Perimeter: 115.472 ft
The perimeter of an enclosure is the length around the enclosure, or the sum of all the outer sides added up (red+blue lines in picture).
1) We are told that the two vertical sides of the enclosure (shown in blue) are each 20ft, making it 20 + 20 = 40ft total. All we need to find is the sum of the lengths of the four sides of the triangles (shown in red).
2) We are given the height of the triangle, 16ft, and the length of the base of the triangle, 20ft. To find the length of the edge we're looking for, we need to find the hypotenuse of the right triangle shaded in green. One of the legs of the triangle is 10ft (half of the base of the larger triangle) and the other leg is 16ft (same as height of the larger triangle).
Use the pythagorean theorem, which says [tex] a^{2} + b^{2} = c^{2} [/tex].
a and b are the length of your legs, 10ft and 16ft, and c is the length of the hypotenuse, or the side we're looking for. Plug your numbers in and find c:
[tex]10^{2} + 16^{2} = c^{2} \\
c^{2} = 356\\
c = 18.868 ft[/tex]
3) Now you know each red side is 18.868 ft. There are 4 red sides, so 4 x 18.868ft = 75.472ft. Add this number to the length of your two blue sides to get the perimeter:
75.472ft + 40ft = 115.472ft.
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Area: [tex]720ft^2[/tex]
The area of the enclosure is shaded in yellow. To find the area of the enclosure, you want to find the sum of the areas of all the shapes. Break the enclosure down into one square (in the middle) with two triangles on the top and bottom, find the area of those three shapes, and add those areas together.
1) The equation for the area of the square is [tex]A=s^2[/tex], where s = length of the side.
You're told that the length, s = 20ft. Plug that in and solve for the area of the square:
[tex]A=s^2\\
A=20^2\\
A = 400 ft^2[/tex]
2) The equation for the area of a triangle is [tex]A = \frac{1}{2} bh[/tex], where b = length of base and h = height of triangle.
You're told that the length of the base, b = 20ft and the height of the triangle, h = 16ft. Plug these values in and solve for area of one of the triangles:
[tex]A = \frac{1}{2} bh\\
A = \frac{1}{2} (20)(16)\\
A = 160 ft^2[/tex]
3) Since you have two identical triangles, multiply the area of one triangle by two to get the total area of the two triangles:
[tex]160 ft^2 \times 2 = 320 ft^2[/tex]
4) Add up the areas of the square and the two triangles to get the total area of the enclosure:
[tex]400 ft ^2 + 320 ft^2 = 720ft^2[/tex]
