Suppose you have an isosceles triangle, and each of the equal sides has a lenght of 1 foot. Suppose the angle formed by those two sides is 45^\circ. Then the area of the triangle is square feet.
.353553
area = (1/2)(1)(1)sin45
= 1/(2√2)
To find the area of the isosceles triangle, we can use the formula A = (1/2) × base × height.
Since the triangle is isosceles and the two equal sides have a length of 1 foot, we can draw an altitude from the top vertex to the midpoint of the base, creating two congruent right triangles.
In each of these right triangles, the base and height are equal, forming a 45-45-90 triangle. In a 45-45-90 triangle, the length of the hypotenuse is √2 times the length of the legs.
Therefore, the length of the base and height of each right triangle is √2/2.
Now we can substitute the values into the formula:
A = (1/2) × (base) × (height)
= (1/2) × (√2/2) × (√2/2)
= (1/2) × (2/4)
= 1/4 square feet
Thus, the area of the isosceles triangle with side lengths of 1 foot and an angle of 45 degrees is 1/4 square feet.
To find the area of the isosceles triangle, we can use the formula:
Area = (1/2) * base * height
In this case, the base of the triangle is one of the equal sides, which has a length of 1 foot.
To find the height, we can use trigonometry, specifically the sine function. Since we know that the angle formed by the two equal sides is 45 degrees, and the triangle is isosceles, we can consider it as a right triangle.
In a right triangle, the sine of an angle is defined as the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the height of the triangle, and the hypotenuse is one of the equal sides, which also has a length of 1 foot.
So, sin(45 degrees) = height / 1 foot
Simplifying, we find:
height = 1 foot * sin(45 degrees)
Using a scientific calculator, we find that sin(45 degrees) is approximately 0.7071.
Therefore, the height of the triangle is:
height = 1 foot * 0.7071 ≈ 0.7071 feet
Now, we can substitute the values into the area formula:
Area = (1/2) * 1 foot * 0.7071 feet
Simplifying, we have:
Area ≈ 0.5 square feet * 0.7071 feet
Area ≈ 0.3536 square feet
Therefore, the area of the isosceles triangle is approximately 0.3536 square feet.