MathGoldMedalist
Sign InRegister

Area Formulas

Lesson · Beginner

Geometry: Plane Geometry

Triangle Area Formulas

1. Base and Height Formula

Triangle ABC with altitude AD to BC
[ABC]=12⋅AD⋅BC[ABC]=\frac12\cdot AD\cdot BC

2. Two Sides and Included Angle Formula

Triangle ABC with sides b and c adjacent to angle A
[ABC]=12bcsin⁡A[ABC]=\frac12bc\sin A

3. Heron's Formula

Triangle ABC with side lengths a, b, and 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 ABC with side lengths a, b, and 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

Quadrilateral ABCD with diagonals AC and BD making angle alpha
[ABCD]=12AC⋅BDsin⁡α[ABCD]=\frac12 AC\cdot BD\sin\alpha

Orthogonal Diagonals

Quadrilateral ABCD with perpendicular diagonals AC and BD
[ABCD]=12AC⋅BD[ABCD]=\frac12 AC\cdot BD

Circle Formulas

Circle Area and Circumference

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

Sector Area and Arc Length

Circular sector AOB 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

Equilateral Triangle

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

Trapezoid

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

Parallelogram

Parallelogram ABCD with base AB and height h
[ABCD]=h⋅AB[ABCD]=h\cdot AB
Back to Toolkit