A stained glass window is composed of 20 triangular sections, each with sides 6, 8, and 7 in. Find the total area of the window (to the nearest square inch).
125 insquared
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 find the area of one triangular section using Heron's formula. Heron's formula allows us to calculate the area of a triangle only using the lengths of its sides.
The formula for the area of a triangle using Heron's formula is:
Area = √(s(s - a)(s - b)(s - c))
where s is the semi-perimeter of the triangle and a, b, and c are the lengths of its sides.
In this case, the lengths of the sides of the triangle are given as 6, 8, and 7 inches.
s = (6 + 8 + 7) / 2 = 21 / 2 = 10.5 inches
Now, we can plug the values into the formula:
Area = √(10.5(10.5 - 6)(10.5 - 8)(10.5 - 7))
= √(10.5 * 4.5 * 2.5 * 3.5)
= √(415.125)
≈ 20.38 square inches (rounded to two decimal places)
Since there are 20 triangular sections in the stained glass window, we can now find the total area by multiplying the area of one section by the total number of sections:
Total Area = 20.38 square inches * 20
= 407.6 square inches
Therefore, the total area of the stained glass window is approximately 407.6 square inches.
To find the total area of the stained glass window, we can start by finding the area of each triangular section and then adding them together.
To calculate the area of a triangle, we can use Heron's formula. Heron's formula states that for a triangle with sides of lengths a, b, and c, the area (A) can be calculated using the formula:
A = √(s(s - a)(s - b)(s - c))
where s is the semiperimeter, calculated by:
s = (a + b + c) / 2
In this case, each triangular section of the stained glass window has sides of lengths 6, 8, and 7 inches. Let's calculate the area of one triangular section using Heron's formula:
a = 6 inches
b = 8 inches
c = 7 inches
s = (6 + 8 + 7) / 2 = 21 / 2 = 10.5 inches
Now, we can calculate the area (A) of the triangle:
A = √(10.5(10.5 - 6)(10.5 - 8)(10.5 - 7))
Using a calculator, we find:
A ≈ 17.146 square inches
Since there are 20 triangular sections in the stained glass window, we can now calculate the total area by multiplying the area of one triangular section (17.146 square inches) by 20:
Total area ≈ 17.146 square inches * 20 = 342.92 square inches
Therefore, the total area of the stained glass window, to the nearest square inch, is approximately 343 square inches.