+0  
 
0
194
2
avatar

Find the area of a regular pentagon with a perimeter of 240km

Guest Mar 17, 2015

Best Answer 

 #2
avatar+91481 
+5

Cut your regular pentagon into 5 isosceles triangles each radiating from the centre.

The cut each of those triangles in half making 10 congruent right angled triangles.

the hypotenuse is the radius of the hypotenuse of the triangles and

One of the sides of the triangle is a half of one side of the pentagon

and the other side of the trianngle is the apothem of the pentagon  I am going to call this side $$a$$

 

The angle radiating from the centre of each right angled triangle is    (360/5)/2 = 36 degrees

the length of the side opposite the centre angle is     0.5(240/5) = 24km

 

Now use tan to get the length of the apothem.     you will get     $$a\approx 33.03km$$

 

 

$$\\Area = 10* $ area of right angled triangle$\\
=10* (0.5*base * height)\\
=10*0.5*24*33.03\\
\approx 3964 km^2$$

 

Melody  Mar 18, 2015
Sort: 

2+0 Answers

 #1
avatar+81077 
+5

The side must be 240/5 = 48km

We have a "formula" to find the area given the side length....I won't derive it for you, but it is given by:

A = 1.7204 s^2   where s is the side length....so we have

A = 1.7204 (48)^2 = 3963.8 km ^2

 

  

CPhill  Mar 17, 2015
 #2
avatar+91481 
+5
Best Answer

Cut your regular pentagon into 5 isosceles triangles each radiating from the centre.

The cut each of those triangles in half making 10 congruent right angled triangles.

the hypotenuse is the radius of the hypotenuse of the triangles and

One of the sides of the triangle is a half of one side of the pentagon

and the other side of the trianngle is the apothem of the pentagon  I am going to call this side $$a$$

 

The angle radiating from the centre of each right angled triangle is    (360/5)/2 = 36 degrees

the length of the side opposite the centre angle is     0.5(240/5) = 24km

 

Now use tan to get the length of the apothem.     you will get     $$a\approx 33.03km$$

 

 

$$\\Area = 10* $ area of right angled triangle$\\
=10* (0.5*base * height)\\
=10*0.5*24*33.03\\
\approx 3964 km^2$$

 

Melody  Mar 18, 2015

33 Online Users

avatar
avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details