Geometry Formulas — All Equations for Geometry (80+)

Geometry is built on a small core of formulas you'll use over and over. The most-searched ones are area, perimeter, volume, and surface area for the standard shapes — plus the Pythagorean theorem, the distance and midpoint formulas, and the polygon angle sum (n − 2) × 180°. Memorise these and you've covered roughly 80% of every geometry problem you'll encounter through high school.

Below you'll find a complete reference of 54+ formulas organized by category. Each entry includes the formula itself, a one-line explanation, and a direct link to a free calculator. No sign-up, no paywall.

Whether you're looking for all equations for geometry, the formulas of geometric figures, specific area formulas for geometry, or just simple basic geometry formulas — every essential equation lives in one of the sections below. From 2D shapes (circles, triangles, polygons, quadrilaterals) to 3D solids (cube, cylinder, sphere, cone, pyramid) and coordinate geometry (distance, midpoint, slope) — this is the single reference you can bookmark for the whole school year.

Triangle Formulas

The most common geometry formulas — area, perimeter, Pythagorean, Law of Sines/Cosines, and Heron's formula.

Formula	Equation	Notes	Calculate
Triangle Perimeter	`P = a + b + c`	Sum of all three sides.	Use
Triangle Area (base × height)	`A = ½ × b × h`	b = base, h = perpendicular height to that base.	Use
Heron's Formula	`A = √(s(s−a)(s−b)(s−c))`	s = (a+b+c)/2 (semi-perimeter). Use when only the 3 sides are known.	Use
Pythagorean Theorem	`a² + b² = c²`	For right triangles only — c is the hypotenuse.	Use
Law of Cosines	`c² = a² + b² − 2ab·cos(C)`	Generalises Pythagoras to any triangle.	Use
Law of Sines	`a / sin(A) = b / sin(B) = c / sin(C)`	Use for ASA, AAS, or SSA triangles.	Use
45-45-90 Triangle	`sides = 1 : 1 : √2`	Right isosceles — hypotenuse = leg × √2.	Use
30-60-90 Triangle	`sides = 1 : √3 : 2`	Special right triangle — long leg = short leg × √3.	Use
Similar Triangle Ratio	`side'/side = scale factor k`	All corresponding sides are proportional by the same k.	Use

Quadrilateral Formulas

Square, rectangle, rhombus, parallelogram, trapezoid — all 4-sided shape area and perimeter formulas.

Formula	Equation	Notes	Calculate
Square Area	`A = s²`	s = side length.	Use
Square Perimeter	`P = 4s`		Use
Rectangle Area	`A = l × w`	l = length, w = width.	Use
Rectangle Perimeter	`P = 2(l + w)`		Use
Parallelogram Area	`A = b × h`	b = base, h = perpendicular height (NOT the slanted side).	Use
Parallelogram Perimeter	`P = 2(a + b)`	a, b = the two different side lengths.	Use
Rhombus Area	`A = ½ × d₁ × d₂`	d₁, d₂ = the two diagonals.	Use
Trapezoid Area	`A = ½ × (b₁ + b₂) × h`	b₁, b₂ = parallel bases, h = perpendicular height.	Use
Trapezoid Midsegment	`m = (b₁ + b₂) / 2`	Average of the two parallel bases.	Use

Circle Formulas

Area, circumference, sector, arc length — every circle calculation derived from π and the radius.

Formula	Equation	Notes	Calculate
Circle Area	`A = π × r²`	r = radius. Equivalent: A = π·d²/4.	Use
Circle Circumference	`C = 2π × r = π × d`	"Perimeter" of a circle. d = 2r = diameter.	Use
Diameter	`d = 2 × r`		Use
Sector Area	`A_sector = ½ × r² × θ`	θ in radians. For degrees: A = (θ°/360) × π × r².	Use
Arc Length	`L = r × θ`	θ in radians. For degrees: L = (θ°/360) × 2π × r.	Use
Standard Equation	`(x − h)² + (y − k)² = r²`	Center (h, k), radius r. Coordinate geometry form.	Use
Inscribed Angle	`∠inscribed = ½ × ∠central`	An angle inscribed in a circle is half the central angle subtending the same arc.	Use

Polygon Formulas

Interior/exterior angles, regular polygon area, and the Shoelace formula for any irregular polygon.

Formula	Equation	Notes	Calculate
Sum of Interior Angles	`S = (n − 2) × 180°`	n = number of sides. Pentagon (n=5) → 540°.	Use
Each Interior Angle (regular)	`a = (n − 2) × 180° / n`	For a regular polygon (all sides equal). Hexagon → 120°.	Use
Sum of Exterior Angles	`360° (always, for any convex polygon)`	Independent of n.	Use
Each Exterior Angle (regular)	`e = 360° / n`	Hexagon → 60°, octagon → 45°.	Use
Number of Sides from Angle Sum	`n = S / 180° + 2`	Inverse: given S, recover n.	Use
Regular Polygon Area	`A = ¼ × n × s² × cot(π/n)`	s = side length. Equivalent: A = ½ × P × apothem.	Use
Shoelace Formula (any polygon)	`A = ½ × \|Σᵢ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)\|`	For irregular polygons defined by vertex coordinates.	Use

3D Solid Formulas

Volume and surface area for cube, rectangular prism, sphere, cylinder, cone, pyramid.

Formula	Equation	Notes	Calculate
Cube Volume	`V = s³`	s = edge length.	Use
Cube Surface Area	`SA = 6s²`		Use
Rectangular Prism Volume	`V = l × w × h`	Box volume.	Use
Rectangular Prism SA	`SA = 2(lw + lh + wh)`		Use
Cylinder Volume	`V = π × r² × h`	r = radius, h = height.	Use
Cylinder Surface Area	`SA = 2πr² + 2πrh`	2 circular caps + lateral rectangle.	Use
Sphere Volume	`V = (4/3) × π × r³`		Use
Sphere Surface Area	`SA = 4 × π × r²`	Equivalent to the area of 4 great circles.	Use
Cone Volume	`V = (1/3) × π × r² × h`	Exactly ⅓ of the cylinder with same base + height.	Use
Cone Surface Area	`SA = πr² + πrl`	l = slant height = √(r² + h²).	Use
Cone Lateral SA	`LSA = π × r × l`	Just the curved side, no base.	Use
Square Pyramid Volume	`V = (1/3) × b² × h`	b = base side.	Use
Space Diagonal of Box	`d = √(l² + w² + h²)`	3D Pythagorean.	Use

Coordinate Geometry Formulas

Distance, midpoint, slope, section formula — analytic geometry essentials.

Formula	Equation	Notes	Calculate
Distance Between Two Points	`d = √((x₂−x₁)² + (y₂−y₁)²)`	2D Pythagorean applied to coordinates.	Use
Midpoint Formula	`M = ((x₁+x₂)/2, (y₁+y₂)/2)`	Exact center of a segment.	Use
Slope of a Line	`m = (y₂ − y₁) / (x₂ − x₁)`	Vertical change over horizontal change. Vertical lines: undefined slope.	Use
Slope-Intercept Form	`y = mx + b`	m = slope, b = y-intercept.	Use
Point-Slope Form	`y − y₁ = m(x − x₁)`	Build a line from one known point + slope.	Use
Section Formula (internal)	`P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))`	Point dividing segment in ratio m : n internally. Class 10 essential.	Use
3D Distance	`d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)`	Adds the z-axis to the 2D distance formula.	Use
Parallel Lines	`m₁ = m₂`	Equal slopes.	Use
Perpendicular Lines	`m₁ × m₂ = −1`	Negative reciprocal slopes.	Use

Frequently Asked Questions

Which geometry formulas do I need to memorize?

For high school: area + perimeter of triangle / rectangle / circle / parallelogram / trapezoid; volume + surface area of cube / cylinder / sphere; the Pythagorean theorem (a² + b² = c²); the distance formula; and the polygon angle sum (n − 2) × 180°. Everything else can be derived from these in seconds.

What's the difference between perimeter, area, and volume?

Perimeter measures the outline (1D — units like cm). Area measures the 2D surface (units like cm²). Volume measures the 3D space inside a solid (units like cm³). A square of side 5 cm has perimeter 20 cm, area 25 cm², and (as a cube) volume 125 cm³.

Why does the polygon angle sum use (n − 2)?

Any polygon with n sides can be cut into (n − 2) non-overlapping triangles by drawing diagonals from one vertex. Each triangle's angles sum to 180°, so the polygon's total angle sum is (n − 2) × 180°. A pentagon (n = 5) splits into 3 triangles → 540°.

When should I use Heron's formula vs ½×base×height for area?

Use ½ × base × height when you have a base and the perpendicular height to that base. Use Heron's formula when you only know the three side lengths (no height available). Heron's: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.

How do I find the surface area of a cone vs the lateral surface area?

Lateral surface area (LSA = πrl) is just the curved side, where l = √(r² + h²) is the slant height. Total surface area (SA = πr² + πrl) adds the circular base. Use lateral when wrapping a cone (e.g. paint, fabric) and total when fully enclosing it.

Are these formulas valid for all coordinate systems?

All formulas above use Euclidean (flat) geometry with Cartesian (rectangular) coordinates. They do NOT apply to spherical geometry (Earth's surface), hyperbolic geometry, or non-Cartesian systems (polar, cylindrical) without conversion. For everyday school and engineering math, Euclidean coverage is sufficient.

Are these calculators free?

Yes. Every formula on this page links to a calculator that's free and unlimited — no sign-up needed. AI-powered step-by-step explanations cost 3 credits each (every account gets 30 free credits on signup).

Geometry Formulas — Complete Reference