Question:

(1) A triangle has vertices at (-3,-3) (7,-3) and (-2,5). What is the area of the triangle?

Is there a way of doing this so that you dont have to graph it? If the x axis is horizontal can i say it is the base and the y-axis is horizontal meaning the height. Would I be able to do this...-3 to 7-->9 spaces, -3 to 5-->7 spaces-----9(7)63/2=31.5

(2) A cube has a surface area of 54 square centimeters. What is the volume of the cube in cubic centimeters?

Hi Jaden,

(1)

You don't have to graph it. There are lots of ways of calculating the area, but the method you've described isn't quite right.

Remember that the area of a triangle is 1/2 its base times its height. In this case, because one side of the triangle is parallel to the x-axis (you can see that because two points have the same value of y), so it is your base. The distance between -3 and +7 is 7-(-3) = 10, so your base is 10 units. The height is from -3 up to +5, so that is
5-(-3) = 8. The area is then (1/2)(8)(10).

If you didn't have one side parallel to an axis, you'd have to use a more complicated method. You can use the "Distance Formula" to find the length between any two planar points (in other words, a side of a triangle) and once you have all three lengths of sides, you can use "Heron's Formula" to find the area. This works for all triangles oriented any which way. Type those into our search box on the home page to get more information about them.

(2)

The surface area (let's call it S of a cube whose side length is x is given by 6 times the area of one side, which is x², so S = 6x². If we re-arrange this and solve for x, we get x = sqrt(S/6). Now you can put 54 in for S and get the length of a side of the cube.

The volume of a cube is just the cube of the side (x³).

Stephen La Rocque.>