A stained glass window is composed of 20 triangular sections, each with sides 6, 8, and 7inches. Find the total are of the window( to the nearest square inch).
To find the total area of the stained glass window, we need to find the area of each triangular section and then sum them up.
The formula to calculate the area of a triangle is given by:
Area of a triangle = (base * height) / 2
In this case, the base is the side measuring 8 inches, and the height is the distance between the base and the opposite vertex. To find the height, we can use the Pythagorean theorem since the triangle is a right triangle.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's calculate the height using the Pythagorean theorem:
Height = sqrt((hypotenuse^2) - (base^2))
Height = sqrt((7^2) - (6^2))
Height = sqrt(49 - 36)
Height = sqrt(13)
Now we can find the area of each triangular section:
Area = (base * height) / 2
Area = (8 * sqrt(13)) / 2
Area = 4 * sqrt(13)
Since there are 20 triangular sections in the stained glass window, we multiply the area of one section by 20 to get the total area:
Total Area = 20 * (4 * sqrt(13))
To get the answer to the nearest square inch, we can calculate the numerical value of the total area:
Total Area ≈ 20 * (4 * sqrt(13)) ≈ 20 * 4 * 3.6056 ≈ 288.448 square inches
Therefore, the total area of the stained glass window, rounded to the nearest square inch, is approximately 288 square inches.
To find the total area of the stained glass window, we need to find the area of each triangular section and then multiply it by the total number of sections.
First, let's use Heron's formula to find the area of one triangular section:
Heron's formula states that the area (A) of a triangle with sides a, b, and c is given by:
A = sqrt(s * (s - a) * (s - b) * (s - c))
where s is the semiperimeter of the triangle, calculated as:
s = (a + b + c) / 2.
For each triangle, a = 6 inches, b = 8 inches, and c = 7 inches.
Calculating s:
s = (a + b + c) / 2
s = (6 + 8 + 7) / 2
s = 21 / 2
s = 10.5 inches
Now, let's use Heron's formula to calculate the area (A) of one triangular section:
A = sqrt(s * (s - a) * (s - b) * (s - c))
A = sqrt(10.5 * (10.5 - 6) * (10.5 - 8) * (10.5 - 7))
A = sqrt(10.5 * 4.5 * 2.5 * 3.5)
A = sqrt(419.0625)
A ≈ 20.47 square inches (rounded to two decimal places)
Since there are 20 triangular sections, we multiply the area of one section by 20 to find the total area of the stained glass window:
Total area = 20 * 20.47
Total area ≈ 409.4 square inches (rounded to the nearest square inch)
Therefore, the total area of the stained glass window is approximately 409.