Deriving the Triangle Area Formula

Deck 1 of 2: Where A = ½ab sin(C) Comes From

In this deck:

• Why the base-height formula sometimes falls short
• Derive A = ½ab sin(C) from scratch
• Prove it works for acute, right, and obtuse triangles

Learning Objectives

By the end of this deck, you should be able to:

1. Derive the formula by drawing an auxiliary altitude
2. Explain why the formula works for acute, right, and obtuse triangles

The Surveying Problem

A surveyor measures two boundary lengths and the angle between them. Can you find the area without measuring the height?

The Base-Height Formula Has a Gap

The standard formula works — but only when height is known:

• What surveyors measure: two boundary lengths + angle between them
• What they don't measure: the perpendicular height
• The goal: express in terms of a side and an angle

If we can write , substitution gives us a new formula.

Setting Up the Derivation

• Angles , , are opposite their respective sides
• We choose side (which is ) as the base
• We need the altitude — the height from vertex perpendicular to

Case 1: the triangle is acute (all angles less than 90°)

Case 1: Acute Triangle — Drawing the Altitude

Step 1: Draw altitude from perpendicular to .

Case 1: Acute Triangle — Applying Sine

In the right triangle at : , so

Case 1: Acute Triangle — The Formula

From the previous slide:

Substitute into :

This is the formula. The entire derivation fits in three lines.

Quick Check: What Did We Derive?

What expression for the height is the key step in the derivation?

• (a)
• (b)
• (c)
• (d)

Choose before advancing.

Answer: Height Equals Side Times Sine

Correct answer: (b)

Why: In the right triangle at vertex :

• Hypotenuse = (side )
• Opposite leg = (the altitude)

Case 2: Right Triangle (A Quick Verification)

What if angle ?

• Sides and serve directly as base and height
• This matches the formula you already knew for right triangles
• The formula passes the sanity check

Case 3: Obtuse Triangle — The Setup

What if angle ?

• The altitude from now falls outside the triangle
• The foot of the altitude is on the extension of beyond

Case 3: Obtuse Triangle — Sine of a Supplement

In the obtuse case, the relevant angle at in the right triangle is :

Therefore:

The height expression is identical — the supplement identity saves us.

Case 3: Obtuse Triangle — Same Formula

From the previous slide:

Substitute into :

A = ½ab sin(C) holds for all three triangle types.

Quick Check: Why Does the Obtuse Case Work?

For an obtuse triangle, the altitude falls outside — yet the formula still holds.

In one sentence: why does remain valid when ?

• (a) Because sin(C) is always positive, the formula can't fail
• (b) Because , so still holds
• (c) Because obtuse triangles have larger altitudes, which compensate
• (d) The formula does not actually work for obtuse triangles

Choose before advancing.

Why Supplementary Angles Have Equal Sines

This identity is the hinge of the obtuse case. Here's why it holds:

• By the supplementary angle identity from the unit circle
• Or: in a right triangle, for supplementary pairs
• Example:

Watch out: This does NOT hold for cosine —

Three Equivalent Forms of the Formula

On triangle , the area can be written three ways:

Rule: Use the form that matches your given information.

The angle must always be the included angle — between the two sides used.

Numerical Example: Using the Formula

Triangle with , , . Find the area.

Step 1: Identify the form — sides and , included angle

Step 2: — verify your calculator is in degree mode

Your Turn: Apply the Formula

Triangle with , , included angle .

Step 1: Write the formula for this case
Step 2: Substitute the values
Step 3: Evaluate — leave your answer to 2 decimal places

Set up and calculate before advancing.

Answer: Worked-Out Area for Practice Problem

Check: . Since , area ✓

Key Takeaways: Deck 1

✓ derives from by writing

✓ The derivation works for all triangle types: acute, right, and obtuse

✓ For obtuse triangles, preserves the formula

The angle must be included — between the two sides you use

Don't drop the ½ — it comes from the base-height formula

Check degree mode — must equal

What Comes Next in Deck Two

In Deck 2 (Triangle Area — Applications and Interpretation):

• Using all three equivalent forms — choosing the right one
• Geometric interpretation: how angle controls area for fixed sides
• Why area is maximized when
• Real-world applications: surveying, parallelograms, land plots
• Formula selection: when to use vs. vs. Heron's formula

Forward connections: HSG.SRT.D.10 (Law of Sines), physics —