How to find the length of the diagonal of a pentagon

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

Questions 1 - 5

1

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

$6+6\sqrt{5}$

$5 + 5\sqrt{6}$

Explanation

Use the formula for the diagonal of a pentagon:

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

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

2

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

$3+3\sqrt{5}$

$2+2\sqrt{5}$

$6+6\sqrt{5}$

$4+4\sqrt{5}$

Explanation

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

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

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

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

3

Find the diagonal for a pentagon with a perimeter of .

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

$\frac{4(1+\sqrt{3})}{2}$

$\frac{7(1+\sqrt{6})}{2}$

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

Explanation

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

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

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

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

4

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

$6+6\sqrt5$

$1+\sqrt5$

$12+12\sqrt5$

$12\sqrt5$

Explanation

Write the diagonal formula for a pentagon.

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

Substitute the side length and find the diagonal.

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

5

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

$1+\sqrt5$

$\frac{1+\sqrt5}{2}$

$\frac{1+\sqrt5}{4}$

$\frac{1+\sqrt5}{8}$

$\frac{2+\sqrt10}{2}$

Explanation

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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