10-5 Trigonometry and Area

Quick Review

You can use trigonometry to find the areas of regular polygons. You can also use trigonometry to find the area of a triangle when you know the lengths of two sides and the measure of the included angle.

Area of a Δ cap delta

= 1 2 · equals , 1 half , middle dot  side length · middle dot  side length · middle dot  sine of included angle

Example

What is the area of Δ cap delta XYZ?

Area = 1 2 · X Y · X Z ·  sin  X   = 1 2 · 15 · 13 · sin 65 °   ≈ 88   .36500924 table with 3 rows and 2 columns , row1 column 1 , cap area , column 2 equals , 1 half , middle dot x y middle dot x z middle dot , sin x , row2 column 1 , , column 2 equals , 1 half , middle dot 15 middle dot 13 middle dot sine , 65 degrees , row3 column 1 , , column 2 almost equal to 88 . .36500924 , end table

The area of Δ X Y Z cap delta x y z  is approximately 88   ft 2 . 88 , ft squared , .

Exercises

Find the area of each polygon. Round your answers to the nearest tenth.

27. regular decagon with radius 5 ft
28. regular pentagon with apothem 8 cm
29. regular hexagon with apothem 6 in.
30. regular quadrilateral with radius 2 m
31. regular octagon with apothem 10 ft
32. regular heptagon with radius 3 ft
33. 
34. 

10-6 Circles and Arcs

Quick Review

A circle is the set of all points in a plane equidistant from a point called the center.

Image Long Description

The circumference of a circle is C = π d c equals , pi d  or C = 2 π r . c equals 2 . pi r .

Arc length is a fraction of a circle's circumference. The length of  A B ⌢ = m A B ⌢ 360 ⋅ 2 π r . . length of . modified eh b with frown above , equals . fraction m , modified eh b with frown above , over 360 end fraction . dot 2 pi r . . .

Example

A circle has a radius of 5 cm. What is the length of an arc measuring 80°?

length of  A B ⌢ = m A B ⌢ 360 ⋅ 2 π r Use the arc length formula.   = 80 360 ⋅ 2 π ( 5 ) Substitute.   = 20 9 π Simplify. The length of the arc is  20 9 π  cm. table with 2 rows and 1 column , row1 column 1 , table with 3 rows and 3 columns , row1 column 1 , length of . modified eh b with frown above , column 2 equals . fraction m , modified eh b with frown above , over 360 end fraction . dot 2 pi r , column 3 cap use the arc length formula. , row2 column 1 , , column 2 equals , 80 over 360 , dot 2 pi open 5 close , column 3 cap substitute. , row3 column 1 , , column 2 equals , 20 over 9 , pi , column 3 cap simplify. , end table , row2 column 1 , cap the length of the arc is . 20 over 9 , pi , cm. , end table

Exercises

Find each measure.

35. m ∠ A P D m angle eh p d
36. m A C ⌢ m , modified eh c with frown above
37. m A B D ⌢ m . modified eh b d with frown above
38. m ∠ C P A m angle c p eh

Find the length of each arc shown in red. Leave your answer in terms of π . pi .

Table of Contents