A saw blade is made by cutting six triangles, each with angles of 30, 60 and 90, out of a regular hexagon. If the length of the longest diagonal of the hexagon is 12, what is the area of the saw
blade?
The area of the saw blade is 24√3.
To find the area of the saw blade, we first need to find the length of each side of the hexagon. Since the hexagon is regular, all sides are of equal length.
Step 1: Finding the length of each side of the hexagon
The hexagon can be divided into six congruent equilateral triangles, each with angles of 60 degrees.
To find the length of each side of the hexagon, we can use the cosine formula:
cos(60) = adjacent / hypotenuse
In this case, the hypotenuse is 12, and we want to find the length of the adjacent side (which is half the length of each side of the hexagon).
Rearranging the formula, we get:
adjacent = cos(60) * hypotenuse
adjacent = cos(60) * 12
Now, let's calculate the value of cos(60) = 0.5:
adjacent = 0.5 * 12
adjacent = 6
So, the length of each side of the hexagon is 6.
Step 2: Finding the area of one of the triangles
Since each triangle has angles of 30, 60, and 90 degrees, it is a special type of right triangle known as a 30-60-90 triangle. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3/2 times the length of the hypotenuse.
In this case, the hypotenuse is the length of each side of the hexagon, which is 6. Therefore, the side opposite the 30-degree angle is 6/2 = 3, and the side opposite the 60-degree angle is (√3/2) * 6 = 3√3.
The area of the triangle can be calculated using the formula:
Area = (1/2) * base * height
The base of the triangle is the side opposite the 30-degree angle, which is 3, and the height of the triangle is the side opposite the 60-degree angle, which is 3√3.
Area = (1/2) * 3 * 3√3
Area = (9/2)√3
Step 3: Finding the area of the saw blade
Since the saw blade is made up of six congruent triangles, the total area of the saw blade is six times the area of one triangle.
Area of the saw blade = 6 * (9/2)√3
Area of the saw blade = 27√3
Therefore, the area of the saw blade is 27√3 units squared.
To find the area of the saw blade, we need to find the area of the regular hexagon and subtract the areas of the six triangles.
Step 1: Find the length of each side of the hexagon.
Since we know that the longest diagonal of the hexagon is 12, we can use this to find the length of one side.
The longest diagonal of a regular hexagon is twice the length of one side. So the length of one side is 12/2 = 6.
Step 2: Find the area of the hexagon.
The area of a regular hexagon can be found using the formula: (3√3 * s^2)/2, where s is the length of one side.
Substituting the value of s, we get:
(3√3 * 6^2)/2 = (3√3 * 36)/2 = 54√3.
Therefore, the area of the hexagon is 54√3.
Step 3: Find the area of each triangle.
Since each triangle has angles of 30, 60, and 90 degrees, we know that it is a 30-60-90 triangle.
In a 30-60-90 triangle, the ratio of the lengths of the sides is 1 : √3 : 2.
Since we know the length of one side of the triangle is 6 (the same as one side of the hexagon), we can find the lengths of the other two sides using the ratio:
Shortest side = 6 * 1 = 6
Medium side = 6 * √3 = 6√3
Longest side = 6 * 2 = 12
Step 4: Find the area of each triangle.
The area of a triangle with sides of lengths a, b, and c can be found using Heron's formula:
Area = √(s * (s - a) * (s - b) * (s - c)), where s is the semi-perimeter of the triangle, given by s = (a + b + c)/2.
For each triangle, the semi-perimeter is (6 + 6√3 + 12)/2 = (18 + 6√3)/2 = 9 + 3√3.
Thus, the area of each triangle is:
√((9 + 3√3) * (9 + 3√3 - 6) * (9 + 3√3 - 6√3) * (9 + 3√3 - 12)) = √((9 + 3√3) * (3 + 3√3) * (3 - √3) * 3).
We can simplify this expression:
√((1278 + 702√3)).
Since there are six triangles, the total area of the triangles is:
6 * √(1278 + 702√3).
Step 5: Find the area of the saw blade.
To find the area of the saw blade, we subtract the total area of the triangles from the area of the hexagon.
Area of the saw blade = Area of hexagon - Total area of triangles
= 54√3 - (6 * √(1278 + 702√3)).
Therefore, the area of the saw blade is 54√3 - (6 * √(1278 + 702√3)).