Just as with perimeter before, we can use coordinate grids to help us determine the area of certain figures. We're going to use the coordinates of the corners of the figures to determine the lengths of the sides and then use that information and apply it to the formula for the area. So let's begin by looking at this square. First, let's determine the dimensions of the square. This side here goes from an x-coordinate of 4 to an x-coordinate of 9. That means it has a length of 5, which is 9 minus 4. And this side here goes from a y-coordinate of 5 to a y-coordinate of 10, so it also has a length of 5. And we could also count the boxes 1, 2, 3, 4, 5 for the width and 1, 2, 3, 4, 5 for the height. Area of a rectangle, which a square is a type of rectangle, the formula we use is area equals width times height. So in this case area is equal to 5 times 5, which is 25 units squared. Area always has units that are squared, so in this case the area is 25 units squared. Let's look at another one.

This one reads, “What is the area of the right triangle shown below?” The formula for the area of a triangle is area equals 1/2 base times height, so first let's determine the length of the base, which goes from an x-coordinate of negative 9 to an x-coordinate of negative 3, which is 6 units. We can verify that 1, 2, 3, 4, 5, 6 units. The height, we go from a y-coordinate of 5 to y-coordinate of 10, which is 5 units. And again we can verify that 1, 2, 3, 4, 5 units. So the area of this triangle is going to be 1/2 times 6 times 5. That gives us an area of 6 times 5 is 30 times 1/2 is 15 units squared. Let's look at another one.

This time we're trying to find the area of an acute triangle. The formula is the same: area equals 1/2 base times height. The base here we go from an x-coordinate of negative 11 to an x-coordinate of negative 3, which that's 8 units: 1, 2, 3, 4, 5, 6, 7, 8. The height is going to be the vertical distance from the top of the triangle to the base, like that. So here our base has a y-coordinate of negative 10 and the top of the triangle has a y-coordinate of negative 4, so that's an increase of 6 units, and we can verify that 1, 2, 3, 4, 5, 6 units. So the area of this triangle is going to be equal to 1/2 times 8 units for the base times 6 units for the height. 1/2 times 8 times 6 is 24 square units. Alright, let's look at one more example.

This one asks us, “What is the area of the obtuse triangle shown below?” Well again, as with the other triangles, the area is equal to 1/2 base times height. Our base goes through an x-coordinate of 4 to an x- coordinate of 8, which is 4 units, 1, 2, 3, 4. The height of this triangle is a little trickier to visualize. So, our base has a y-coordinate of negative 10, and the tip of this triangle, the uppermost point, has a y- coordinate of negative 4. That's this distance right here, which is 6 units. We can verify that again 1, 2, 3, 4, 5, 6. So the area of this triangle is going to be equal to 1/2 times the base which is 4 units times the height which is 6 units. 1/2 times 4 times 6 is 12 units squared.