Below, we include some additional geometry practice problems.
1. What is the area of ? Of ?
We use the Pythagorean Theorem on triangle to get that . Then, again by the Pythagorean Theorem, we get that Thus, the area of And since we know that and are similar, we get that Thus, the area of is
2. In the following diagram, What is the area of ?
Since and are similar, and since we know that Then, by the Pythagorean Theorem, we get that Again, by the Pythagorean Theorem, we get that Thus, we can add up the areas of and to get the area of So we get:
3. What is the area of ?
We know, since and are similar triangles, that . Thus, by the Pythagorean Theorem, we get and So we get that the area of
4. The radius of the below circle is 3. Find the area of the shaded region.
We know that the shaded area is part of a sector of the circle that takes up a fourth of its total area. Now, we just need to subtract the area of the triangle. We know that the triangle's base and height are just radii of the circle, so they equal 3. Thus, we get:
5. The circle below has a radius of . Find the ratio of the shaded region's area to the area of the square below.
This is just a generalization of the previous problem. Let be the radius of the circle. Then, we get (following the same method from the previous problem) that the area of the shaded region is Now, we just need to find the area of the square. Each side (by the Pythagorean Theorem) has a length of So its area must be Thus, we get that the ratio of the shaded region to the area of the square is to Or, simplifying, we get to .