By the end of this lesson, you should be able to:

2. Find the perimeter and area of triangles, rectangles, parallelograms, trapezoids, and circles.

3. Calculate the sum of interior angles of a polygon.

4. Classify triangles.

5. Apply postulates to determine congruency and similarity of triangles.

6. Identify right triangles and their hypotenuses.

7. Use the Pythagorean Theorem to determine the length of any side of a right triangle given the lengths of the other two sides.

8. Correctly name types of quadrilaterals.

9. Correctly name types of polygons.

10. Identify a circle's definition, center, radius, diameter, chord, secant, and tangent.

11. Calculate a circle's circumference and area given the radius or diameter.

12. Calculate the volumes of simple geometric solids.

Plane Geometry

Geometry of items that exist on a two-dimensional plane is called plane geometry.

Points and Lines

Points are locations. They have no length, volume or mass. They are invisible, but we represent them by drawing large dots.

Similarly, lines do not have thickness, but we draw representations of them with a pencil or a pen that leaves a mark of a given thickness. Remember that a line extends in both directions, where as a ray has a beginning and extends in only one direction, while a line segment has two end points.

Common Formulas

Perimeter refers to the sum of the length of the line sides of a figure.

Recall the formulas for the area of common geometric figures:

Polygons

Polygons are closed plane figures formed from line segments that contain only one enclosed interior region. Unlike a convex polygon, a concave polygon has an interior angle greater than 180^o.

The sum of the interior angles of a polygon is

(n - 2) * 180^o

where n is the number of sides. Thus, in a regular polygon (i.e., where all sides are of equal length) the measure of each interior angle is

((n - 2) * 180^o) / n

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Triangles are 3-sided polygons.

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Types

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Sum of Interior Angles

The interior angles of a triangle add to 180^o.

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Congruency

Two triangles are congruent (i.e., they have all three sides and angles of equal measurement) if the have any of the following three in common:

• Three sides (SSS)
• Side-Angle-Side (SAS)
• Angle-Side-Angle (ASA)
• Angle-Angle-Side (AAS)

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Similarity

Two triangles are similar (i.e., with the same angles but not necessarily the same side lengths) if the share the following in common:

• Two angles (AA)

A triangle with one right angle (90^o) is called a right triangle. The side opposite the right angle is the hypotenuse.

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Pythagorean Theorem

The Pythagorean theorem can be used to find the length of any side of a right triangle if the other two sides are known. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

C² = A² + B²

This works out with whole numbers for a 3-4-5 triangle (or a 5-12-13 triangle, etc.)

Problem 1: A guy wire is holding a tower in place. The wire is attached to the tower 100 feet of the (level) ground, and the guy wire is anchored 75 feet from the center of the base of the tower. If you did not account for the wire's sag or its attachment, how long would it be?

Solution 1: 125 feet. This is because 100' ² + 75' ² = 15625 square feet, and the square root of that is 125 feet. (Or you could have used the 3-4-5 trick.)

Quadrilaterals are 4-sided, closed, geometric figures. They include:

Other polygons include:
 

Name	Number of Sides
pentagon	5
hexagon	6
heptagon	7
octagon	8
nonagon	9
decagon	10
n-gon	n

Circles

A circle is the locus of points equidistant from a given point on a coordinate plane. The radius is the distance from the center to the circle. Any straight line intersecting the circle at only one point is a tangent. Any straight line intersecting the circle at two points is a secant. A chord is a line segment with both endpoints on the circle. Chords that pass through a circle's center are diameters (or the diameter is the length of chords that pass through the center.)

The circumference (i.e., perimeter) of a circle is equal to pi times the diameter: 

pd or 2pr

The area within a circle is equal to

pr²

Solid geometry

We can describe a one-dimensional (length) environment as a line.

A two-dimensional environment (area, plane geometry) can be illustrated by a second line perpendicular to the first.

A three-dimensional environment (volume, solid geometry) can be illustrated by a third line, perpendicular to the first two.

So a four-dimensional environment can be illustrated by .....

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To simplify this, consider just three cases.

• The sphere, where the volume is 4/3 (p r ³), is the first case.
• The second case includes all objects the have a constant cross section (and parallel, congruent bases), like rectangular prisms and cylinders; here the volume is found by multiplying the area of the base by the perpendicular height.
• The third case covers objects that come to a point, including cones, triangular prisms, pyramids, and other such shapes; Here the volume is found by multiplying 1/3 of the area of the base by the height perpendicular to that base.