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\(h\) \(b\)

Deriving the
Triangle Area Formula

Discover how the formula \( A = \tfrac{1}{2}ab\sin C \) connects trigonometry and geometry to find the area of any triangle.

Geometry

Why Triangle Area Matters

Triangles are the building blocks of all polygons: any polygon can be divided into triangles. Knowing how to find triangle area unlocks the area of every polygon, curved surface approximation, and real-world applications from architecture to navigation.

The Base-Height Formula

A B C \(h\) \(b\)

The Problem with Heights

B C A \(a\) \(c\) B \(h = ?\)

In many real-world problems you know two sides and the included angle, but not the perpendicular height. Surveyors, pilots, and engineers encounter this constantly. We need a formula that bypasses the height entirely.

The Trigonometric Insight

B C A \(h\) \(c\) B

Deriving \( A = \tfrac{1}{2}ab\sin C \)

1
Start with the base-height formula: \( A = \tfrac{1}{2} \times a \times h \)
2
Express \( h \) using trigonometry: From the right triangle, \( \sin C = \frac{h}{b} \), so \( h = b\sin C \).
3
Substitute: \( A = \tfrac{1}{2} \times a \times b\sin C \)
4
Result: \( \boxed{A = \tfrac{1}{2}ab\sin C} \) where \( a \) and \( b \) are any two sides, and \( C \) is the included angle.

The SAS Area Formula

A B C \(c\) \(b\) A \( \text{Area} = \tfrac{1}{2} \cdot b \cdot c \cdot \sin A \)

Included vs. Non-Included Angle

✓ Included Angle
The angle is between the two known sides. You can apply \( \tfrac{1}{2}ab\sin C \) directly.
✕ Non-Included Angle
The angle is opposite one of the known sides. You must first find an included angle using the Law of Sines or other methods.

When Does the Formula Apply?

Acute Triangles
\( \sin C \) is positive, height falls inside the triangle.
Right Triangles
\( \sin 90^\circ = 1 \), so \( A = \tfrac{1}{2}ab \): the legs are the base and height.
Obtuse Triangles
\( \sin C \) is still positive for \( 0^\circ < C < 180^\circ \). The formula works perfectly.

How to Apply the Formula

1
Identify two sides and the included angle from the given information.
2
Write the formula: \( A = \tfrac{1}{2}ab\sin C \)
3
Substitute the known values for \( a \), \( b \), and \( C \).
4
Compute \( \sin C \) (using a calculator or known value), then multiply.
5
Include units: If sides are in cm, area is in cm\(^2\).

Example 1: Acute Triangle

\(30^\circ\) B C A 10 cm 8 cm

Example 2: Right Triangle Check

B C A 8 6 \(90^\circ\)

Example 3: Obtuse Triangle

\(120^\circ\) P Q R 5 cm 7 cm

Real-World Application

A surveyor measures two boundary lines from a corner post: one is 120 m and the other is 85 m, with an angle of \( 72° \) between them. What is the area of the triangular plot?

1
Identify: \( a = 120 \), \( b = 85 \), \( C = 72° \)
2
Apply: \( A = \tfrac{1}{2}(120)(85)\sin 72° \)
3
Compute: \( A = \tfrac{1}{2}(10{,}200)(0.9511) \approx \) 4{,}850.6 m\(^2\)

Common Mistakes

✕ Forgetting the \( \tfrac{1}{2} \)
Writing \( A = ab\sin C \) instead of \( \tfrac{1}{2}ab\sin C \). The half is essential: a triangle is half a parallelogram.
✕ Wrong Angle
Using an angle that is not between the two chosen sides. The angle must be the included angle.
✕ Degrees vs. Radians
Calculator set to radians when the angle is in degrees, giving \( \sin(30) \approx -0.99 \) instead of \( 0.5 \).

Comparing Area Formulas

Base-Height SAS (Trig)
Formula \( \tfrac{1}{2}bh \) \( \tfrac{1}{2}ab\sin C \)
Required Info Base + perpendicular height Two sides + included angle
Works for All Triangles? ✓ (if height known) ✓ (always)
Best Used When Height is given or easy to find Height is unknown; SAS data available
Relationship Both are equivalent: \( h = b\sin C \) bridges them

Question 1 of 5

What is the area of a triangle with sides \( a = 12 \) and \( b = 9 \), and an included angle of \( 60° \)?

Question 2 of 5

In the formula \( A = \tfrac{1}{2}ab\sin C \), what does \( C \) represent?

Question 3 of 5

A triangle has sides \( 10 \) and \( 6 \) with an included angle of \( 90° \). What is its area?

Question 4 of 5

Why does the formula \( A = \tfrac{1}{2}ab\sin C \) still work when \( C \) is obtuse?

Question 5 of 5

A triangular garden has two edges measuring \( 15 \) m and \( 20 \) m with an included angle of \( 45° \). What is the garden's area?