The document explains how to calculate the area of a triangle when two sides and the included angle are given, using the formula 1/2 × a × b × sin(c). It provides several examples with specific measurements and solutions, demonstrating the application of the formula. The document concludes with a variety of scenarios for triangle area calculation.

1. Area of Triangles
Given two sides and the included angle.

2. Formula
 To find the area of a triangle when given two sides and
the included angle, the formula to be used is
𝟏
𝟐
× 𝒂 × 𝒃 × 𝐬𝐢𝐧 𝑪
where 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠 𝑎𝑛𝑑 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛𝑐𝑙𝑢𝑑𝑒𝑑.

3. Example 1
Find the area of the triangle: 𝐿𝑒𝑡 𝑎 = 4 𝑐𝑚 , 𝑏 = 5 𝑐𝑚 𝑎𝑛𝑑 𝜃 = 30°
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
1
2
× 4 × 5 × sin 30
= 𝟓 𝒄𝒎𝟐

4. Example 2
Find the area of the triangle
to the nearest tenth. 𝐿𝑒𝑡 𝑎 = 4 𝑐𝑚 , 𝑏 = 6 𝑐𝑚 𝑎𝑛𝑑 𝜃 = 25°
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
1
2
× 4 × 6 × sin 25
= 𝟓. 𝟏 𝒄𝒎𝟐

5. Example 3
Find the area of the triangle
to two decimal places. 𝐿𝑒𝑡 𝑎 = 4.5 𝑐𝑚 , 𝑏 = 6 𝑐𝑚 𝑎𝑛𝑑 𝜃 = 40°
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
1
2
× 4.5 × 6 × sin 40
= 𝟖. 𝟔𝟖 𝒄𝒎𝟐

6. Examples
Example 4
Tony is preparing a triangular sandbox
for his 2 year old daughter. Two sides are
6m and 7m in length and the angle
between those two sides is 37°. What
will be the area of the sandbox to 1
decimal place?
𝐿𝑒𝑡 𝑎 = 6 𝑚 , 𝑏 = 7 𝑚 𝑎𝑛𝑑 𝜃 = 37°
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
1
2
× 6 × 7 × sin 37
= 𝟏𝟐. 𝟔 𝒎𝟐
Example 5
Janice’s favourite picture frame is shaped
like a triangle with an angle of 120° between
two sides that are
5.5 cm and 8 cm long. What is its area to the
nearest whole number?
𝐿𝑒𝑡 𝑎 = 5.5 𝑐𝑚 , 𝑏 = 8 𝑐𝑚 𝑎𝑛𝑑 𝜃 = 120°
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
1
2
× 5.5 × 8 × sin 120
= 𝟏𝟗 𝒄𝒎𝟐

7. THE END.