IXL - Area of a triangle: Law of Sines (Geometry practice)

Find the area of

△

39 cm

34°

125°

Write your answer as an integer or as a decimal rounded to the nearest tenth.

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Learn with an example

question

Find the area of

△

41 km

48°

112°

Write your answer as an integer or as a decimal rounded to the nearest tenth.

key idea

The Law of Sines states

sin

(

)

sin

(

)

sin

(

)

where each lowercase letter (like

is the length of the side opposite the vertex labeled with the same capital letter (like

A).

If you know the measures of two angles and the length of one's opposite side (like

∠

and

a),

you can use the Law of Sines to solve for the length of the other angle's opposite side (like

b).

The area of

△

is half of the product of two side lengths and the sine of their included angle:

Area

sin

(

)

Area

sin

(

)

Area

sin

(

)

where

and

are the lengths of the sides opposite the angles

and

solution

You are given two angle measures

(

∠

and

∠

)

and the length of one of their opposite sides

(

To find the area of

△

you must know two side lengths and the measure of their included angle.

So, find

∠

Then use the Law of Sines to find

the length of the side opposite

∠

Find

∠

Solve for

∠

using the fact that the angle measures of a triangle sum to 180°.

∠

180°

48°

112°

∠

180°

160°

∠

180°

∠

20°

So,

∠

20°

Find

Since you know the measure of

∠

the length

of its opposite side, and the measure of

∠

use the Law of Sines to find the length

of the side opposite

∠

sin

(

)

sin

(

)

sin

112°

sin

48°

0.9271…

0.7431…

44.2199…

0.7431…

32.8618…

So,

32.8618… km

Find the area.

Now that you know two side lengths of the triangle and the measure of the included angle

(

and

∠

plug the values into the area formula.

Area of

△

sin

(

)

32.8618…

sin

20°

32.8618…

0.3420…

230.4077…

To the nearest tenth, the area of

△

is 230.4 square kilometers.

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Work it out

Not feeling ready yet? These can help:

R.4Area of a triangle: sine formula

R.1Law of Sines

F.2Triangle Angle-Sum Theorem

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