Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you...
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Transcript of Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you...
![Page 1: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/1.jpg)
Areas of Regular PolygonsLesson 11.5
![Page 2: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/2.jpg)
Equilateral Triangle
Remember: drop an altitude and you create two 30-60-90 triangles.What is the measure of the sides and altitude in terms of one side equaling s?
Altitude = s√3 2
![Page 3: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/3.jpg)
Given: ∆ CAT is equilateral, and TA = s
Find the area of ∆CAT T A
C
S
A∆CAT =
=
=
1
2s(s
23)
1
2bh
S
43
2
![Page 4: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/4.jpg)
Theorem 106: Area of an equilateral triangle = the product of 1/4 the square of a side and the square root of 3. Where s is the length of a side
Aeq∆ =
S
43
2
![Page 5: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/5.jpg)
An equilateral triangle has a side of 10 cm long. Find the area of the triangle.
A = 102(√3) 4
A = 25√3 cm2
![Page 6: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/6.jpg)
Area of a regular polygon:
Remember all interior angles are congruent and all sides are equal. Regular pentagon:
O is the center
OA the radius
OM is an apothem
N
T
A
O
M
E
P
![Page 7: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/7.jpg)
You can make 5 isosceles triangles in a pentagon.
Any regular polygon:
Radius: is a segment joining the center to any vertex
Apothem: is a segment joining the center to the midpoint of any side.
![Page 8: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/8.jpg)
Apothems:1. All apothems of a regular polygon are
congruent.
2. Only regular polygons have apothems.
3. An apothem is a radius of a circle inscribed in the polygon.
4. An apothem is the perpendicular bisector of a side.
5. A radius of a regular polygon is a radius of a circle circumscribed about the polygon.
6. A radius of a regular polygon bisects an angle of the polygon.
![Page 9: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/9.jpg)
Theorem 107: Areg poly = ½ ap
Area of a regular polygon equals one-half the product of the apothem and the perimeter.
Where a = apothem
p = perimeter
![Page 10: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/10.jpg)
A regular polygon has a perimeter of 40 cm and an apothem of 5 cm. Find the polygon’s area.
A = ½ap = ½(5)(40) = 100 cm2
![Page 11: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/11.jpg)
Find the area of a regular hexagon whose sides are 18 cm long.
1. Draw the picture
2. Write the formula
3. Plug in the numbers
4. Solve and label units
![Page 12: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/12.jpg)
18cm
Find the perimeter
Find each angle
Find the apothem
Write the formula, and solve.
P = 18(6) = 108 cmAngles = 720º/6 angles = 120º per angleRadius breaks it into 60º angles.30-60-90 triangle, apothem = 9√3 cm
![Page 13: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/13.jpg)
A = ½ apA = ½ (9√3)108A = 486√3 cm 2
![Page 14: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/14.jpg)
Team Challenge:
A square is inscribed in an equilateral triangle as shown. Find the area of the shaded region.
![Page 15: Areas of Regular Polygons Lesson 11.5. Equilateral Triangle Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the.](https://reader034.fdocuments.us/reader034/viewer/2022042717/56649e385503460f94b28f0b/html5/thumbnails/15.jpg)
2x + x√3 = 12x = 12 2 + √3x = 12(2 – √3)
A (shaded) = ½ (12)(6√3) – [12(2 – √3)√3]2
= 1764√3 - 3024
x x
x√3
x√3