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All courses Geometry · G-SRT.9 57 of 57
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
Find the area of the triangle shown with sides \(8\) and \(10\) and included angle \(30°\).
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 included-angle area data

\[a=8;~b=10;~C=30^{\circ}\]

The two supplied sides meet at the marked 30-degree angle.

02

Use the trigonometric area formula

\[K=(1/2)ab*\sin(C)\]

One side can act as base while the sine component of the other supplies perpendicular height.

03

Substitute the exact sine value

\[K=(1/2)(8)(10)\sin(30^{\circ})=40(1/2)\]

Since sin(30 degrees)=1/2, no decimal approximation is needed.

04

State the area

\[20\]

The triangular area is 20 square units; equivalently the altitude is 8sin30=4 and one half of 10 times 4 is 20.

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Another way

  1. Find height h=8sin30=4, then use K=(1/2)(10)(4)=20.

!

Common mistake

Do not use 8*10 without the one-half factor. A triangle occupies half the corresponding base-height parallelogram.

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

Solution walkthrough video