Geometría CECC: marzo 2014

sábado, 22 de marzo de 2014

Circle

A circle is easy to make:

Draw a curve that is "radius" away
from a central point.

And so:

All points are the same distance from the center.

You Can Draw It Yourself

Put a pin in a board, put a loop of string around it, and insert a pencil into the loop. Keep the string stretched and draw the circle!

Radius, Diameter and Circumference

The Radius is the distance from the center to the edge.
The Diameter starts at one side of the circle, goes through the center and ends on the other side.
The Circumference is the distance around the edge of the circle.
And here is the really cool thing:

When you divide the circumference by the diameter you get 3.141592654...
which is the number π (Pi)

So when the diameter is 1, the circumference is 3.141592654...

We can say:

Circumference = π × Diameter

Example: You walk around a circle which has a diameter of 100m, how far have you walked?

Distance walked = Circumference = π × 100m

= 314m (to the nearest m)

Also note that the Diameter is twice the Radius:

Diameter = 2 × Radius

And so this is also true:

Circumference = 2 × π × Radius

Remembering

The length of the words may help you remember:

Radius is the shortest word
Diameter is longer (and is 2 × Radius)
Circumference is the longest (and is π × Diameter)

Definition

The circle is a plane shape (two dimensional):
And the definition of a circle is:

The set of all points on a plane that are a fixed distance from a center.

Area

The area of a circle is π times the radius squared, which is written:

A = π r²

To help you remember think "Pie Are Squared"
(even though pies are usually round)

Or, in relation to Diameter:

A = (π/4) × D²

Example: What is the area of a circle with radius of 1.2 m ?

A = π × r²
A = π × 1.2²
A = π × (1.2 × 1.2)
A = 3.14159... × 1.44 = 4.52 (to 2 decimals)

Area Compared to a Square

A circle has about 80% of the area of a similar-width square.
The actual value is (π/4) = 0.785398... = 78.5398...%

Names

Because people have studied circles for thousands of years special names have come about.
Nobody wants to say "that line that starts at one side of the circle, goes through the center and ends on the other side" when a word like "Diameter" would do.
So here are the most common special names:

Lines

A line that goes from one point to another on the circle's circumference is called a Chord.
If that line passes through the center it is called a Diameter.
A line that "just touches" the circle as it passes by is called a Tangent.
And a part of the circumference is called an Arc.

Slices

There are two main "slices" of a circle
The "pizza" slice is called a Sector.
And the slice made by a chord is called a Segment.

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

	Quarter of a circle is called a Quadrant. Half a circle is called a Semicircle.

Inside and Outside

A circle has an inside and an outside (of course!). But it also has an "on", because you could be right on the circle.
Example: "A" is outside the circle, "B" is inside the circle and "C" is on the circle.

miércoles, 5 de marzo de 2014

The surface area and the volume of pyramids, prisms, cylinders and cones

The surface area is the area that describes the material that will be used to cover a geometric solid. When we determine the surface areas of a geometric solid we take the sum of the area for each geometric form within the solid.
The volume is a measure of how much a figure can hold and is measured in cubic units. The volume tells us something about the capacity of a figure.

A prism is a solid figure that has two parallel congruent sides that are called bases that are connected by the lateral faces that are parallelograms. There are both rectangular and triangular prisms.

To find the surface area of a prism (or any other geometric solid) we open the solid like a carton box and flatten it out to find all included geometric forms.

To find the volume of a prism (it doesn't matter if it is rectangular or triangular) we multiply the area of the base, called the base area B, by the height h.
$V=B\cdot h$
A cylinder is a tube and is composed of two parallel congruent circles and a rectangle which base is the circumference of the circle.

Example

The area of one circle is:
$\left.\begin{matrix}\, \, A=\pi r^{2}\\ \; \: \: \: \, A=\pi \cdot 2^{2}\\ \; \:\, \, A=\pi \cdot 4\\ \, \, \, \, A\approx 12.6 \end{array}$
The circumference of a circle:
$\left.\begin{matrix}\, \, \,\, C=\pi d\\ \; \: \: \: \, C=\pi \cdot 4\\ \, \, \,\, \, \, C\approx 12.6 \end{array}$
The area of the rectangle:
$\left.\begin{matrix}\, \, A=C\cdot h\\ \, \, \, \, \, \, \, A=12.6 \cdot 6\\ A\approx 75.6 \end{array}$
The surface area of the whole cylinder:
$A=75.6+12.6+12.6=100.8\, units^{2}$
To find the volume of a cylinder we multiply the base area (which is a circle) and the height h.
$V=\pi r^{2}\cdot h$
A pyramid consists of three or four triangular lateral surfaces and a three or four sided surface, respectively, at its base. When we calculate the surface area of the pyramid below we take the sum of the areas of the 4 triangles area and the base square. The height of a triangle within a pyramid is called the slant height.
Pyramid

The volume of a pyramid is one third of the volume of a prism.
$V=\frac{1}{3}\cdot B\cdot h$
The base of a cone is a circle and that is easy to see. The lateral surface of a cone is a parallelogram with a base that is half the circumference of the cone and with the slant height as the height. This can be a little bit trickier to see, but if you cut the lateral surface of the cone into sections and lay them next to each other it's easily seen.
Cone

The surface area of a cone is thus the sum of the areas of the base and the lateral surface:
$A_{base}=\pi r^{2}\: and\: A_{LS}=\pi rl$
$A=\pi r^{2}+\pi rl$
Example
Cone

$\\\begin{matrix} A_{base}=\pi r^{2}\: \: &\, \, and\, \, & A_{LS}=\pi rl\: \: \: \: \: \: \: \\ A_{base}=\pi \cdot 3^{2} & & A_{LS}=\pi \cdot 3\cdot 9\\ A_{base}\approx 28.3\: \: && A_{LS}\approx 84.8\: \: \: \: \: \\ \end{matrix}$
$\\\\\\A=\pi r^{2}+\pi rl=28.3+84.8=113.1\, units^{2}\\$
The volume of a cone is one third of the volume of a cylinder.
$V=\frac{1}{3}\pi \cdot r^{2}\cdot h$
Example
Find the volume of a prism that has the base 5 and the height 3.
$\left.\begin{matrix}\, \, B=3\cdot 5=15 \end{array}$
$\left.\begin{matrix}\, \, V=15\cdot 3=45\: units^{3} \end{array}$
Video lesson: Find the surface area of a cylinder with the radius 4 and height 8

Geometric solid

Solid Geometry is the geometry of three-dimensional space,
the kind of space we live in ...

Three Dimensions

It is called three-dimensional, or 3D because there are three dimensions: width, depth and height.

Simple Shapes

Let us start with some of the simplest shapes:

Common 3D Shapes

Properties

Solids have properties (special things about them), such as:

volume (think of how much water it could hold)
surface area (think of the area you would have to paint)
how many vertices (corner points), faces and edges they have