Overview

Choose the area formula that matches the information given. Some problems are fast with base-height, others with Heron or trigonometric area.

Key Ideas

• Base-height: $A = \frac{1}{2}bh$.
• Heron: $A = \sqrt{s(s-a)(s-b)(s-c)}$ with $s=\frac{a+b+c}{2}$.
• Trig area: $A = \frac{1}{2}ab\sin C$.
• Inradius/circumradius: $A = rs$ and $A = \frac{abc}{4R}$.

Core Skills

Pick the Fast Formula

Use base-height when a height is given or easy to find, Heron when all three sides are known, and trigonometric area when two sides and the included angle are given.

Convert Between Forms

Area can unlock missing heights: if $A$ is known and the base is known, then $h = 2A/b$. This is often faster than building new triangles.

Radius Connections

If the inradius or circumradius is given, use $A=rs$ or $A=abc/(4R)$ directly instead of creating additional equations.

Worked Example

A triangle has sides $13,14,15$. Find its area.

Compute $s = (13+14+15)/2 = 21$. Then $A = \sqrt{21\cdot 8\cdot 7\cdot 6} = \sqrt{7056} = 84$.

More Examples

Example 1: Trigonometric Area

Two sides of a triangle are $7$ and $9$ with included angle $60^\circ$. Find the area.

$A = \frac{1}{2}\cdot 7\cdot 9\cdot \sin 60^\circ = \frac{63\sqrt{3}}{4}$.

Example 2: Height from Area

A triangle has base $12$ and area $30$. Find the corresponding height.

$h = \frac{2A}{b} = \frac{60}{12} = 5$.

Example 3: Inradius

A triangle has inradius $3$ and semiperimeter $14$. Find the area.

$A = rs = 3\cdot 14 = 42$.

Strategy Checklist

• Match the formula to the given data before computing.
• For Heron, compute $s$ first and simplify factors.
• Use $A=rs$ or $A=abc/(4R)$ only when radii are known.
• Keep track of units and avoid negative square roots.

Common Pitfalls

• Forgetting to compute the semiperimeter in Heron's formula.
• Using $\sin$ on the wrong included angle.
• Mixing a height from a different base.

Practice Problems