what is the length of any diagonal of a regular pentagon with a side of 20mm ?
what is the length of any diagonal of a regular pentagon with a side of 20mm
first find the size of each interior angle
angle = 180(5-2)/5 = 108°
let the length of the diagonal be x
by cosine law
x^2 = 20^2 + 20^2 -2(20)(20)cos 108°
I am sure you can finish it from there.
To find the length of any diagonal of a regular pentagon, you can use the formula:
Diagonal Length = Side Length * √(5 – 2√5)
In this case, the side length is given as 20mm. So we can substitute this value into the formula:
Diagonal Length = 20mm * √(5 – 2√5)
To simplify this expression, we need to evaluate √(5 – 2√5):
1. Let's define a value: Let (√5 - 1) = x
Now, (5 - 2√5) = x^2
2. We can substitute this value back into the expression:
Diagonal Length = 20mm * √x^2
3. Since we defined x = (√5 - 1), we can substitute that value back in:
Diagonal Length = 20mm * √(√5 - 1)^2
4. Simplify (√5 - 1)^2:
(√5 - 1)^2 = (x)^2
= (x)(x)
= x^2
= (√5 - 1)(√5 - 1)
= 5 - 2√5 + 1
= 6 - 2√5
5. Substitute this value back into the expression:
Diagonal Length = 20mm * √(6 - 2√5)
Now, we have the simplified formula to find the length of any diagonal of a regular pentagon: Diagonal Length = 20mm * √(6 - 2√5)
To get the specific numerical answer, we substitute √(6 - 2√5) into a calculator to find its decimal approximation.