March 29: How do we find the surface area and lateral area of pyramids & cones? (AIM)

Aim: How do we find the surface area and lateral area of pyramids & cones?

• So like last time's review, the surface area is the outside of a solid. So here we'll be working with solids like pyramids & cones. The similarity between these two solids is that they both have triangle(s). 

• Pyramids have triangles as the lateral area and only one face. This face can be any shape, from a circle (cone) to a heptagon (7 sides). 

• A pyramid in general is a solid that connects a polygon base to a point, called the apex. 

• For the surface area & lateral area this is the formula. 

• But remember this is only for a square pyramid. For any base use the formula of 1/2pl (perimeter x slant length)  + B (area of the base). 

• For a cone it's the same, but its for a circular base. 

• Well that's about it. So try this problem !

What is the total surface area of this cone, rounded to 

the nearest tenth ?

Sources Cited: 

1. John Schnatterly AKA -->  http://johnschnatterly.blogspot.com/
2. http://www.mathsisfun.com/geometry/pyramids.html
3. http://www.mrlarkins.com/geometry/InteractiveTextbook/Ch10/10EM/PH_Geom_ch10_Review.html
4. http://hotmath.com/help/gt/genericprealg/section_9_5.html

March 28: How do we calculate the surface area of cylinders? (AIM)

Aim: How do we calculate the surface area of a cylinder? 

• To find the surface area of a any figure, you must have the right idea of what surface area is. 

• Surface area if one of the properties to a solid (which is 3-Dimensional). It covers the outside of the figure, unlike the volume (which is the inside).

• When calculating the surface area of a cylinder, you should remember that it's always easier to break the figure down into smaller shapes. For example:

So this figure shows
how there are 2
circles & a rectangle
in a cylinder.

• So for surface area you'll basically need the formula of a rectangle + 2 times the formula for a circle. Which gives you the SA formula. 

• So if that's how you find surface area, then how do you find lateral area? Well you basically just subtract the the bases. 

So lateral area is
the rectangle of
the cylinder. 

• Now that this is somewhat down-packed, try this problem. Don't worry its not hard ?! 

A cylinder has a radius of 8 inches and a height of 12 inches. What are the surface area of this cylinder?

1. SA = 320π
2. SA = 208π
3. SA = 480π

Sources Cited:

1. John Schnatterly AKA -->  http://johnschnatterly.blogspot.com/
2. http://www.mathsteacher.com.au/year8/ch10_geomcons/09_cones/cylinder.html
3. http://teamshen.blogspot.com/2010/04/how-does-surface-area-of-cylinder.html
4. http://www.mathguide.com/lessons/SurfaceArea.html

March 17: How do we find the area of regular polygons? (AIM)

Aim: How do we find the area of regular polygons? 

• Regular polygons are polygons with all sides and all angles equal.

• Some examples usually used as regular polygons are:

• Pentagons - 5 (1/2 x a s)

There are 5 equal
triangles, with equal
side lengths. 

• Hexagons - 6 (1/2 x a s)

 

• Heptagons - 7 (1/2 x a s) 

• But there's a shorter way to know this. Also remember that in any case n is number of sides, s is side length, and a is apothem [distance from center of polygon to the side].

This table shows that
the formula is 1/2 times
the # of sides, apothem,
and side length. 

• Also there's another rule for finding the area of a regular polygon. It's 1/2 x P x a. In this case P is the perimeter and a is the apothem. 

• Now try this problem:

1. What is n and the areas of this figure? 

Sources Cited:

1. John Schnatterly AKA -->  http://johnschnatterly.blogspot.com/
2. http://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/jenna2.html
3. http://www.kathykwylie.com/blog/2010/10/geometric-shapes-in-quilts-hexagons/
4. http://mryipsmathclass.blogspot.com/2010_11_01_archive.html
5. http://www.trueknowledge.com/q/facts_about__pentagon

March 20: How do we find the area of a circle? (AIM)

Aim: How do we find the area of a circle?

• A circle is known as a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed center. 

• There are circles EVERYWHERE, seriously. Look at anything right now, & i will guarantee you can find a type of circle. For example is you have the iPhone, here's some circles on your phone (that you probably never let go of) :

These are just SOME circles
on the iPhone. Even the circles
i drew on there are CIRCLES man !

• Now when finding the area of a circle, its just a little tricky. This is because there's 2 formulas for a circle. There's circumference (perimeter for circles) & area, but they're 2 different things. But they do have one thing the same, PI. 

• For circumference, you multiply the radius by 2 or multiply the diameter by Pi. Or go by the rule of either 2πr or πd. Either way they're both the same. 

• For area you'll square the radius and multiply it by Pi. Or the rule is  πr².

 

The pink/purple shaded
part of the circle (inside),
is the area of a circle.

• Here's an example of finding the area & circumference of a circle :

 

• So now that you've got everything packed down on area & circumference ... try this problem out :

1. What is the area of the colored part this circle ? 

 

Sources Cited:

1. John Schnatterly AKA -->  http://johnschnatterly.blogspot.com/
2. www.apple.com
3. http://www.mathsisfun.com/geometry/circle.html
4. http://www.free-online-art-classes.com/images/copencilball.jpg
5. http://www.udjuni.com/learningcontent/gui/displaycontent.php?content=7&contentid=83

March 15: How do we find the area of parallelograms, kites, and trapezoids ? (AIM)

Aim: How do we find the area of parallelograms, kites, and trapezoids ?

• The area of a shape is the amount of space inside the boundary of a flat (2-dimensional) object. 

• Here's a video link that can help you understand it more:  http://www.brainpop.com/math/geometryandmeasurement/areaofpolygons/

• On that link you can watch the video, try the activity, take a quiz on it, & etc. But I will still try to explain it here. 

• The area for a parallelogram is probably the easiest. Just because you only have to multiply length and height (l x w). But there is an easier way to look at it. 

 

• Now the area of a kite is very different. You have to multiply the 2 diagonals and then divide that by 2. ((d1 x d2) / 2)

 

• The area for a trapezoid is kind of similar to the kite's, but instead of diagonals there are bases & a hieght. In this case you add the 2 bases and then multiply that by the half of the height. 

 

• So after all this wonderful information, try this problem here !

 

Sources Cited:

1. http://savannahsmathblog.blogspot.com/2010/04/area-and-volume.html 
2. http://www.k6-geometric-shapes.com/kite-shape.html
3. http://www.algebra-class.com/area-formula.html
4. John Schnatterly AKA -->  http://johnschnatterly.blogspot.com/

March 12: How do we calculate the area of rectangles and triangles? (AIM)

Aim: How do we calculate the area of rectangles and triangles?

 

• Well for starters, the are of a rectangle is very different from the area of a triangle. 

• The area of a triangle has a formula like this:

Area would equal
1/2 times the base
and height.

• The area of a rectangle has a formula like this:

The length is multiplied
by the width.

• Sometimes in a question, you'll get a shape that looks kind of funny. But if you break it down, there might be a triangle or/and a rectangle. For example in this problem:

What would be the area
for this figure? 

• So you would break it down like this:

Let's break it down into two parts.
Part A is a rectangle,
& Part B is a triangle. 

• Part A is a square:

1. Area of A = w x h = (20m) × (20m) = 400m²

• Part B is a triangle: 

       Area of B = ½(b × h) or (b × h)/2 = ½ × (20m × 14m) = 140m²

• Now to find the area of that figure you saw earlier, you add the 2 areas:

       Area = (Area of A) + (Area of B) = 400m² + 140m² = 540m² 

 

                                                      Now you try it !

• What is the area of this triangle ?

1. 30
2. 90
3. 56
4. 28

Sources Cited:

1. http://www.math-worksheets.info/2010/06/worksheets-area-triangle/
2. http://ed101.bu.edu/StudentDoc/current/ED101fa09/arl2013/Area.html
3. http://www.mathsisfun.com/area.html
4. John Schnatterly AKA -->  http://johnschnatterly.blogspot.com/

 

March 5: How do we find the locus of points? (AIM)

Aim: How do we find the locus of points?

• When finding the locus or the locus of points, you should go by this definition:

                                            - the set of all points that satisfy a given condition.

• The locus is usually graphed on a general graph of a given equation. but this is only sometimes, depending on the question (what is asked). 

A single point:

• When graphing the locus of points that are equidistant from a single point, it'll create a circle with the original point at it's center point. Also with the set of points in the line of the circle !

2 points:

• When graphing the locus of points that are equidistant from 2 points , it'll be the perpendicular bisector of the line segment connecting the 2 points. This forms a line through the middle of the 2 points.

The 2 points which are P & Q
have a line that makes a perpendicular
bisector through the line that P & Q.

A line:

• When graphing the locus of points that are equidistant from a line are 2 lines (that are on opposite sides), it'll be the same distance [equidistant] and parallel to the line. This leaves 2 parallel lines on opposite sides of the original line. 

2 lines:

• When graphing the locus of points that are equidistant from 2 parallel lines is another line (half-way between both lines), then each of the them will be parallel. So it'll be a line through the middle of the 2 lines. 

These locus of points are equidistant
from the y-axis. They are on the
lines x=-1.5 and x=1.5. 

2 intersecting lines:

• When graphing the locus of points that are equidistant from 2 intersecting lines, it'll bisect the angles formed by the original lines. That means that the 2 intersecting lines are half-way between the 2 original lines.  

Line 1 and Line 2 intersect at the origin. 

Sources Cited: 

1. http://www.cramster.com/definitions/locus-of-points/490
2. http://regentsprep.org/Regents/math/geometry/GL1/What.htm 
3. http://www.regentsprep.org/regents/math/geometry/GL1/ans5.htm
4. John Schnatterly AKA -->  http://johnschnatterly.blogspot.com/