Area of a Triangle (½bh)

Statement

The area of a triangle can be calculated by multiplying its base and perpendicular height, then halving the result. The formula is:

\[ A = \tfrac{1}{2} b h \]

Here, \(b\) is the length of the chosen base, and \(h\) is the perpendicular height (altitude) from that base to the opposite vertex.

Why it’s true (short reason)

• A rectangle has area base × height.
• A triangle is exactly half of a rectangle with the same base and height.
• Therefore, triangle area is \(\tfrac{1}{2} b h\).

Recipe (how to use it)

1. Choose a base \(b\) (any side of the triangle).
2. Find the perpendicular height \(h\) from the opposite vertex to that base.
3. Multiply base and height together.
4. Divide the result by 2.
5. Give the final answer with correct square units.

Spotting it

Use this formula when:

• The triangle’s base and perpendicular height are given directly.
• The question provides dimensions with a right angle marked between them.
• The problem says “altitude” or “perpendicular distance.”

Common pairings

• Pythagoras’ theorem, when height is hidden but can be calculated.
• Trigonometry, when the perpendicular height must be found using sine or cosine.
• Other triangle area formulas, such as \(A = \tfrac{1}{2} ab \sin C\), which generalise this method.

Mini examples

1. Given: Triangle with base 10 cm, height 6 cm. Find: Area. Answer: \(A = \tfrac{1}{2} \times 10 \times 6 = 30\ \text{cm}^2\).
2. Given: Triangle with base 12 m, height 8 m. Find: Area. Answer: \(A = \tfrac{1}{2} \times 12 \times 8 = 48\ \text{m}^2\).

Pitfalls

• Using a slanted side instead of the perpendicular height.
• Forgetting to halve the product.
• Confusing units (e.g. base in cm, height in mm).
• Choosing a base without ensuring height is perpendicular to it.

Exam strategy

• Underline the base and height in the diagram to confirm they are perpendicular.
• If no height is drawn, construct the altitude with a dotted line.
• Check units carefully before multiplying.
• Always write the formula before substitution to secure method marks.

Summary

The formula \(A = \tfrac{1}{2} b h\) is the most fundamental method for finding the area of a triangle. It works for all triangles as long as you can find a perpendicular height. It is simple, reliable, and forms the basis of many advanced geometry methods.