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Toolkit 54

Area Formulas and Important Geometry Formulas

Triangle Area Formulas

1. Base and Height Formula

Triangle with altitude h from A to BC at D
[ABC]=12⋅AD⋅BC[ABC]=\frac{1}{2}\cdot AD\cdot BC

2. Two Sides and Included Angle Formula

Triangle with sides a, b, c
[ABC]=12bcsin⁡A[ABC]=\frac{1}{2}bc\sin A

3. Heron's Formula

Triangle with sides a, b, c
s=a+b+c2s=\frac{a+b+c}{2}
[ABC]=s(s−a)(s−b)(s−c)[ABC]=\sqrt{s(s-a)(s-b)(s-c)}

4. Expanded Heron's Formula

Triangle with sides a, b, c
16[ABC]2=4a2b2c2−(a2+b2−c2)216[ABC]^2=4a^2b^2c^2-(a^2+b^2-c^2)^2

Quadrilateral Area Formulas

General Quadrilateral

Convex quadrilateral with diagonals intersecting at angle alpha
[ABCD]=12AC⋅BDsin⁡α[ABCD]=\frac{1}{2}AC\cdot BD\sin\alpha

Orthogonal Diagonals

Quadrilateral with perpendicular diagonals
[ABCD]=12AC⋅BD[ABCD]=\frac{1}{2}AC\cdot BD

Circle Formulas

Circle Area and Circumference

Circle with radius r and center O
Area=πr2\text{Area}=\pi r^2
Circumference=2πr\text{Circumference}=2\pi r

Sector Area and Arc Length

Circular sector with radius r and central angle alpha

If α\alpha is in degrees:

Area=α360πr2\text{Area}=\frac{\alpha}{360}\pi r^2
Length of arc AB=α360⋅2πr\text{Length of arc }AB=\frac{\alpha}{360}\cdot2\pi r

Special Right Triangles

30-60-90 Triangle

30-60-90 right triangle
Sides are a, a32, a2\text{Sides are } a,\ \frac{a\sqrt3}{2},\ \frac{a}{2}

45-45-90 Triangle

45-45-90 right isosceles triangle
Sides are a22, a22, a\text{Sides are }\frac{a\sqrt2}{2},\ \frac{a\sqrt2}{2},\ a

Equilateral Triangle

Equilateral triangle with altitude AM
AM=a32AM=\frac{a\sqrt3}{2}
[ABC]=a234[ABC]=\frac{a^2\sqrt3}{4}

Trapezoid

Trapezoid ABCD with height h
[ABCD]=h(AB+CD)2[ABCD]=\frac{h(AB+CD)}{2}

Parallelogram

Parallelogram ABCD with height h
[ABCD]=h⋅AB[ABCD]=h\cdot AB
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