Area of a triangle
Calculate the area of any triangle using ; choose the appropriate formula for a given configuration, including using the cosine rule to find an unknown angle before applying the area formula.
Worked examples
Finding area given two sides and an included angle
Straightforward
Problem
In triangle , cm, cm and . Find the area.
1
Identify the two known sides and the included angle (the angle between them).
Here the included angle is , between sides cm and cm.
2
Apply the formula .
3
Evaluate and the area.
.
Answer
The area of triangle is approximately 61.8 cm².
Finding the included angle given area and two sides
Moderate
Problem
The area of triangle is 56 cm², cm and cm. Find the acute angle .
1
Write the area formula with the unknown angle.
2
Simplify the right side.
3
Solve for .
4
Find the angle.
Answer
The acute angle is approximately 39.5°.
Multi-step — cosine rule then area formula
Challenging
Problem
A triangular paddock has sides 30 m, 40 m and 50 m. Find the area.
1
We need an included angle before we can use the area formula. Use the cosine rule to find the angle between the 30 m and 40 m sides (call it , opposite the 50 m side).
2
Find angle .
3
Recognise the triangle type and apply the area formula.
This is actually a right-angled triangle (a 3-4-5 Pythagorean triple scaled by 10). The area formula confirms:
4
Check the area using the standard base-height formula.
The area can also be checked using the standard m².
Answer
The area of the paddock is 600 m².
Practise
Q1·Straightforward
In triangle , cm, cm and . Find the area of the triangle to 2 decimal places.
Explanation
Q2·Straightforward
In triangle , cm, cm and . Find the area of the triangle to 2 decimal places.
Explanation
Q3·Straightforward
In triangle , m, m and . Find the area to 2 decimal places.
Explanation
Note: , so an obtuse included angle still gives a positive area.
Q4·Straightforward
The area of triangle is 30 cm², cm and cm. Find the included angle to the nearest degree.
Explanation
Rearranging the area formula:
Q5·Moderate
A triangular garden has two sides of 15 m and 20 m with an included angle of 68°. Find the area of the garden to 2 decimal places.
Explanation
Q6·Moderate
In triangle , cm, cm and . Find the area to 2 decimal places.
Explanation
Q7·Moderate
Two sides of a triangle are 9 cm and 14 cm. The area of the triangle is 45 cm². Find the **obtuse** angle between the two sides, to 1 decimal place.
Explanation
Rearranging the area formula:
The acute solution:
The obtuse solution:
Since the question asks for the obtuse angle, the answer is .
The acute solution:
The obtuse solution:
Since the question asks for the obtuse angle, the answer is .
Q8·Moderate
A triangular plot of land has sides m and m, with . Find the area of the plot to the nearest square metre.
Explanation
Q9·Moderate
The area of triangle is 84 cm², angle , and side cm. Find side to 2 decimal places.
Explanation
Substituting into the area formula:
Q10·Challenging
A triangular plot has sides 45 m and 60 m with an included angle of 108°. Find the area to the nearest square metre.
Explanation
Q11·Challenging
A triangular block of land has sides 25 m, 35 m and 28 m. Find the area of the block to the nearest square metre.
Explanation
Step 1: Find the angle between sides m and m (opposite side m).
Step 2: Find the area.
Step 2: Find the area.
Q12·Challenging
In triangle , sides cm, cm and cm. Find the area of the triangle to 2 decimal places.
Explanation
Step 1: Find angle opposite side cm.
Step 2: Find the area.
Step 2: Find the area.
Open Math
Area of a triangle
Measurement · MS-M6
Name:
Date:
Q1Straightforward
In triangle , cm, cm and . Find the area of the triangle to 2 decimal places.
Q2Straightforward
In triangle , cm, cm and . Find the area of the triangle to 2 decimal places.
Q3Straightforward
In triangle , m, m and . Find the area to 2 decimal places.
Q4Straightforward
The area of triangle is 30 cm², cm and cm. Find the included angle to the nearest degree.
Q5Moderate
A triangular garden has two sides of 15 m and 20 m with an included angle of 68°. Find the area of the garden to 2 decimal places.
Q6Moderate
In triangle , cm, cm and . Find the area to 2 decimal places.
Q7Moderate
Two sides of a triangle are 9 cm and 14 cm. The area of the triangle is 45 cm². Find the **obtuse** angle between the two sides, to 1 decimal place.
Q8Moderate
A triangular plot of land has sides m and m, with . Find the area of the plot to the nearest square metre.
Q9Moderate
The area of triangle is 84 cm², angle , and side cm. Find side to 2 decimal places.
Q10Challenging
A triangular plot has sides 45 m and 60 m with an included angle of 108°. Find the area to the nearest square metre.
Q11Challenging
A triangular block of land has sides 25 m, 35 m and 28 m. Find the area of the block to the nearest square metre.
Q12Challenging
In triangle , sides cm, cm and cm. Find the area of the triangle to 2 decimal places.
Worked solutions and answers at openmath.au/year-12/standard-2/non-right-angled-trigonometry/area-of-a-triangle