The with B base area, and h height can be calculated using the formula below.
V=Bh
We are given a in horizontal position with a height of 4 and its base is a regular whose side length is 2.
The area of a is half the product of its and . Note that we are only given the side length.
We will first need to find the and the apothem of the pentagon. Then we will use the formula A= 12ap to find the area.
Finding the Perimeter
In a regular pentagon all five sides have the same length. Therefore, we can obtain its perimeter by multiplying the length of a side by 5.
5* 2= 10
The perimeter of the given polygon is 10 yards.
Finding the Apothem
Let's now find the apothem. To do so, we will start by drawing the of the pentagon. Be aware that the radii divide a regular pentagon into five congruent .
Since corresponding of are , the vertex angles of the isosceles triangles formed by the radii are congruent. Moreover, since a full turn measures 360^(∘), we can divide 360 by 5 to obtain their measures.
360/5=72
The vertex angles of the isosceles triangles measure 72^(∘) each.
The apothem both the vertex angle of the isosceles triangle and the opposite side of the vertex, which is a side of the pentagon. As a result, a 36^(∘)-54^(∘)-90^(∘) triangle is created. The length of its shorter leg is 2÷ 2=1.
To find the apothem we can use the .
cot36=a/1 ⇔ a=cot36≈ 1.38
Therefore, the length of the apothem is approximately 1.38 yards. However, keep in mind that, to find the exact value of the volume, we will substitute cot36 for a in the following operations.
Finding the Area
To find the area of the given regular polygon, we will substitute a=cot36 and p= 10 in the formula A= 12ap.
A=1/2ap
A=1/2(cot36)( 10)
A=1/210 cot36
A=10 cot36/2
A=5 cot36
The area of the base of the given prism is about 5cot36 squared yards.
Volume
We calculated that the area of the base B is 5cot36 squared yards. We are also given that the height equals 4 yards. Let's substitute these values into the formula of a volume of a prism and solve for V.
V=Bh
V= 5cot36( 4)
V=4(5cot36)
V= 20cot36
V=27.52763...
V≈ 27.53
The volume of the prism is about 27.53 cubic yards.
