I need help finding the area of a pentagon as part of a larger problem. I only know the side length is 2 cm. Please help explain!
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I am assuming that the pentagon is regular; that is, all five sides have a length of 2 cm.
Split the pentagon into five congruent triangles. To find the area of the pentagon, we need to find the area of one of these triangles and multiply it by 5.
The interior angles in a pentagon sum to 180(5 - 2) = 540 degrees, so each angle is 540/5 = 108 degrees. The triangles split each angle in half, so the base angles of each triangle measure 108/2 = 54 degrees and the apex angle is 180 - 108 = 72 degrees.
To find the area of the triangle, we need to find the length of the two sides opposite to the base angles. These two sides have the same length since the base angles have the same measure. Using the Law of Sines, we see that if s is the length of the side, then:
sin(72°)/2 = sin(54°)/s ==> s = 2sin(54°)/sin(72°) ≈ 1.701 cm.
Now, we know the measure of two sides (the sides opposite to the base angles) and we know their included angle (the apex). Using A = (1/2)ab*sin(C) (a and b are the two sides and C is the included angle), the area of the triangle is:
A ≈ (1/2)(1.701)(1.701)sin(72°) ≈ 1.376 cm^2.
Therefore, the area of the trapezoid is 5A ≈ 6.880 cm^2.
I hope this helps!
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A pentagon is not made up of a square and a triangle, people.
Split the pentagon into five congruent triangles. To find the area of the pentagon, we need to find the area of one of these triangles and multiply it by 5.
The interior angles in a pentagon sum to 180(5 - 2) = 540 degrees, so each angle is 540/5 = 108 degrees. The triangles split each angle in half, so the base angles of each triangle measure 108/2 = 54 degrees and the apex angle is 180 - 108 = 72 degrees.
To find the area of the triangle, we need to find the length of the two sides opposite to the base angles. These two sides have the same length since the base angles have the same measure. Using the Law of Sines, we see that if s is the length of the side, then:
sin(72°)/2 = sin(54°)/s ==> s = 2sin(54°)/sin(72°) ≈ 1.701 cm.
Now, we know the measure of two sides (the sides opposite to the base angles) and we know their included angle (the apex). Using A = (1/2)ab*sin(C) (a and b are the two sides and C is the included angle), the area of the triangle is:
A ≈ (1/2)(1.701)(1.701)sin(72°) ≈ 1.376 cm^2.
Therefore, the area of the trapezoid is 5A ≈ 6.880 cm^2.
I hope this helps!
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A pentagon is not made up of a square and a triangle, people.
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Is it A=(1/2)(2)(1.701) ? I thought you would include the hypotenuse in the area calculation =/ why the 2 equal sides?
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Well your answer is right according the calculator online so it must be (1.701)(1.701)(1/2),
thanks again!
thanks again!
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I used the formula A = (1/2)ab*sin(C) on the two sides.
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Split it up into a triangle and a square.
Find the area of the triangle (1/2)Bh
Then find the area of the square (s^2) where s = a side.
Pentagon = Triangle + Square
P = T+S
For the triangle
So T = (1/2)2(2)
T = 2
S = 2^2
S = 4
2+4 = 6
So the area is 6 cm
Find the area of the triangle (1/2)Bh
Then find the area of the square (s^2) where s = a side.
Pentagon = Triangle + Square
P = T+S
For the triangle
So T = (1/2)2(2)
T = 2
S = 2^2
S = 4
2+4 = 6
So the area is 6 cm
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Break down the pentagon into a triangle and a square.
Area of Pentagon = Area of square + Area of triangle
Area of Pentagon = Area of square + Area of triangle