Question Video: Finding the Area of a Composite Figure

Using 3.14 as an estimate for 𝜋, find, to the nearest tenth, the area of the shown figure, given that each square represents 10 square centimeters.

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Video Transcript

Using 3.14 as an estimate for 𝜋, find, to the nearest tenth, the area of the shown figure, given that each square represents 10 square centimeters.

The figure we’re given is a composite figure. It’s made up of multiple shapes. In order to find the area, we’ll need to break this figure up into shapes we know how to find the area of. We recognize that we have a half circle. And then for the bottom piece, we have some choices. If we draw a line here, the two figures we’ve created are trapezoids. And that means you could find the area of both of these trapezoids.

However, if you didn’t remember that formula or you wanted to break this shape up into smaller pieces, you could break it up into triangles and rectangles or squares. For the rectangle and square here, we could either count the number of squares in the area or we could use the formula for finding the area of a rectangle or a square.

We also need to deal with the fact that each square represents 10 square centimeters. Probably the easiest way to deal with this is waiting until our last step. We can find the area in units squared first and then multiply that by square centimeters.

So let’s get started. We have a half circle. To find the area of a half circle, we take one-half 𝜋 times the radius squared. The radius is the distance from the center to any point on the outside of the circle. There’s a distance of six units from the center to the top or from the center to the outside. This confirms our radius of six. So we can multiply one-half by six squared. And then we’re using 3.14 for 𝜋. When we do this multiplication, we get 56.52, and we’ll leave that as units squared for now.

Next, we can find the area of our two triangles. We know that the area of a triangle is one-half times the height times the base. If we let this be our first triangle, it has a base of six and a height of six. To find the area then, we multiply one-half times six times six. And we say that the area of the larger triangle is 18 units squared. And then for our smaller triangle, we’ll need to find the area. It has a height of three units and a base of three units. So the area will be one-half times three times three, which is four and a half units squared. And for a rectangle, the area is length times width. This rectangle has a length of eight and a width of six. When we multiply those together, we get 48 units squared. And finally, for a square, the area equal side squared. This square has a side of three, so three squared equals nine units squared.

Now, we have broken this composite figure up into five smaller figures. And we need to add these values together. We’ll combine these five to find the area of our composite figure in units squared, which will give us 136.02 units squared. Now, if every unit squared in our figure equals 10 centimeters squared, we have 136.02 of these square units. And to convert that value into centimeters squared, we just need to multiply by 10, which gives us 1360.2 centimeters squared. And that is already rounded to the nearest tenth.

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