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Section 7.2 - Area of a Triangle

In this section, we’ll use a familiar formula and a new formula to find the area of a triangle. You have probably used the formula

1 1( )( )

2 2Area base height bh

to find the area of a triangle, where b is the length of the base of the triangle and h is the height of the triangle. Example:

If the base is 10b and height is 4h , then the area is: 1

10 4 202

A .

We’ll use this formula in some of the examples here, but we may have to find either the base or the height using trig functions before proceeding.

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Here’s another approach to finding area of a triangle using trigonometry. Consider this triangle:

The area of the triangle ABC is: 1

sin( )2

Area bc A

It is helpful to think of this as Area = ½*side*side*sine of the included angle. Other versions (depending on which angle you would like to use):

1sin( )

2Area ab C

1

sin( )2

Area ac B

b a

c

A

C

B

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Example 1: Find the area of the triangle.

14

6

450

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Example 2: Find the area of the triangle.

12029

1712

20

3

Special Cases Right triangles:

01 1 1sin( ) sin(90 )

2 2 2Area ab C ab ab

Or: Area of a right triangle = ½ * leg1 * leg2.

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Equilateral Triangles:

0 2 21 1 3 3sin(60 )

2 2 2 4Area a a a a

Or: Area of an equilateral triangle is side squared times 3

4

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POPPER for Section 7.2: Question#1: Find the area of an equilateral triangle whose sides are 8 units each.

a) 64 3

b) 16 3

c) 32 3 d) 64 e) 32 f) None of these

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Example 3: Find the area of an isosceles triangle with legs measuring 12 inches and base angles measuring 52 degrees each. Round to the nearest hundredth.

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Example 4: In triangle ABC; 12, 20a b and 42.0)sin( C . Find the area of the triangle.

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Example 5: In right triangle ABC; 24AB and 1

sin( )3

B . Find the area

of this triangle.

C

A

B

6

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Example 6: In triangle ABC; 10, 12AC AB , x is an acute angle with 1

sin( )4

x . Given that 2m A x , find the area of this triangle.

C

A

B

12 10

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POPPER for Section 7.2: Question#2: In triangle ABC, 6AB , 10AC and 030m A . Find the area of this triangle.

a) 60 b) 30 c) 15 d) 30 3

e) 15 3 f) None of these

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Example 7: In triangle KLM, k = 10 and m = 8. If the area of this triangle is 20 and L is an acute angle, find the measure of angle L.

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Example 8: In triangle KLM, k = 10 and m = 8. Find all possible measures of the angle L if a) the area of the triangle is 20 unit squares. b) the area of the triangle is 25 unit squares. c) the area of the triangle is 80 unit squares.

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Formula for Area of a Regular Polygon Given a Side Length

2

4 tan

S NA

N

, where S = length of a side, N = number of sides.

For reference, a pentagon has 5 sides, a hexagon has 6 sides, a heptagon has 7 sides, an octagon has 8 sides, a nonagon has 9 sides and a decagon has 10 sides. Example 9: A regular hexagon is inscribed in a circle of radius 12. Find the area of the hexagon.

2

4 tan

S NA

N

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Area of a segment of a circle You can also find the area of a segment of a circle. The shaded area of the picture is an example of a segment of a circle.

A

O B

To find the area of a segment, find the area of the sector with central angle θ and radius OA. Then find the area of OAB . Then subtract the area of the triangle from the area of the sector. Area of segment = Area of sector AOB - Area of AOB

= )sin(2

1

2

1 22 rr

Example 10: Find the area of the segment of the circle with radius 8 inches

and central angle measuring 4

.

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POPPER for Section 7.2: Question#3: In triangle ABC, 6AB , 10AC and area of this triangle is 15 2 . If A is an acute angle, find the measure of angle A. triangle.

a) 300 b) 600 c) 150 d) 750 e) 450 f) None of these