so the length is 6 feet.

Now you may use the formula for finding the diagonal:

$d = \sqrt{l^{2} + w^{2} + h^{2}}$ . So, $d = \sqrt{6^{2} + 3^{2} + 2^{2}} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7$ .

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Question

A right, rectangular prism has a length of meters, width that is meters longer than the length, and a height of meters. The volume of the prism is cubic meters. Find the diagonal of the prism.

Answer

First, use the volume formula,

to find the missing length and width.

$V = x\cdot(3+x)\cdot10\rightarrow 100 = (3x + x^{2})10$

$\rightarrow 100 = 30x + 10x^{2}$

$\rightarrow 10x^{2} + 30 x - 100 = 0 \rightarrow (5x - 10)(2x + 10) = 0$

$\rightarrow x = 2, -5$

Since cannot be a negative value is it represents a length of a prism, we know . So the length is 2 meters, and therefore the width is 5 meters.

Now you can plug in the length, width and height into the formula for finding the diagonal of the prism.

$d = \sqrt{2^{2} + 5^{2} + 10^{2}} = \sqrt{4 + 25 + 100} = \sqrt{129}$ .

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Question

A right, rectangular prism has a volume of cubic centimeters. Its width is cm and its length is three times its height. Find the length of the diagonal of the prism.

Answer

First, use the volume to find the missing height and length.

Since the length is three times the height, use to represent the length and to represent the height.

So, $V = lwh \rightarrow 96 = 3x\cdot8\cdotx \rightarrow 96 = 24x^{2} \rightarrow 4 = x^2 \rightarrow 2 = x$ .

So the height of the prism is 2 centimeters, and the length is 6 centimeters.

Use these values to now solve for the diagonal distance.

$d = \sqrt{6^{2} + 8^{2} + 2^{2}} = \sqrt{36 + 64 + 4} = \sqrt{104} = 2\sqrt{26}$ .

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Question

The surface area of a right, rectangular prism is square inches. The height is inches and the length is times the width. Find the diagonal distance of the prism.

Answer

Use the surface area, 280 square inches, and the formula for finding the surface area of a right, rectangular prsim to find the missing length and width measurements.

So,

$\rightarrow 280 = 10x^{2} + 100x + 20x \rightarrow 280 = 10x^{2}+ 120x$

$\rightarrow 10x^2 +120x - 280 = 0 \rightarrow 10(x^{2} + 12x - 28) = 0$

$\rightarrow 10(x-2)(x+14) = 0) \rightarrow x=2, -14$

Since represents the length of a solid figure, we must assume , rather than the negative value.

So, the width of the figure is 2 inches and the length is 10 inches.

Now, use the formula for finding the diagonal of a right, rectangular prism:

$d = \sqrt10^{2} + 2^{2} + 10^{2} = \sqrt{100 + 4 + 10} = \sqrt{204} = 2\sqrt{51}$ .

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Question

A right, rectangular prism has a surface area of square meters. Its width is twice its length, and its height is four times its length. Find the diagonal distance of the prism.

Answer

Use the surface area of the prism to find the missing length, width and height.

$2(x)(2x) + 2(x)(4x) + 2(2x)(4x) = 28 \rightarrow 4x^{2} + 8x^{2} + 16x^{2} = 28$

$\rightarrow 28x^{2} = 28 \rightarrow x^{2} = 1 \rightarrow x = 1$

So, the prism's length is 1 meter, the width is 2 meters and the height is 4 meters.

Now you can find the diagonal distance using those values.

$d = \sqrt{1^{2} + 2^{2} + 4^{2}} = \sqrt{1 + 4 + 16} = \sqrt{21}$ .

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Question

The above cube has edges of length 1. True or false: The dashed line has length $\sqrt{2}$ .

Answer

Examine the diagram below.

$\bigtriangleup ABC$ is a right triangle with legs of length , so it is an isosceles - or 45-45-90 - right triangle. By the 45-45-90 Triangle Theorem its hypotenuse measures

$AC = 1 \cdot \sqrt{2} = \sqrt{2}$

$\bigtriangleup DAC$ is a right triangle with legs of lengths and $AC = \sqrt{2}$ , so the length of its hypotenuse is

$DC = \sqrt{(AD)^{2}+ \left ( AC\right )^{2}}$

$DC = \sqrt{1^{2}+ \left ( \sqrt{2}\right )^{2}}$

$DC = \sqrt{1 +2}$

$DC = \sqrt{3}$ .

$\overline{DC}$ , the diagonal in question, has length $\sqrt{3}$ , not $\sqrt{2}$ .

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Question

The surface area of a sphere is $100\pi ; in^{2}$ . What is the diameter of the sphere?

Answer

The surface area of a sphere is given by $SA=4\pi r^{2}$

So the equation to sovle becomes $4\pi r^{2}=100\pi$ or $r^{2}=25$ so

To answer the question we need to find the diameter:

$d=2r=2\cdot 5=10; in$

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Question

A company wants to construct an advertising balloon spherical in shape. It can afford to buy 28,000 square meters of material to make the balloon. What is the largest possible diameter of this balloon (nearest whole meter)?

Answer

This is equivalent to asking the diameter of a balloon with surface area 28,000 square meters.

The relationship between the surface area and the radius is:

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

To find the radius, substitute for the surface area, then solve:

$28,000=4\pi r^{2}$

$r^{2}=\frac{28,000}{4\pi }$

$r=\sqrt{\frac{28,000}{4\pi }}\approx47$

To find the diameter , double the radius—this is 94.

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Question

If the volume of a sphere is , what is the sphere's diameter?

Answer

Write the formula for the volume of a sphere:

$V=\frac{4}{3}\pi r^3$

Plug in the volume and find the radius by solving for :

$2=\frac{4}{3}\pi r^3$

Start solving for by multiplying both sides of the equation by :

$2\cdot3=\frac{4}{3}\pi r^3\cdot3$

$6=4\pi r^3$

Now, divide each side of the equation by $4\pi$ :

$\frac{6}{4\pi}=\frac{4\pi r^3}{4\pi}$

$\frac{6}{4\pi}=r^3$

Reduce the left side of the equation:

$\frac{3}{2\pi}=r^3$

Finally, take the cubed root of both sides of the equation:

$\sqrt[3]{\frac{3}{2\pi}}=r$

Keep in mind that you've solved for the radius, not the diameter. The diameter is double the radius, which is: $2\sqrt[3]{\frac{3}{2\pi}}$ .

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