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Area is basically the size of a surface, or how much space it takes up Area is measured in square units, such as square centimeters (cm 2 ), square inches(in 2 ), square feet(ft 2 ), square kilometers (km 2 ), etc. 1 square meter or 1 m 2 1 meter
This activity is going to look into the basic shapes of geometry (quadrilaterals, triangles, circles, polygons) and try to make some connections between them. By the end of the activity, we should have a good grasp of these basic areas, and be able to use what we learn today, and what we already know, to solve some more complicated problems. Lets get started with rectangles…
Rectangles are perhaps the simplest of two- dimensional figures, and you probably already know how to find their areas, but lets do a quick review
Rectangles have two measurable linear dimensions which were going to call the base and the height Base
To find the area of a rectangle, you simply multiply the base times the height A rectangle = b×h 18 cm 10 cm Area = 18 cm × 10 cm = 180 cm 2
So, the point of this presentation is to find out how the areas of most two dimensional shapes relate to this simple formula Well be looking at: ParallelogramsTrapezoids RhombusesTriangles KitesEven Circles!
Parallelograms are really just slanty rectangles Same Base! Same Height! = Same Area!
Just remember, its the height of the parallelogram that matters, NOT the length of the other side Use this… not this
We can find the areas of rectangles and parallelograms A rectangle = b×hA parallelogram = b×h
Rhombuses and Kites work in similar ways. Which would you like to start with (click one)?
A very useful property of rhombuses (and kites) is that their diagonals are perpendicular. In a rhombus, they also bisect each other. We can use this property to cut them up and rearrange the pieces into one of the shapes we already know. Click on the top left piece to see what happens. A Rectangle!
Diagonal 1 (d 1 ) Diagonal 2 (d 2 ) ½ Diagonal 1 (d 1 ) Diagonal 2 (d 2 ) The new rectangle we created has a base equal to the long diagonal of the rhombus, and a height equal to ½ of the short diagonal of the rhombus, giving us the equation:
Diagonal 1 (d 1 ) Diagonal 2 (d 2 ) A rhombus = ½ d 1 × d 2 Move on Learn about kites
A very useful property of kites (and rhombuses) is that their diagonals are perpendicular. In kites, one of the diagonals also gets bisected (both diagonals are bisected in rhombuses). We can use this property to cut them up and rearrange the pieces into one of the shapes we already know. Click on the top left piece to see what happens. A rectangle!
Diagonal 1 (d 1 ) Diagonal 2 (d 2 ) ½ Diagonal 1 (d 1 ) Diagonal 2 (d 2 ) The new rectangle we created has a height equal to the long diagonal of the kite, and a base equal to ½ of the short diagonal of the kite, giving us the equation:
Area of a kite… Diagonal 1 (d 1 ) Diagonal 2 (d 2 ) A kite = ½ d 1 × d 2 Learn about rhombuses
We can find the areas of rectangles, parallelograms, kites, and rhombuses A rectangle = b×hA parallelogram = b×h A kite/rhombus = ½ d 1 × d 2
Trapezoids have three measurements that well be using to find their areas: Remember, just like parallelograms, the height is the important measurement, not the length of the legs. Base 1 (b 1 ) Base 2 (b 2 ) Height (h)
Click on the trapezoid to see if we can figure their areas out. Copy it..Flip it And put the pieces together to form…
Our old friend, the parallelogram, with a combined base of (b 1 + b 2 ) So, the area of this parallelogram is (b 1 + b 2 )×h, but we only want half of that (for only one of the trapezoids), giving the formula: b2b2 b1b1 h +
A trapezoid = ½ (b 1 + b 2 )h Base 1 (b 1 ) Base 2 (b 2 ) Height (h)
Lets move away from quadrilaterals and look at triangles. For area, the two important measurements of a triangle are base and height, just like a parallelogram. Again, this is the actual height, not necessarily the length of a side. Base (b) Height (h)
To find the area of a triangle, follow the method we used for trapezoids (click on the first triangle) Copy it… Flip it… And put the pieces together to form a parallelogram
Again, we only want half of the parallelogram, which gives us the following formula for the area of a triangle: b h A triangle = ½ b×h
Circles only have one measurement to worry about, the radius (r). The radius can be used to find the Circumference by using the formula C=2πr. Watch the video on the next page to see how the area of a circle can be represented by a parallelogram. Radius
So, if we cut a circle into wedges, we can rearrange them as a parallelogram where The base is half the circumference of the circle The height is the radius of the circle This gives us the formula: A circle = π×r×r=πr 2
1. We know rectangles 2. Parallelograms are just slanty rectangles 3. We can cut up kites and rhombuses through their diagonals, then put the pieces together to make rectangles 4. Two trapezoids put together make a parallelogram 5. Two triangles put together make a parallelogram 6. Even a circle, if its cut up into many wedges, resembles a parallelogram
Here are the formulas weve learned so far… A rectangle = b×h A parallelogram = b×h A kite/rhombus = ½ d 1 × d 2 A trapezoid = ½ (b 1 × b 2 )h A triangle = ½ b×h A circle = π×r×r=πr 2
For some practice, try the problems starting on the next slide.
The following three slides can be used as a template for multiple choice questions. Instructors can decide how many (if any) of these questions to include
What is the area of the following figure (click on the correct answer): 8 cm 15 cm 10 cm 80 cm 2 120 cm 2 150 cm 2 50 cm 2