Chapter 5Perimeter and Area

5-1: Perimeter and Area5-2: Areas of Triangles, Parallelograms, and Trapezoids

5-3: Circumferences and Areas of Circles5-4:The Pythagorean Theorem

5-5: Special Triangles and Areas of Regular Polygons5-6: The Distance Formula and the Method of Quadrature

5-7: Proofs Using Coordinate Geometry5-8: Geometric Probability

5.1 Perimeter and Area

• Perimeter—the distance around an object

• To find the perimeter of a polygon, add the lengths of all of its sides.

Area—the surface encompassed by a polygon

Area of Triangle=1/2bh, where height, h, is perpendicular to the base

Area of Square=s2 where s is the length of the side

Area of Rectangle = l x w, where l is the length and w is the width

5.2 Areas of Triangles, Parallelograms, and Trapezoids

• Area of Triangle=1/2bh, where h is perpendicular to the base, b; h does not have to touch the base

Area of Parallelogram=b x h, where height, h, is perpendicular to base, b

Area of Trapezoid=1/2(h)(b1+b2) or A=h(midsegment); h is height

perpendicular to and intersecting both bases (b1and b2)

5.3 Circumferences and Areas of Circles

• Circumference—perimeter of a circle

• C=2πr or C=2d, where r is the radius and d is the diameter

Area of Circle =πr2

If they give you the area or circumference and ask for the

radius, you must use algebra to solve for r.

5.4 The Pythagorean Theorem

• Pythagorean Theorem: Given a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse

• hyp2=leg2+leg2 or c2=a2+b2

• Your longest leg is always c!

Pythagorean Triples: Sets of lengths that always make a right

triangle3,4,5

5,12, 137,24,25

9, 40, 4111, 60, 61

Using Side Lengths to Determine if a Triangle is Right, Acute, or Obtuse

Converse of the Pythagorean Theorem:

If c2=a2+b2, then you have a right triangle.If c2<a2+b2, then you have an acute triangle.

If c2>a2+b2, then you have an obtuse triangle.

5.5 Special Triangles and Areas of Regular Polygons

• Special Triangle 1: 30-60-90•Take an equilateral triangle with sides of length 2a and split it in half. This leaves you with a 30-60-90 triangle.

•Since everything was split in half, the base of this 30-60-90 triangle is half of the original, or a.

•Use the Pythagorean Theorem to get the height:

Special Triangle 1: 30-60-90

• So, in a 30°-60°-90° triangle, the side opposite 30° is a, the side opposite 60° is a√3, and the side opposite 90° is 2a

a√3

•When solving problems with a 30-60-90 triangle, set the known side length equal to its ratio.

•Solve for a.

•Then substitute into the other ratios to find the missing side lengths.

Special Triangle 2: 45-45-90

• Take a 45-45-90 triangle with sides of length a. (Since the angles are congruent, this triangle is isosceles.

• Use the Pythagorean Theorem to find the length of the hypotenuse.

Special Triangle 2: 45-45-90

So, in a 45°-45°-90° triangle, the sides opposite 45° are a, and the side opposite 90° is a√2

•When solving problems with a 45-45-90 triangle, set the known side length equal to its ratio.•Solve for a.•Then substitute into the other ratios to find the missing side lengths.

Finding the Area of a Regular Polygon

• Let’s say we have a hexagon with side length 10.

1. Put the hexagon into a circle. How many triangles do we get? What are the angles inside?

2. Draw the height of the triangles (from the vertex of the circle to the middle of the edge). This is called your apothem in a regular polygon.

Finding the Area of a Regular Polygon

3. Use special triangles to find the length of the apothem and calculate the area of the triangle.

4. Multiply this area by the number of triangles in the hexagon to get the total area.

Our side length=10, so a=5, and the height must be 5√3.

Finding the Area of a Regular Polygon

Formula for the area of a regular polygon:

• A=1/2(apothem)(perimeter)

• p=(number of sides)(length of side)

5.6 The Distance Formula and the Method of Quadrature

• Distance—the length between two points

• Given points (x1, y1) and (x2, y2)

Choose point one and point two and substitute in this formula to find distance.

5.7 Proofs Using Coordinate Geometry

• Midpoint—the point which divides a segment into two congruent parts

• Given endpoints (x1, y1) and (x2, y2)

Slope=

• Use properties of polygons to find determine lengths of sides and points of vertices.

5.8 Geometric Probability

• Probability is the likelihood that an event will happen. It is always between 0 and 1. (0 means that it is impossible, and 1 means that it always has to happen.)

• P= (#favorable outcomes)

(#possible outcomes)• In geometry, probability is usually the percent of

area an “event” represents—like the area represented by blue on a spin dial.