Andromeda
Note

Area of a Triangle

Definition

Beyond the basic 12bh\frac{1}{2}bh formula, trigonometry provides methods to calculate the area of any triangle using side lengths and angles.

Why It Matters

Since any polygon can be decomposed into triangles, this formula is the fundamental atom of all area calculations. It allows us to measure any complex surface by knowing only a few side lengths and angles.

Core Concepts

  • Foundational Formula (base × height) Area=12bh\text{Area} = \frac{1}{2} b h

    • How to read: “The area A equals one-half the base b times the height h.”
    • Meaning: The definition of triangular area. Every other formula is a device to find the product bh when h is not given directly.

  • SAS Area Formula Area=12bcsin⁡α=12acsin⁡β=12absin⁡γ\text{Area} = \frac12 b c \sin \alpha = \frac12 a c \sin \beta = \frac12 a b \sin \gamma

    • How to read: “The area A equals one-half b c times the sine of the included angle alpha.”
    • Meaning: See Triangle Area formulas. Use when two sides and the angle between them are known.

  • Heron’s Formula (SSS) Area=s(s−a)(s−b)(s−c),s=a+b+c2\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}

    • How to read: “The area A equals the square root of the product of s and the quantities s minus a, s minus b, and s minus c”; “the semiperimeter s equals the sum a plus b plus c, all over two.”
    • Meaning: See dedicated Heron’s Formula note.

  • Inradius form Area=12rP=r⋅s\text{Area} = \frac12 r P = r \cdot s

    • How to read: “The area A equals one-half the inradius r times the perimeter P, or the inradius r times the semiperimeter s.”
    • Meaning: Area is the sum of areas of three triangles from incenter to sides. Useful when the incircle radius is known or easy to construct.

  • Equilateral special case A=s234=h233A = \frac{s^2 \sqrt{3}}{4} = \frac{h^2 \sqrt{3}}{3}

    • How to read: “The area A equals the side s squared times the square root of three, all over four.”
    • Meaning: Constant factor derived from 30-60-90 triangles. Memorize for speed.

  • ASA form Area=a2sin⁡Bsin⁡C2sin⁡A\text{Area} = \frac{a^2 \sin B \sin C}{2 \sin A}

    • How to read: “The area A equals a squared times the sine of angle B times the sine of angle C, all over two times the sine of angle A.”
    • Meaning: After finding third angle via 180° sum and applying law of sines. See triangle area formulas.

Connected Concepts

Connected notes

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