How to find the length of the diagonal of a pentagon - ACT Math

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Question

If the side of a pentagon is , what is the diagonal of the pentagon?

Answer

Write the formula for finding the diagonal of a pentagon.

$d=\frac{1+\sqrt5}{2}s$

Plug in the side length and simplify.

$d=\frac{1+\sqrt5}{2}(2)= 1+\sqrt5$

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Question

If a side of the pentagon is , what is the diagonal of the pentagon?

Answer

Write the diagonal formula for a pentagon.

$d=\frac{1+\sqrt5}{2}s$

Substitute the side length and find the diagonal.

$d=\frac{1+\sqrt5}{2}(12) = (1+\sqrt5)(6) = 6+6\sqrt5$

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Question

Find the length of the diagonal for a pentagon with a a side length of .

Answer

Use the formula for the diagonal of a pentagon:

$d=\frac{s(1+\sqrt{5})}{2}$

$d=\frac{12(1+\sqrt{5})}{2}=6+6\sqrt{5}$

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Question

Find the length of the diagonal for a pentagon with a perimeter of .

Answer

1. Use the formula for perimeter to solve for the side length:

2. Use the formula for the diagonal of a pentagon:

$d=\frac{s(1+\sqrt{5})}{2}$

$d=\frac{6(1+\sqrt{5})}{2}=3+3\sqrt{5}$

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Question

Find the diagonal for a pentagon with a perimeter of .

Answer

1. Use the formula for perimeter to solve for the side length:

2. Use the formula for the diagonal of a pentagon:

$d=\frac{s(1+\sqrt{5})}{2}$

$d=\frac{5(1+\sqrt{5})}{2}$

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