How to find the diagonal of a cube - GRE Quantitative Reasoning
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The surface area of a cube is 486 units. What is the distance of its diagonal (e.g. from its front-left-bottom corner to its rear-right-top corner)?
The surface area of a cube is 486 units. What is the distance of its diagonal (e.g. from its front-left-bottom corner to its rear-right-top corner)?
First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x2. This yields the equation:
6x2 = 486, which simplifies to: x2 = 81; x = 9.
Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:
d = √((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2)
For our data, this will be:
√( (–4.5 – 4.5)2 + (–4.5 – 4.5)2 + (4.5 + 4.5)2) =
√( (–9)2 + (–9)2 + (9)2) = √(81 + 81 + 81) = √(243) =
√(3 * 81) = √(3) * √(81) = 9√(3)
First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x2. This yields the equation:
6x2 = 486, which simplifies to: x2 = 81; x = 9.
Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:
d = √((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2)
For our data, this will be:
√( (–4.5 – 4.5)2 + (–4.5 – 4.5)2 + (4.5 + 4.5)2) =
√( (–9)2 + (–9)2 + (9)2) = √(81 + 81 + 81) = √(243) =
√(3 * 81) = √(3) * √(81) = 9√(3)
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You have a rectangular box with dimensions 6 inches by 6 inches by 8 inches. What is the length of the shortest distance between two non-adjacent corners of the box?
You have a rectangular box with dimensions 6 inches by 6 inches by 8 inches. What is the length of the shortest distance between two non-adjacent corners of the box?
The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is 
.
The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is
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What is the length of the diagonal of a cube with side lengths of 
each?
What is the length of the diagonal of a cube with side lengths of
The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:
, or 
, or 
Now, if the the value of 
is 
, we get simply 
The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:
Now, if the the value of
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What is the length of the diagonal of a cube that has a surface area of 
?
What is the length of the diagonal of a cube that has a surface area of
To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of 
squares. Therefore, its surface area is:
, where 
is the length of a side.
Therefore, for our data, we have:
Solving for 
, we get:
This means that 
Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:
, or 
, or 
Now, if the the value of 
is 
, we get simply 
To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of
Therefore, for our data, we have:
Solving for
This means that
Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:
Now, if the the value of
Compare your answer with the correct one above