Area of a pentagon given radius.?

Question 1: A regular pentagon inscribed in a circle whose radius measures 9 inches. Find the area of the pentagon.





Question 2: A landscaper wants to plant begonias along the edges of a triangular plot of land in Winton Woods Park. Two of the angles of the triangle measure 95 degrees and 40 degrees. The side between these two angles is 80 feet long.


A. Find the measure of the third angle.


B. Find the length of the other two sides of the triangle.


What is the perimeter of this triangular plot of land?

Area of a pentagon given radius.?
1) Since the Pentagon can be divided into 5 equal triangles, and each of the angle opposite the sides of the pentagon is 360/5 = 72 degree.





Also, since each of the 5 triangles from the divided pentagon is an isoceles triangle (same 2 sides = radius), the 2 base angles are equal. Base angle = (180 - 72)/2 = 54.





Using Sine Law to calculate side of the Pentagon:


Side/sin(72) = radius /sin(54) = 9/sin(54)


Side = 10.58





Therefore each triangle has an area of:


= 0.5(side)(radius)sin(54) = 0.5(10.58)(9)sin(54)


= 38.52





Total area of Pentagon = 5 x 38.52 = 192.60





2) Use Sine Law:


The 3rd angle = 180 - 95 - 40 = 45 (A)


Since 80 is the side opposite the 45 degree angle above





80/sin(45) = side1/sin(95)


side1 = 112.7





80/sin(45) = side2/sin(40)


side 2 = 72.7





Perimeter = 80+112.7+72.7 = 265.4
Reply:This is right out of my Pre calculus book helps a lot though Report It

Reply:1)So regular pentagon inscribed in a circle.





Immediately you know those 5 sides are equal. Radius is 9 inches. Therfore if you divide the pentago into 1 triangle and 1 trapezoid. So the triangle will have a base of diameter length and the trapezoid will have a base of diameter length as well.





Also, a pentagon has 5 sides so to find the total degrees you use (n-2) times 180





n represents the number of sides





total is 540 degrees divided by 5 give 108 degrees for each angle





Now the triangle you have now is an isosceles triangle with a base of 18 inches and 2 equal base angles of 30 degrees





to find one of the sides of the pentagon you would use


A/Sin A=B/Sin B





A is a side you want to know and A is opposite angle


B is the side you know (18 inches) and B is opposite angle





A/Sin 30=18/Sin 120





A=10.39





Area of Pent=1/2(perimeter)(radius/opatholem) I don't know if thats the right spelling





Area=233.78 sq inches





2)


A.





Angle1=95 deg


Angle2=40 deg


Angle 3 therefore=45deg (180 deg-total of 2 known angles)





B.


Side1=80 feet





A/SinA=B/SinB





80/Sin45=B/Sin40





B=72.72 feet





Now the 3rd side needs to be found which is C





A/SinA=C/Sinc





80/Sin45=C/Sin95





C=112.71





So perimeter is 265.43 feet.
Reply:split the pentagon into 5 equal triangles


Then the area of one triangle is


1/2(radius^2)(sin360/5degrees)


or 1/2(r^2)(sin72)


= 81/2(sin72)


= 38.51778891


so area pentagon = 192.59 square inches





As the two known angles add to 135, then the third angle is 45 degrees.


B. If you use the sine rule then one side = 80(sin95)/sin(45)


and the other is 80(sin40)/sin45


I make the perimeter 265.43 feet.