NYC GEOMETRY TUTORING

GEOMETRY TUTORING

Module 1: Plane Geometry

Topic A: Introduction to Geometry

Plane Geometry is about flat shapes like lines, circles, and triangles … shapes that can be drawn on a piece of paper.
Point, Line, Plane and Solid
A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)

Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper).

Let us start with a point. A point has no dimensions.
A point really has no size at all! But we show them as dots so we can see where they are.

Now let’s allow the point to move in one direction. We get a line.
We need just one value to find a point on that line. So we have one dimension. A line is one-dimensional.

Now let’s allow the point to move in a different direction. And we get a plane. We need two values to find a point on that plane. So we have two dimensions, or “2D”.
Circles, triangles, squares, and more are plane shapes.

We let that point move in another completely different direction, and we have three dimensions.
Spheres, cubes, cylinders, and more are 3-dimensional or “3D”. We also call them solid shapes.

The world we live in is 3D.
We can have more dimensions (such as this 4D Tessaract) in mathematics, but they are hard to draw!

2D Shapes

A polygon is a plane (2D) shape with straight sides. To be a regular polygon, all the sides and angles must be the same:

Other Common Polygons

These are 2D shapes, and they are polygons, but not regular polygons:

Curved Shapes

These are also 2D shapes but are not polygons because they have curves:

Triangles

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane. It can be classified into three: Equilateral, Isosceles, and Scalene.

There are three special names given to triangles that tell how many sides (or angles) are equal. There can be 3, 2, or no equal sides/angles:

How to remember? Alphabetically they go 3, 2, none:

• Equilateral: “equal”-lateral (lateral means side) so they have all equal sides
• Isosceles: means “equal legs”, and we have two legs, right?   Also isosceles has two equal “Sides” joined by an “Odd” side.
• Scalene: means “uneven” or “odd”, so no equal sides.

What Type of Angle?

Triangles can also have names that tell you what type of angle is inside:

Combining the Names

Sometimes a triangle will have two names, for example,

Right Isosceles Triangle
Has a right angle (90°), and also two equal angles

Topic B: Perimeter

Perimeter is the distance around a two-dimensional shape.

Example: the perimeter of this rectangle is 7+3+7+3 = 20

Example: the perimeter of this regular pentagon is:

3 + 3 + 3 + 3 + 3 = 5×3 = 15

The perimeter of a circle is called the circumference:
Circumference = 2π × radius

Perimeter Formulas

Triangle
Perimeter = a + b + c

Square
Perimeter = 4 × a
a = length of side

Rectangle
Perimeter = 2 × (a + b)

Quadrilateral
Perimeter = a + b + c + d

Circle
Circumference = 2πr
r = radius

Sector
Perimeter = r(θ+2)
r = radius
θ = angle in radians

Ellipse
Perimeter = very hard!

Topic C: Area

Area

Area is the quantity that expresses the extent of a region on the plane or a curved surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.

Example: What is the area of this rectangle?

The formula is:
Area = w × h
w = width
h = height

We know w = 5 and h = 3, so:
Area = 5 × 3 = 15
Example: What is the area of this circle?

Radius = r = 3
Area
= π × r2
= π × 32
= π × (3 × 3)
= 3.14159… × 9
= 28.27 (to 2 decimal places)

Example: What is the area of this triangle?

Height = h = 12
Base = b = 20

Area = ½ × b × h = ½ × 20 × 12 = 120

Word Problem:
Example: Sam cuts grass at $0.10 per square meter How much does Sam earn cutting this area:

Let’s break the area into two parts:

Part A is a square:
Area of A = a2 = 20m × 20m = 400m2
Part B is a triangle. Viewed sideways, it has a base of 20m and a height of 14m.
Area of B = ½b × h = ½ × 20m × 14m = 140m2
So the total area is:
Area = Area of A + Area of B = 400m2 + 140m2 = 540m2

Sam earns $0.10 per square meter
Sam earns = $0.10 × 540m2 = $54

TESTIMONIALS

We Also Offer:

• NYC Pre-Algebra Tutoring
• NYC Calculus Tutoring
• NYC Math Tutoring K-12