Triangle Area

TipLearning Objectives
  • Compute triangle area using multiple formulas.
  • Identify base and height correctly.
  • Use area relationships in coordinate geometry.

Key Ideas

Main Formula

\[ A = \frac{1}{2}bh \]

  • Base and height must be perpendicular.

Special Cases

  • Right triangles: legs act as base and height.
  • Coordinate geometry: use vertical or horizontal distances.
  • Heron’s Formula (optional):
    \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

Triangle with height drawn perpendicular to the base.

Common Problem Types

Right Triangle Area

Use legs:
\[A = \frac12 ab.\]

Using Coordinates

Compute vertical or horizontal height.

Non-Right Triangle With Altitude

Height drawn from vertex to base.

Using Area to Solve for Missing Measurements

Set up equation \(A = \frac12 bh\).

Strategies

  • Identify a perpendicular height — draw it if not given.
  • For coordinate problems, use differences in x or y.
  • Check whether a triangle is right before calculating.

Worked Examples

Example 1 — Basic

Base = 10, height = 6
\[ A = \frac12(10)(6)=30 \]

Example 2 — Coordinate

Points A(1,2), B(1,7), C(6,2) form a right triangle with legs 5 and 5: \[A=\frac12(5)(5)=12.5.\]

WarningCommon Mistakes
  • Using incorrect height (not perpendicular).
  • Using side lengths without verifying right triangle.
  • Forgetting the 1/2 factor.

Practice Problems

  1. Base = 12, height = 4.
  2. Right triangle legs: 5 and 8.
  3. Coordinates: A(0,0), B(6,0), C(6,4).
  4. Area = 24, base = 8 → find height.
  1. \(A=24\)
  2. \(A=20\)
  3. \(A=\frac12(6)(4)=12\)
  4. \(24=\frac12(8)h → h=6\)

Summary

  • Area formula: \(A=\frac12 bh\).
  • Height must be perpendicular.
  • Right triangles simplify area greatly.
  • Draw altitudes for non-right triangles.
  • In coordinates, use horizontal/vertical distances.