10.1: Areas of Parallelograms and Triangles
Important Vocabulary: By the end of this lesson, you should be able to define these terms:
Base of a Parallelogram, Altitude of a Parallelogram, Height of a Parallelogram,
Base of a Triangle, Height of a Triangle
What does it mean to find the area of a figure?
When using formulas, should units be the same before you multiply?
Examples: Determine the area of the following figures.
1)
2)
How are these two formulas alike?
Examples: Find the area of the following figures.
3.
4.
Example: Find h in the following.
5.
In a right triangle, what parts represent the base and height?
Examples: Find the area of the following triangles.
6)
7)
Example: Find the area of the following figure.
8)
10.2: Areas of Trapezoids, Rhombuses and Kites
Important Vocabulary:
Height of a trapezoid
Trapezoid: a quadrilateral that has one pair of parallel sides, known as bases
Area of a Trapezoid
Area = ½ h(b1 + b2)
h = height of the trapezoid (perpendicular segment between bases)
b1 and b2 = lengths of the two parallel sides
Examples: Find the area of the following trapezoids.
1)
2)
3)
4)
Rhombus: quadrilateral with all four sides congruent
Kite: quadrilateral with 2 pairs of consecutive sides congruent and no parallel sides
Remember:
1) You need each diagonal length.
2) The angle formed in the middle is a right angle.
3) Diagonals of a rhombus bisect each other. WHY?
Examples: Find the area of each kite or rhombus.
5)
6)
7)
8)
10.3: Areas of Regular Polygons
Regular Polygon: polygon that is equilateral and equiangular
Radius of a Regular Polygon:
Apothem:
Label the center, side and apothem on the lines. Then draw in the radius.
Finding Angle Measures:
1. Divide 360 by the number of sides. (This finds the central angle)
2. Divide this by 2, this gives you the angle at the center when the apothem is drawn.
3. The apothem forms a right angle with a side. Subtract the number you found and 90 from 180
to find the third angle.
Examples: Find the measure of each numbered angle inside the regular polygon.
1)
2)
Postulate 10-1: If 2 figures are congruent, then their areas are equal.
Theorem 10-6: The area of a regular polygon is
.
A = ½ Pa
P= perimeter; a = apothem
the product of the perimeter and
Examples: Find the area of the each regular polygon with the given side length and apothem.
3) Pentagon w/ a = 1.6 in and s = 2.4 in
4) Decagon w/ a = 3.8 in and s = 2.5 in
5) Find the area of a regular pentagon with an 8cm apothem and sides 11.6cm.
Using 45-45 and 30-60 within Regular Polygons
Squares: Use 45-45 to find missing pieces if you know the apothem or radius and “use your smarts”.
Triangles and Hexagons: Apply 30-60 techniques to find missing pieces.
Examples: Find the area of the following regular polygons.
6)
7)
8)
10.4: Perimeters and Areas of Similar Figures
REMEMBER TO SIMPLIFY FRACTIONS!!!!
Ratios can be used to compare perimeters and areas of similar figures.
Similar Figures: figures with the same angle measures, but sides are proportional
Radius of a Polygon: segment from the center of a polygon to a vertex
Example: Use the figure to answer the following questions.
1) What segment is NOT a radius of the polygon?
2) Are the radii congruent?
3) Are the triangles formed by the radii congruent?
Examples:
4) What is the ratio called that compares two corresponding sides of similar figures?
5) Assume that the scale factor is 2:3.
a) What is the ratio of the perimeters from small to large?
b) What is the ratio of the areas from small to large?
Examples: Assume the figures are similar. For the following, find the scale factor from the first to the
second figure. Then, find the ratio of their perimeters and areas.
6)
7)
If you can determine the ratio of the perimeter or area of the figures, a proportion can be used to find
missing values.
Examples: The following figures are similar. The area of one is given. Find the area of the other to the
nearest whole number.
8)
9)
Area of Smaller = 54 in2
Area of Larger = 100 cm2
Examples: Find the scale factor and the ratio of perimeters for each pair of similar figures.
10) two regular pentagons with areas 50 in.2 and 162 in.2
11) two rectangles with areas 8 m2 and 98 m2
12) two regular pentagons with areas 45 ft2 and 20 ft2
10.5: Trigonometry and Area
What formula allows us to find the area of a regular polygon?
What do the variables stand for in the previous equation?
Using Trigonometry
What is the acronym to remember the trig functions?
How can you find the central angle of a regular polygon?
Example: Find the area in the given situations.
1) What is the area of the given polygon?
2) What is the area of a hexagon with radius of 4 m?
3)
Area of a Triangle Given SAS
The area of a triangle is half the product of the lengths of two sides and the sine of the included
angle.
Examples: Find the area of the given triangles. Round your answers to the nearest 10th.
4)
5)
6)
10.6: Circles and Areas
Circle:
Center:
Diameter:
Radius:
Congruent Circles:
Central Angle:
Identify the previous vocabulary terms on the given circle. Name the circle as well.
Name:
Center:
Diameter:
Radius:
One Central Angle:
Semicircle:
Major Arc:
Minor Arc:
Naming Semicircles and Arcs:
Major Arcs and Semicircles:
Minor Arcs:
1) Pick starting point and end point.
1) List start and end.
2) Choose one point in between those.
2) Use symbol.
3) List from start, middle to end.
4) Use arc symbol above the letters.
Examples: Use the figure to answer the questions.
1) What is the diameter of the figure?
2) Name a semicircle.
3) Name three minor arcs.
4) Name three major arcs.
Measures of Arcs
1) A semicircle has a measure of 180.
2) A minor arc has the measure of its central angle.
3) A major arc has the measure of 360 minus the related minor arc or the sum of the central angles.
Examples: Find the measures for the given circle.
5) mEDI
6)
7)
8) mIDH
9)
10)
11) What is the value of “a”?
Circumference of a Circle
𝑪 = 𝟐𝝅𝒓 𝒐𝒓 𝑪 = 𝝅𝒅
r = radius and d = diameter
Examples: Find the area of each circle. Leave your answer in terms of pi.
12)
13)
Arc Length
The length of an arc of a circle is found by the formula:
̂
𝑚𝐴𝐵
̂ =
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐵
∙ 2𝜋𝑟
360
You must know the arc measure (central angle).
You must know the radius of the circle.
PLUG and CHUG
Examples: Find the length of the darkened arc. Leave your answer in terms of pi.
14)
15)
16)
10.7: Areas of Circles and Sectors
Area of a Circle
The area of a circle is the product of 𝜋 and the square of the radius.
𝐴 = 𝜋𝑟 2
Examples: Find the area of the circle. Leave answers in terms of pi.
1)
Sector of a Circle: region bounded by the arc of a circle and the two radii that form the arc
Area of a Sector of a Circle
The area of a sector of a circle is found by the following formula:
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆𝑒𝑐𝑡𝑜𝑟 𝐴𝑂𝐵 =
̂
𝑚𝐴𝐵
∙ 𝜋𝑟 2
360
Examples: Find the area of the shaded sector. Leave answers in terms of pi.
2)
3)
4)
Segment of a Circle: part of a circle bounded by an arc and the segment joining its endpoints.
Examples: Find the area of the shaded segment to the nearest tenth.
5)
6)
7)
8)
10.8: Geometric Probability
What is probability?
Geometric Probability: the ratio that involves geometric lengths or area
In other words: the probability that a point lies on a segment is the length of what you want to
happen over the length of all that can happen.
Examples: A point on AM is chosen at random. Find the probability that the point lies on the given
segment.
1. DJ
2. JL
3. BE
4. CK
5. AJ
Examples: A point in the figure is chosen at random. Find the probability that the point lies in
the shaded region.
6)
7)