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Lesson 10.1 ‐10.2: Perimeter and Area of Common Geometric Figures
Focused Learning Target: I will be able to
Solve problems involving perimeter and area of common geometric figures.
Compute areas of rectangles, triangles, rhombuses, parallelograms, trapezoids, and kites.
CA Std 8.0: Students know, derive, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
CA Std 10.0: Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
Vocabulary: • Base of a parallelogram
Height of a parallelogram
Height of a triangle
Altitude of a parallelogram
Base of a triangle
Height of a trapezoid
Base & height (altitude of a parallelogram)
Base & height of a triangle
Bases & Height of a trapezoid
Perimeter and area formulas for common geometric figures:
Rectangle P= 2b + 2h A= bh
Square P = 4s A = s2
Triangle P = a + b + c
Circle
Parallelogram
P = 2a + 2b A= bh
Trapezoid
Rhombus P = 4s
Kite P = 2a+2b
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Example 1: Using the perimeter to find area
I’ll do one: The perimeter of a rectangle is 48 ft and the base is 8ft. What is the area?
We’ll do one together: The perimeter of a rectangle is 34 inches and the height is 10 inches. Find the area of the rectangle.
You try one: The perimeter of a rectangle is 36cm and the base is 10cm. Find the area of the rectangle.
Example 2: Using area to find perimeter.
I’ll do one: A square and a rectangle have equal areas. If the rectangle is 9cm by 4cm. What is the perimeter of the square?
We’ll do one together: A square and a rectangle have equal areas. If the rectangle is 27 inches by 3 inches. What is the perimeter of the square?
You try one: A square and a rectangle have equal areas. If the rectangle is 16cm by 9cm. What is the perimeter of the square?
Example 3: Finding areas parallelograms.
I’ll do one: Find the area of each parallelogram.
We’ll do one together: Find the area of a parallelogram with the base 12 m and height 9 m.
You try one: Find the area of each parallelogram.
Example 4: Finding a missing dimension.
I’ll do one: Find the value of h in the parallelogram
We’ll do one together: Find the value of h in the parallelogram
You try one: Find the value of h in the parallelogram
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Example 5: Finding the area of the shaded region (triangles):
I’ll do one:
We’ll do one together:
You try one:
Example 6: Finding the area of trapezoid.
I’ll do one: Find the area of trapezoid ABCD.
We’ll do one together: What is the area of trapezoid PQRS? (Using a right triangle 30°‐60°‐90°)
You try one: Find the area of trapezoid.
Example 7: Finding areas of rhombuses and kites:
I’ll do one: a) Finding the area of a rhombus:
We’ll do one together: a) Finding the area of a rhombus:
You try one: a) Finding the area of a rhombus:
b) Find the area of a kite:
b) Find the area of a kite:
b) Find the area of a kite:
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10‐3 Areas of Regular Polygons:
Focused Learning Target: I will be able to
Find the area of a regular polygon
Vocabulary:
Radius of a regular polygon
Apothem
CA Std 10.0: Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
Radius is the distance from the center to a vertex. Apothem is the perpendicular distance from the center to a side
Example 1: Finding angle measures:
I’ll do one: Find the measure of each numbered angle.
We’ll do one together: A portion of a regular hexagon has apothem and radii drawn. Find the measure of each numbered angle.
You try one: Find the measure of each numbered angle.
Example 2: Finding the area of a regular polygon:
I’ll do one: Finding the area of each regular polygon. Round your answer to the nearest tenth.
We’ll do one together: Finding the area of each regular polygon. Round your answer to the nearest tenth.
You try one: Finding the area of each regular polygon. Round your answer to the nearest tenth.
Notice: Regular hexagons have 30o‐60o‐90o triangles within their 6 equilateral triangles.
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10‐4 Perimeters and Areas of Similar Figures:
Focused Learning Target:
I will be able to find the perimeters and areas of similar figures
CA Std 11.0: Students determine how changes in dimensions affect the perimeter and area of common geometric figures.
Finding Ratios in Similar Figures: I’ll do one:
The trapezoids below are similar. The ratio of the lengths of the corresponding sides is 6
9or 2
3.
a) Find the ratio (smaller to larger) of the perimeters. b) Find the ratio (smaller to larger) of the areas.
We’ll do one together:
The pentagons below are similar. a) Find the ratio of the lengths of the corresponding sides. b) Find the ratio (smaller to larger) of the perimeters. c) Find the ratio (smaller to larger) of the areas.
You try:
The triangles below are similar. a) Find the ratio of the lengths of the corresponding sides. b) Find the ratio (smaller to larger) of the perimeters. c) Find the ratio (smaller to larger) of the areas.
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If you know the area of one of two similar figures, you can find the other by using proportions. Finding areas using similar figures:
The area of the smaller regular pentagon is about 227.5cm . Approximate the area of the larger regular pentagon.
We’ll do one:
The two triangles below are similar. The area of the larger triangle is about 2625m . Approximate the area of the smaller triangle.
You Try:
The two trapezoids below are similar. The area of the larger trapezoid is about 210in . To the nearest tenth, find the area of the smaller trapezoid.
Application Problems: I’ll do one:
For some medical imaging, the scale is 3:1. That means that if an image is 3cm long, then the corresponding
length on the person’s body is 1cm. Find the actual area of a lesion if its image has area of 22.7cm
We’ll do one:
Two similar rectangles have areas 227in and 248in . The length of one side of the larger rectangle is 16 in, what are the dimensions of both rectangles?
You Try:
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Two rectangles are similar. The smaller has an area of 260in and the shorter side is 6in long. The larger has a shorter side that is 8 in long. Find the area of the larger rectangle.
Challenge problem:
Draw a square with an area 28in . Draw a 2nd square with an area that is four times as large. What is the ratio of their perimeters?
10‐6 Circles and Arcs
Focused Learning Target: I will be able to
Find the measures of central angles and arcs
Find circumference and arc length
CA STD 7.0 – Students prove and use theorems involving the properties of circles. CA STD 8.0 – Students know, derive, and solve problems involving circumference.
Vocabulary:
Circle
Center
Radius
Congruent circles
Diameter
Central Angle
Semicircle
Minor Arc
Major Arc
Circumference
Pi ( )
Concentric Circles
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Example – Identifying Arcs I’ll do one: We’ll do one together: You try:
From the picture to the left, identify a minor arc: A semicircle: A major arc:
From the picture to the left, identify a minor arc: A semicircle: A major arc:
From the picture to the left, identify a minor arc: A semicircle: A major arc:
I’ll do one: We’ll do one together: You try:
From the picture to the left, find the measure of each arc:
a.
b.
From the picture to the left, find the measure of each arc:
a.
b.
From the picture to the left, find the measure of each arc:
a.
b.
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I’ll do one: Find the circumference of the circle. Leave your answer in terms of
We’ll do one together: The diameter of a bicycle wheel is 22 in. To the nearest whole number, how many revolutions does the wheel make when the bicycle travels 100 ft?
You try one: The radius of a tire is 12 in. To the nearest whole number, how many revolutions does the tire make when the car travels 5280 ft?
As we’ve seen so far, the measure of an arc is in degrees out of 360o. Arc length is different because it refers to the distance around a circle; a fraction of a circle’s circumference.
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I’ll do one: Find the length of the minor arc XY:
We’ll do one together: Find the length of the major arc XPY:
You try one: Find the length of a semicircle with radius 1.3 m. Leave your answer in terms of .
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10‐7 Areas of Circles and Sectors
Focused Learning Target: I will be able to
Find the areas of circles and sectors Find segments of circles
CA STD 8.0 – Students know, derive, and solve problems involving circumference and area of common geometric figures
Vocabulary:
Sector of a circle
Segment of a circle
Sector of a circle is a region bounded by an arc of the circle and the two radii to the arc’s endpoints. Segment of a circle is a part of a circle bounded by an arc and the segment joining its endpoints
I’ll do one:
How much more pizza is in a 12 in pizza compared to a 10 in. pizza?
We’ll do one:
Two sprinklers spray water in a circular path. One sprinkler sprays in a circle with 6 ft. diameter, the larger sprays with an 8 ft diameter. How much more ground is covered by the larger sprinkler?
You try:
How much more pizza does a 14 in. pizza have compared to a 12 in. pizza
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I’ll do one:
Find the area of sector ZOM. Leave your answer in terms of .
We’ll do one:
Find the area of the shaded sector of a circle. Leave your answer in terms of .
You Try:
Find the area of the shaded sector of a circle. Leave your answer in terms of .
A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle. To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.
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I’ll do one:
Find the area of the shaded segment. Leave your answer in terms of .
We’ll do one together: Finding the area of shaded segment. Round your answer to the nearest tenth.
You Try: Finding the area of shaded segment. Round your answer to the nearest tenth.
I’ll do one:
Find the shaded region. Leave your answer in terms of .
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We’ll do one:
Find the shaded region. Leave your answer in terms of .
Challenge A:
Find the area of the shaded region. Leave your answer in terms of .
Challenge B:
Find the total area of the shaded segments. Round your answer to the nearest tenth.