Calculating the diagonal of a prism - GMAT Quantitative Reasoning

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Question

If a rectangular prism has a length of , a width of , and a height of , what is the length of its diagonal?

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle formed by the height of the prism and the diagonal of its bottom face. First apply the Pythagorean Theorem to find the length of the diagonal of the bottom face, and then apply the Pythagorean Theorem again with this side and the height of the prism to find the length of its diagonal:

$C^2=52\rightarrow C=\sqrt{52}$

$3^2+(\sqrt{52})^2=D^2$

$D^2=61\rightarrow D=\sqrt{61}$

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Question

Calculate the diagonal length for a rectangular prism with a length of , a width of , and a height of .

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle, where the other two sides are the height of the prism and the diagonal of either its top or bottom face. This means we can find the length of the prism's diagonal using the Pythagorean Theorem, but first we must apply this theorem to find the diagonal of the prism's top or bottom face, which forms the base of the right triangle whose hypotenuse is the diagonal of the entire prism. After finding the face diagonal, we apply the Pythagorean Theorem again to calculate the answer:

$c^2=89\rightarrow c=\sqrt{89}$

$3^2+(\sqrt{89})^2=d^2$

$d^2=98\rightarrow d=\sqrt{98}=\sqrt{2\cdot 49}=7\sqrt{2}$

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Question

Calculate the diagonal length for a rectangular prism with a length of , a width of , and a height of .

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle whose other two sides are the height of the prism and the diagonal of either its top or bottom face. We start by finding the diagonal of either the prism's top or bottom face, as this is the base of the right triangle for which the diagonal of the prism is the hypotenuse:

$c^2=25\rightarrow c=5$

Now we apply the Pythagorean Theorem again, this time using the prism height and the face diagonal calculated above, and the hypotenuse we're left with is the same as the diagonal length of the rectangular prism:

$c^2=29\rightarrow c=\sqrt{29}$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{12^{2} +4^{2}}=c$

$\sqrt{144 +16}=c$

$\sqrt{160}=c$

$\sqrt{16\times 10}=c$

$4\sqrt{10}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{9^{2} +(\sqrt{160})^{2}}=d$

$\sqrt{81 +160}=d$

$\sqrt{241}=d$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{8^{2} +10^{2}}=c$

$\sqrt{64 +100}=c$

$\sqrt{164}=c$

$\sqrt{41\times 4}=c$

$2\sqrt{41}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{6^{2} +(\sqrt{164})^{2}}=d$

$\sqrt{36 +164}=d$

$\sqrt{200}=d$

$\sqrt{100\times 2}=d$

$10\sqrt{2}=d$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{7^{2} +4^{2}}=c$

$\sqrt{49 +16}=c$

$\sqrt{65}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{11^{2} +\sqrt{65}^{2}}=d$

$\sqrt{121+65}=d$

$\sqrt{186}=d$

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Question

Find the diagonal of a rectangular prism whose base is $3\times4$ and has a base of .

Answer

To solve, simply solve for the base diagonal which will become the side of the other triangle, whose hypotenuse is the diagonal we are looking for.

$c1^2=3^2+4^2\Rightarrow c=5$

$diagonal=\sqrt{5^2+12^2}=13$

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