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All courses Math III · G-SRT.9 44 of 55
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.

Find the area of a triangle from two sides and the included angle

Problem
Which area is correct for a triangle with sides 8 and 10 and included angle 30 degrees?
Triangle with side lengths 8 and 10 meeting at a 30-degree included angle and a dashed perpendicular altitude labeled h. Open full size
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Hint

Use the triangle area formula with the two given sides and their included angle: \(A=\tfrac12 ab\sin(C)\).

For a triangle with sides 8 and 10 and included angle \(30^\circ\), use \(A=\tfrac12(8)(10)\sin(30^\circ)\). Also, \(\sin(30^\circ)=\tfrac12\).

Solution walkthrough

01

Identify the correct area formula

\[A~=~\tfrac12~\text{ab}\sin(C)\]

You are given two sides, 8 and 10, and the included angle, \(30^\circ\). That is exactly when the trigonometric area formula applies.

02

Substitute the given values

\[A~=~\tfrac12(8)(10)\sin(30^\circ)\]

Here, \(a=8\), \(b=10\), and the included angle is \(C=30^\circ\). Plug those directly into the formula.

03

Evaluate the sine and simplify

\[A~=~\tfrac12(8)(10)~\left(\tfrac12\right)~=~\tfrac12(80)~\left(\tfrac12\right)~=~20\]

Since \(\sin(30^\circ)=\tfrac12\), the multiplication becomes straightforward. First, \(8\cdot 10=80\), and then \(\tfrac12\cdot 80\cdot \tfrac12=20\).

04

State the area

\[\text{area }~=~20\]

So, using the trigonometric area formula for sides 8 and 10 with included angle \(30^\circ\), the area is 20.

+

Another way

  1. You can verify by finding an altitude first: if 10 is the base, then the height is \(8\sin(30^\circ)=4\). Then \(A=\tfrac12(10)(4)=20\).

  2. You could also use 8 as the base. Then the height is \(10\sin(30^\circ)=5\), so \(A=\tfrac12(8)(5)=20\).

!

Common mistake

A common mistake is to ignore the sine and do \(\tfrac12(8)(10)=40\). The formula needs the included angle factor, so you must multiply by \(\sin(30^\circ)=\tfrac12\), which gives 20 instead.

The same triangle with height 4 and the resulting area 20 shown only in answer support.