MathIsimple

Geometry Formulas

Complete reference of all geometry formulas: triangles, quadrilaterals, circles, vectors, and trigonometry.

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Triangle Formulas

Angle Sum

∠A+∠B+∠C=180°\angle A + \angle B + \angle C = 180°

Area (Base × Height)

S=12bhS = \frac{1}{2}bh

Heron's Formula

S=s(s−a)(s−b)(s−c)S = \sqrt{s(s-a)(s-b)(s-c)}

where s=a+b+c2s = \frac{a+b+c}{2}

Area (SAS)

S=12absin⁡CS = \frac{1}{2}ab\sin C

Pythagorean Theorem

a2+b2=c2a^2 + b^2 = c^2

(right triangle, c = hypotenuse)

Equilateral Area

S=34s2S = \frac{\sqrt{3}}{4}s^2

Law of Sines

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Perimeter

P=a+b+cP = a + b + c
Quadrilateral Formulas

Angle Sum

∑angles=360°\sum \text{angles} = 360°

Rectangle Area

A=l×wA = l \times w

Rectangle Perimeter

P=2(l+w)P = 2(l + w)

Square Area

A=s2A = s^2

Square Perimeter

P=4sP = 4s

Parallelogram Area

A=b×hA = b \times h

Rhombus Area

A=12d1×d2A = \frac{1}{2}d_1 \times d_2

Trapezoid Area

A=12(a+b)×hA = \frac{1}{2}(a + b) \times h

Rectangle Diagonal

d=l2+w2d = \sqrt{l^2 + w^2}
Circle Formulas

Circumference

C=2πr=πdC = 2\pi r = \pi d

Area

A=πr2A = \pi r^2

Diameter

d=2rd = 2r

Arc Length (radians)

s=rθs = r\theta

Sector Area (radians)

A=12r2θA = \frac{1}{2}r^2\theta

Arc Length (degrees)

s=θ°360°×2πrs = \frac{\theta°}{360°} \times 2\pi r

Sector Area (degrees)

A=θ°360°×πr2A = \frac{\theta°}{360°} \times \pi r^2

Semicircle Area

A=12πr2A = \frac{1}{2}\pi r^2

Pi (π)

π≈3.14159\pi \approx 3.14159
Vector Formulas

Magnitude

∣a⃗∣=a12+a22|\vec{a}| = \sqrt{a_1^2 + a_2^2}

Addition

a⃗+b⃗=(a1+b1,a2+b2)\vec{a} + \vec{b} = (a_1+b_1, a_2+b_2)

Scalar Multiplication

ka⃗=(ka1,ka2)k\vec{a} = (ka_1, ka_2)

Dot Product

a⃗⋅b⃗=a1b1+a2b2\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2

Angle Formula

cos⁡θ=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}

Unit Vector

a^=a⃗∣a⃗∣\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Perpendicular Test

a⃗⊥b⃗  ⟺  a⃗⋅b⃗=0\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0

Position Vector

AB⃗=(xB−xA,yB−yA)\vec{AB} = (x_B - x_A, y_B - y_A)

Dot Product Property

a⃗⋅a⃗=∣a⃗∣2\vec{a} \cdot \vec{a} = |\vec{a}|^2
Angle Relationships

Complementary

∠A+∠B=90°\angle A + \angle B = 90°

Supplementary

∠A+∠B=180°\angle A + \angle B = 180°

Vertical Angles

∠1=∠3\angle 1 = \angle 3

Polygon Angle Sum

(n−2)×180°(n-2) \times 180°

Regular Polygon Interior

(n−2)×180°n\frac{(n-2) \times 180°}{n}

Exterior Angle (Triangle)

∠ext=∠1+∠2\angle_{ext} = \angle_1 + \angle_2
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