How to find the length of the diagonal of a pentagon - Geometry

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Question

What is the length of a diagonal of a pentagon with side length of 20 centimeters (to the nearest tenth)?

Answer

Let be the length fo a diagonal.

One interior angle of a pentagon measures

$C=\frac{180(5-2)}{5}=108^{\circ }$

A diagonal of this pentagon, along with two sides, forms an isosceles triangle with sides 20, 20 and ; the side of length is opposite the angle of measure $108^{\circ }$ , and each side of length 20 is opposite an angle of measure $\frac{180-108}{2}=36^{\circ }$ .

The Law of Sines helps us to determine :

$\frac{d }{sin;108^{\circ}}=\frac{20 }{sin;36^{\circ}}$

$d=\frac{20\cdot sin;108^{\circ} }{sin;36^{\circ}}\approx\frac{20\cdot 0.9511 }{0.5878}\approx 32.4$

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Question

In the pentagon below, the length of is one-third that of . Find the length of diagonal .

Answer

Start by using the Pythagorean theorem to find the length of .

$AF=\sqrt{(AE)^2-(EF)^2}$

$AF=\sqrt{22^2-18^2}=\sqrt{324}$

Since is one-third the length of , we know that the length of must be $3\sqrt{340}$ .

Now, notice that the pentagon is made up of one rectangle and two right triangles.

Focus on rectangle . We can find the length of by adding together the lengths of and .

$\text{AD}=\sqrt{324}+3\sqrt{324}=4\sqrt{324}$

Since we are working with a rectangle, we know that . Thus, we can use the Pythagorean Theorem again to find the length of .

$AC=\sqrt{(AB)^2+(BC)^2}$

$AC=\sqrt{36^2+(4\sqrt{324})^2}=\sqrt{6480}=80.50$

Make sure to round to two places after the decimal.

Compare your answer with the correct one above

Question

What is the length of a diagonal of a pentagon with side length of 20 centimeters (to the nearest tenth)?

Answer

Let be the length fo a diagonal.

One interior angle of a pentagon measures

$C=\frac{180(5-2)}{5}=108^{\circ }$

A diagonal of this pentagon, along with two sides, forms an isosceles triangle with sides 20, 20 and ; the side of length is opposite the angle of measure $108^{\circ }$ , and each side of length 20 is opposite an angle of measure $\frac{180-108}{2}=36^{\circ }$ .

The Law of Sines helps us to determine :

$\frac{d }{sin;108^{\circ}}=\frac{20 }{sin;36^{\circ}}$

$d=\frac{20\cdot sin;108^{\circ} }{sin;36^{\circ}}\approx\frac{20\cdot 0.9511 }{0.5878}\approx 32.4$

Compare your answer with the correct one above

Question

In the pentagon below, the length of is one-third that of . Find the length of diagonal .

Answer

Start by using the Pythagorean theorem to find the length of .

$AF=\sqrt{(AE)^2-(EF)^2}$

$AF=\sqrt{22^2-18^2}=\sqrt{324}$

Since is one-third the length of , we know that the length of must be $3\sqrt{340}$ .

Now, notice that the pentagon is made up of one rectangle and two right triangles.

Focus on rectangle . We can find the length of by adding together the lengths of and .

$\text{AD}=\sqrt{324}+3\sqrt{324}=4\sqrt{324}$

Since we are working with a rectangle, we know that . Thus, we can use the Pythagorean Theorem again to find the length of .

$AC=\sqrt{(AB)^2+(BC)^2}$

$AC=\sqrt{36^2+(4\sqrt{324})^2}=\sqrt{6480}=80.50$

Make sure to round to two places after the decimal.

Compare your answer with the correct one above

Question

What is the length of a diagonal of a pentagon with side length of 20 centimeters (to the nearest tenth)?

Answer

Let be the length fo a diagonal.

One interior angle of a pentagon measures

$C=\frac{180(5-2)}{5}=108^{\circ }$

A diagonal of this pentagon, along with two sides, forms an isosceles triangle with sides 20, 20 and ; the side of length is opposite the angle of measure $108^{\circ }$ , and each side of length 20 is opposite an angle of measure $\frac{180-108}{2}=36^{\circ }$ .

The Law of Sines helps us to determine :

$\frac{d }{sin;108^{\circ}}=\frac{20 }{sin;36^{\circ}}$

$d=\frac{20\cdot sin;108^{\circ} }{sin;36^{\circ}}\approx\frac{20\cdot 0.9511 }{0.5878}\approx 32.4$

Compare your answer with the correct one above

Question

In the pentagon below, the length of is one-third that of . Find the length of diagonal .

Answer

Start by using the Pythagorean theorem to find the length of .

$AF=\sqrt{(AE)^2-(EF)^2}$

$AF=\sqrt{22^2-18^2}=\sqrt{324}$

Since is one-third the length of , we know that the length of must be $3\sqrt{340}$ .

Now, notice that the pentagon is made up of one rectangle and two right triangles.

Focus on rectangle . We can find the length of by adding together the lengths of and .

$\text{AD}=\sqrt{324}+3\sqrt{324}=4\sqrt{324}$

Since we are working with a rectangle, we know that . Thus, we can use the Pythagorean Theorem again to find the length of .

$AC=\sqrt{(AB)^2+(BC)^2}$

$AC=\sqrt{36^2+(4\sqrt{324})^2}=\sqrt{6480}=80.50$

Make sure to round to two places after the decimal.

Compare your answer with the correct one above

Question

What is the length of a diagonal of a pentagon with side length of 20 centimeters (to the nearest tenth)?

Answer

Let be the length fo a diagonal.

One interior angle of a pentagon measures

$C=\frac{180(5-2)}{5}=108^{\circ }$

A diagonal of this pentagon, along with two sides, forms an isosceles triangle with sides 20, 20 and ; the side of length is opposite the angle of measure $108^{\circ }$ , and each side of length 20 is opposite an angle of measure $\frac{180-108}{2}=36^{\circ }$ .

The Law of Sines helps us to determine :

$\frac{d }{sin;108^{\circ}}=\frac{20 }{sin;36^{\circ}}$

$d=\frac{20\cdot sin;108^{\circ} }{sin;36^{\circ}}\approx\frac{20\cdot 0.9511 }{0.5878}\approx 32.4$

Compare your answer with the correct one above

Question

In the pentagon below, the length of is one-third that of . Find the length of diagonal .

Answer

Start by using the Pythagorean theorem to find the length of .

$AF=\sqrt{(AE)^2-(EF)^2}$

$AF=\sqrt{22^2-18^2}=\sqrt{324}$

Since is one-third the length of , we know that the length of must be $3\sqrt{340}$ .

Now, notice that the pentagon is made up of one rectangle and two right triangles.

Focus on rectangle . We can find the length of by adding together the lengths of and .

$\text{AD}=\sqrt{324}+3\sqrt{324}=4\sqrt{324}$

Since we are working with a rectangle, we know that . Thus, we can use the Pythagorean Theorem again to find the length of .

$AC=\sqrt{(AB)^2+(BC)^2}$

$AC=\sqrt{36^2+(4\sqrt{324})^2}=\sqrt{6480}=80.50$

Make sure to round to two places after the decimal.

Compare your answer with the correct one above

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