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Finding Area
Area: The space within a two dimensional region. Examples: Square, Triangle, Rectangle, Parallelogram, Trapezoid, Circle NOT: Cubes, Spheres, Prisms, Cylinders, and other 3-D objects |
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Area of a Rectangle
Area= Base x Height Area is the number of units on one side of a rectangle (base) multiplied by the number of units of the adjacent side of the rectangle (height). In this video we will look at why this is using doubles. **Remember: Height isn't always how tall something is and base isn't always the bottom. They can be any side we choose.
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Area of a Parallelogram
Area= base x height To find the area, we need to make this parallelogram a rectangle, or our "home base". How might we do this? Imagine the parallelogram as a rectangle with a triangle on either side. Now cut one of those triangles off and move it to the other side. Now you have a simple rectangle. From this point, we can use our equation for area of a rectangle:
base x height. |
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Area of a Triangle
Area= 1/2 (base x height) To find the area of a triangle, we have to get back to "home base"--the rectangle. To do that, we can make an exact copy of our triangle and push it up against our existing triangle to make a rectangle. From here, we can use our base x height equation. We will get the area of the whole rectangle. Remember, we only need half of this area because our triangle is half. So, we can divide the answer by two (which is the same as multiplying by 1/2). In simple form, this gives us the equation for area of a triangle: 1/2(base x height).
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Area of a Trapezoid
Area= 1/2 x height x (base 1 +base 2) To find the area of a trapezoid, we need to make it look like our "home base" which is the rectangle. The trapezoid is unique--it has to different bases. So, we need to label our two bases as base 1 and base 2. We can also label our height. This will be a perpendicular line (it makes a right angle) from either one of our bases. Now we can take that trapezoid and make a copy of it. If we flip it and push it up against the original, we have one big parallelogram.
We remember how to find the area of a parallelogram. The formula is the same as the formula for the area of a rectangle. Area= base x height. We can add our base 1 and base 2 together to get the base of our parallelogram and multiply it by the height (which never changed). Remember! We are not looking for the area of the new parallelogram! We only need half of it, our original trapezoid. So, if we divide our area of the parallelogram in half, we will have the area of our trapezoid.
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Area of a Circle
Area=πr^2 To find area of a circle, we must first cover a few new facts: Perimeter of a circle= circumference Circumference= 2πr π= 3.14 Radius: line from the middle of a circle to the outer edge We need to get to "home base". Somehow, this circle needs to become a rectangle. If we cut our circle like a pie into a bunch of slices, we can take those slices and rearrange them to look something like this: From the bottom image, we can see how the rearranged slices make a shape that is similar to a parallelogram. We can also see that since half of the slices are on the top and half are on the bottom, the circumference of the circle has been cut in half. So, the base of this parallelogram is half of 2πr, which equals πr. In addition, the remaining side of the parallelogram is the same as one side of a slice of the circle. This side is the same measurement as the radius.
We know that the area of a parallelogram is base x height. So, the area of this parallelogram equals πr x r. The "r"s can be combined to make r^2. So, area of this parallelogram= πr^2. This parallelogram is really just our circle only rearranged so the area of a circle= πr^2 as well! |
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Volume of a Cylinder
Volume= (πr^2)h 3 Dimensional Objects: If you can pick a corner and draw three lines from it, the shape is 3-D. Volume: the capacity of a 3-D shape. The empty space in a 3-D object. Cylinder: 3-D shape with two congruent faces (or bases) and parallel lines to connect them. A cylinder doesn't have to have a circular base! A cylinder will be our "home base" for finding volume. When finding volume, we use the formula: length x width x height. Remember that these names are just placeholders for different sides. To find the volume of a cylinder with a circular base, we must find the length and width of the base first. This is the same as finding the area of that base. In this case, it is a circle. We remember that the area of a circle= πr^2.
Next, we must multiply that by the height of the cylinder. This will give us a formula that looks like this: Volume of a cylinder= (πr^2)h |
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Volume of a Cone and Pyramid
Volume of a cone: 1/3 (πr^2)h Volume of a pyramid: 1/3 (length x width x height) To find the volume of a cone, we have to imagine that we have a cone and a cylinder with bases that are exactly the same. If they are exactly the same, they will fit inside each other like so: If we fill up the cone with water and pour it into the cylinder, it will fill up a third of the cylinder. This means we could put three cone-fuls of water into the cylinder. Therefore, the volume of the cone is 1/3 of the volume of the cylinder. We know the volume of the cylinder= (πr^2)h. So, if the volume of a cone is 1/3 of this, we can multiply that equation by 1/3 to get the cone's volume.
We can follow this same thinking to find the volume of a pyramid. If we have a cube and a pyramid with the same base, they will fit perfectly in one another, just like the cylinder and the cone. The cube can also be filled with three pyramid-fuls of water. So, the volume of the pyramid is 1/3 of the volume of the cube. The volume of a cube= length x width x height. If we multiply this by 1/3, we get the volume of the pyramid.
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