What is the area of this triangle?

Question

its a 45-45-90 with sides of 3,3,3squarerootsof2

Answer 1

4.5

Answer 2

Because your triangle contains a right angle, it is a right triangle. The formula for the area of a triangle is:

A = b h / 2

where A is area, b is the length of the base, and h is the height of the triangle. In your case, the base and height are both 3, so the area is:

A = 3 * 3 / 2

A = 9 / 2 = 4.5

You can calculate the area of any triangle (not just a right triangle) by using the more general Heron's formula:

s = (a + b + c) / 2

A = √(s(s - a)(s - b)(s - c))

where s is the semiperimeter of the triangle (the perimeter divided by two, as shown in the first equation), A is the area of the triangle, and a, b, and c are the three side lengths.

If we plug in the values for your triangle, we see that once again we arrive at the same answer:

s = (3 + 3 + 3√2) / 2 = 3 + (3√2) / 2

A = √((3 + (3√2) / 2)(3 + (3√2) / 2 - 3)(3 + (3√2) / 2 - 3)(3 + (3√2) / 2 - 3√2))

A = √((3 + (3√2) / 2)((3√2) / 2)((3√2) / 2)(3 - (3√2) / 2))

A = √((9 - 18 / 4)(18 / 4))

A = √(81 / 2 - 81 / 4)

A = √(81 / 4)

A = 9 / 2 = 4.5