9.2 The Area of a Triangle

Objective To find the area of a triangle given the lengths of two sides and the measure of the included angle.

The Area of a TriangleWhen the lengths of two sides of a triangle and the measure of the included angle are known, the triangle is uniquely determined. Therefore, the area of the triangle is unique.

1.

2K bh

sin .h

Ca

sinh a C

1sin

2K ab C

As we know, the area of a triangle, denoted as K, is

By the right triangle trigonometry, we know that

So

Then the area of a triangle is

If some other pair of sides and the included angle of ABC were known, we could repeat the procedure for finding the area and thereby obtain two other area formulas.

The area of ABC is given by:

1 1 1sin sin sin

2 2 2K ab C bc A ac B

The Area of a Triangle

Example 1: Two sides of a triangle have lengths 7 cm and 4 cm. The angle between the sides measures 73º. Find the area of the triangle.

7

473

17 4 sin 73

2K

13.4K 2So, the area is about 13.4 cm .

[Solution]

1 1 1sin sin sin

2 2 2K ab C bc A ac B

Example 2: The area of PQR is 15. If p = 5 and q = 10, find all possible measures of R.

1sin

2K pq R 15 25sin R

0.6 sin R R can be acute or obtuse.

1sin 0.6 36.9R or 180 36.9 143.1R

From this example, we can see that the converse of the following statement is NOT TRUE!

When the lengths of two sides of a triangle and the measure of the included angle are known, the triangle is uniquely determined. Therefore, the area of the triangle is unique.

115 5 10 sin

2R

A segment of a circle is the region bounded by an arc of the circle and the chord connecting the endpoints of the arc.

What is the length of the chord AB?A

BO

CDraw line OC chord AB at C, then OC is the bisector of chord AB and AOB. In the right AOC, The length of AC is

sin2

AC r

Then the length of chord AB is

2 2 sin2

AB AC r

And the area of AOB is

21sin

2K r

Practice: Express the area of AOB in a different way.

After we learn the double angle formula, we will know the two formulas are the same.

A

BO

C

cos2

OC r

Then the length of chord AB is

2 2 sin2

AB AC r

And the area of AOB is

21 12 sin sin ccos

2 2 2 2os

2 2K AB OC r r r

And OC, the height of AOB, is

2 21si sin cos

2 2n

2rK r

We have already known that the area of a sector is

A

BO

C

What is the area of a segment?

Obviously, the area of the segment is the difference of the area of the sector and that of central AOB.

2 2

2

2

2 2

1 1sin

2 21

sin cos2 2

1( sin )

2

sin cos2 22 2

segmentA r r r

rr r

21

2A r

Example 3. The diagram shows an end view of a cylindrical oil

tank with radius 1 m. Through a hole in the top, a vertical rod is

lowered to touch the bottom of the tank. When the rod is removed,

the oil level in the tank can be read from the oil mark on the rod.

3Where on the rod should marks be put to show that the tank is

41 1

full, full, and full?2 4

Example 4: Given ABC with an inscribed circle as shown below. Show that the radius r of the circle is given by

In addition, find the radius of the inscribed

circle if a = 14, b = 12, and c = 8. (Hint: Heron’s formula)

2 ABCKr

a b c

r

rr

B

A

O

C

[Proof] The area of ABC , KABC , is the sum of the area of AOB, BOC, and AOC. Since the radii of inscribed circle are perpendicular to all three sides. So the areas of AOB, BOC, and AOC are respectively

1

2AOBK cr 1

2BOCK ar 1

2AOCK br

1 1 1( )

2 2 2 2AOB BOC AOC

rK K K cr ar br a b c

( )2ABC

rK a b c 2 ABCK

ra b c

Example 4: Given ABC with an inscribed circle as shown below. Show that the radius r of the circle is given by

In addition, find the radius of the inscribed

circle if a = 14, b = 12, and c = 8. (Hint: Heron’s formula)

2 ABCKr

a b c

r

rr

B

A

O

C

[Solution] The area of ABC , KABC , can be calculated by Heron’s formula:

We then calculate the perimeter (a + b + c) = 14 + 12 + 8 = 36, so s = 18. Thus the area of ABC is

2 12 30 2 30

36 3r

( )( )( ), where ( ) / 2.K s s a s b s c s a b c

18(18 14)(18 12)(18 8) 18(4)(6)(10) 12 30ABCK

Example 5: Given ABC with an inscribed circle as shown below. Show that the radius r of the circle is given by

(Hint: Heron’s formula)r

rr

B

A

O

C

[Proof] Since s = (a + b + c)/2, then a + b + c = 2s. By applying the conclusion and Heron’s formula from the last example, we have

2 ( )( )( )2

2ABC s s a s b s cK

ra b c s

( )( )( ), where ( ) / 2.

s a s b s cr s a b c

s

2

( )( )( ) ( )( )( )s s a s b s c s s a s b s c

s s

2

( )( )( ) ( )( )( )s s a s b s c s a s b s c

s s

Assignment

P. 342 #1 – 13, 19, 21