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.

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.

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:

The circumference of a circle:

The area of the rectangle:

The surface area of the whole cylinder:

To find the volume of a cylinder we multiply the base area (which is a circle) and the height 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.

The volume of a pyramid is one third of the volume of a prism.

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.


The surface area of a cone is thus the sum of the areas of the base and the lateral surface:


Example



The volume of a cone is one third of the volume of a cylinder.

Example
Find the volume of a prism that has the base 5 and the height 3.


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

Mis mejores deseos en este nuevo año y siempre.

Special Parallelograms

A parallelogram may be equiangular (rectangle), equilateral (rhombus) or both equiangular and equilateral. An example of a special parallelogram that is both equiangular and equilateral is our friend the square.

Rectangle

A rectangle is a parallelogram with 4 right angles. A rectangle has the following rules:
(1) All the rules of a parallelogram.
(2) Four right angles. Remember that a right angle measures 90 degrees.
(3) Diagonals which are congruent (they have the same length).
The picture of rectangle ABCD below shows all three rules listed above.

Sample:
In rectangle ABCD below, diagonals AC and BD intersect at point R. If AR = 2x - 6 and CR = x + 10, find BD.

Since the diagonals of a rectangle bisect each other, we can say that
AR = CR.
We equate the values of AR and CR and solve for x.
2x - 6 = x + 10
2x - x = 10 + 6
x = 16
Use either of the given equations to determine that each segment equals 26. Since they are all equal, BD = 26.

Rhombus

A rhombus is a parallelogram with 4 congruent or equal sides. A rhombus has the following rules:
(1) All the rules of a parallelogram.
(2) Four sides that have the same length.
(3) Diagonals that intersect at right angles.
(4) Diagonals that bisect opposite pairs of angles.

Sample:
Given that ABCD is a rhombus and the measure of angle D = 60 degrees.
Find the measure of angles A and B.

Solution:
Triangle ABC is isosceles since line segment AB is congruent to line segment BC. Then we can say that the base angles of triangle ABC must be congruent or equal. Since we know the diagonal bisects the angles A and C, we must have two congruent triangles here. If so, then the measure of angle D = the measure of angle B = 60 degrees.

In triangle AEB, angle AEB is a right triangle because the diagonals of a rhombus are perpendicular to each other. Since the sum of the degree measures of the angles of a triangle is 180 degrees, we can say that the measure of angle A must be 30 degrees. How do we get 30?

Measure of angle A = 180 - 90 degrees - 60 degrees
Measure of angle A = 30 degrees

Square

A square is a parallelogram with 4 right angles and 4 sides that have the same length. A square has all the rules of a rectangle and a rhombus as shown in square ABCD below.

Sample:
In square ABCD, AB = x + 4. What is the perimeter of square ABCD?
Solution:
A square has the same length on all 4 sides.
We can use the formula P = side times 4, or P = 4s, where P = perimeter and s = side of square.
P = 4s
P = 4(x + 4)....We apply the distributive rule here and get
P = 4x + 16
Our perimeter is 4x + 16.

Exterior Angles of Polygons

The Exterior Angle is the angle between any side of a shape,
and a line extended from the next side.

Note: when you add up the Interior Angle and Exterior Angle you get a straight line, 180°. (See Supplementary Angles)

Polygons

A Polygon is any flat shape with straight sides

The Exterior Angles of a Polygon add up to 360°
		In other words the exterior angles add up to one full revolution (Exercise: try this with a square, then with some interesting polygon you invent yourself.) Note: This rule only works for simple polygons

Here is another way to think about it: Each lines changes direction until you eventually get back to the start: