Find the area of the shaded regions.

Question

d

Answer 1

Hi there Jessica,

I'll answer the one on the left first. Given the triangle in question we can see that it's a 45-45-90 triangle, which means the length of the diameter will be 4√2. This means that the radius will be 2√2. Now let's find the area of the top half of the circle and subtract it from the area of the triangle. Since we're focusing on just half of the circle our equation for area will be (1/2)πr². Plugging in our radius will give us an area of 4π. Since the triangle is a right triangle the two legs are the height and base, meaning that the area will be given by (1/2)x4x4 (remember it's a 45-45-90 so both legs are the same length). This will give us an area of 8 for the triangle. So the area of the shaded region will be 4π-8 cm.

For the second one what we'll do is find the area of the large circle, then subtract the area of the medium white circle, and finally add the last small grey circle to find the total area. The area of the large circle follows πr²→π8²=64π. Next the area of the medium white circle is going to be π4²=16π. The radius of 4 comes from the fact that from O to B is another radius of 8 and the dashed lines tell us that it's evenly split in half, giving us a radius of 4. Finally the area of the little circle is going to equal π2²=4π. Once again the dashed lines tell us that the diameter of the little circle is 4 which means the radius is equal to 2. Finally we follow our equation from the start, 64π-16π+4π= 52π cm²