Geometry formulas — 108 solvers

Annulus Area (Ring)

A = \pi (R^2 - r^2)

GeometryArea of the ring between two concentric circles.

Arc Length

s = r \theta

GeometryTrigonometryLength of a circular arc as radius times central angle in radians.

Area Moment of Inertia — Rectangle

I = \frac{b h^{3}}{12}

Strength of MaterialsMechanicsGeometryArea moment of inertia of a rectangle about its centroidal axis, I = bh³/12, returned in m⁴ and entered as a plain number.

Area Moment of Inertia — Solid Round Bar

I = \frac{\pi d^{4}}{64}

Strength of MaterialsMechanicsGeometryArea moment of inertia of a solid round bar about a diameter, I = πd⁴/64, returned in m⁴ and entered as a plain number.

Area of a Circle

A = \pi r^{2}

GeometryArea enclosed by a circle of radius r.

Area of a Three-Sided Parcel by Coordinates

A = \tfrac{1}{2}\left[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right]

Civil & SurveyingGeometryShoelace area of a parcel from the coordinates of its three corners, listed counter-clockwise so the result comes out positive.

Area of a Triangle

A = \tfrac{1}{2} b h

GeometryArea from base and perpendicular height.

Circle Radius from Centre and a Point

r = \sqrt{(x - h)^2 + (y - k)^2}

AlgebraGeometryRadius of a circle passing through a given point, measured from its centre using the distance formula inside the circle equation.

Circular Sector Area

A = \frac{1}{2} r^{2} \theta

GeometryTrigonometryArea of a pie-slice sector of a circle from its radius and central angle in radians.

Circular Segment Area

A = \frac{r^2}{2}(\theta - \sin\theta)

GeometryArea of the segment cut off by a chord — the sector minus the triangle, from radius and central angle.

Circumference of a Circle

C = 2 \pi r

GeometryDistance around a circle of radius r.

Common Rafter Length from Run and Pitch

L = R\sqrt{1 + \left(\frac{p}{12}\right)^2}

Trades & ConstructionGeometryThe line length of a common rafter: the level run from ridge to wall plate, stretched by the slope factor for a pitch of p in 12.

Concrete Volume with Waste Allowance

V = L\,W\,T\left(1 + \frac{w}{100}\right)

Civil & SurveyingGeometryOrders concrete for a rectangular slab, footing or column by adding a percent waste allowance to the neat volume, in cubic yards.

Cone Frustum Volume (Truncated Cone)

V = \frac{\pi h}{3}\left(R^2 + Rr + r^2\right)

GeometryVolume of a cone with its tip cut off parallel to the base — the shape of a bucket, a lampshade or a hopper.

Cone Lateral Surface Area

A = \pi r l

GeometryCurved surface of a right circular cone from its base radius and slant height, excluding the base disc.

Cone Slant Height

l = \sqrt{r^2 + h^2}

GeometrySlant height of a right circular cone by Pythagoras on its axial cross-section, from base radius and vertical height.

Cone Total Surface Area

A = \pi r (r + l)

GeometryWhole surface of a solid right circular cone: the curved side pi*r*l plus the base disc pi*r^2.

Cone Volume

V = \frac{1}{3} \pi r^{2} h

GeometryVolume of a right circular cone — one-third of the matching cylinder — using π ≈ 3.14159265.

Crest Vertical Curve Length for Sight Distance

L = \frac{A\,S^{2}}{200\left(\sqrt{h_1} + \sqrt{h_2}\right)^{2}}

Civil & SurveyingGeometryCrest curve length needed to see an object over the hill, for the case where sight distance is shorter than the curve.

Cross Product z-Component of Two 2D Vectors

(\vec{a}\times\vec{b})_z = a_x b_y - a_y b_x

Vectors & MatricesAlgebraGeometryComputes the signed out-of-plane cross product of two plane vectors from their components, whose sign reveals their turning direction.

Cube Face Diagonal

d = s\sqrt{2}

GeometryDiagonal measured across one square face of a cube — shorter than the space diagonal, which cuts through the interior.

Cube Space Diagonal

d = a \sqrt{3}

GeometryLength of the interior diagonal joining opposite corners of a cube.

Cube Surface Area

S = 6 a^{2}

GeometrySurface area of a cube as six times the area of one square face.

Cube Volume

V = a^{3}

GeometryVolume of a cube as its edge length raised to the third power.

Cylinder Lateral Surface Area

A = 2\pi r h

GeometryArea of a cylinder's curved wall only, with no end caps — the label on a can, the insulation around a pipe.

Cylinder Surface Area

S = 2 \pi r^{2} + 2 \pi r h

GeometryTotal surface area of a closed cylinder — two end caps plus the wrapped side — using π ≈ 3.14159265.

Cylinder Volume

V = \pi r^{2} h

GeometryWater TreatmentVolume of a right circular cylinder from its radius and height, using π ≈ 3.14159265.

Distance Formula (2D)

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

GeometryAlgebraStraight-line distance between two points in the coordinate plane.

Distance from a Point to a Line

d = \frac{\left| A x_0 + B y_0 + C \right|}{\sqrt{A^2 + B^2}}

AlgebraGeometryPerpendicular distance from a point to a line written in general form Ax + By + C = 0, the shortest gap between the two.

Earthwork Volume by Average End Area

V = \frac{L\,(A_1 + A_2)}{2}

Civil & SurveyingGeometryVolume between two cross sections, averaging their end areas over the distance between them; the answer is reported in cubic yards.

Earthwork Volume by the Prismoidal Formula

V = \frac{L\,(A_1 + 4A_m + A_2)}{6}

Civil & SurveyingGeometrySimpson's rule applied to earthwork, weighting the middle cross section four times the ends; the answer is reported in cubic yards.

Elastic Section Modulus (S = I/c)

S = \frac{I}{c}

Strength of MaterialsMechanicsGeometryElastic section modulus S = I/c, in m³, the single number that turns a bending moment straight into a bending stress.

Elevation from Grade and Distance

E_2 = E_1 + \frac{G\,L}{100}

Civil & SurveyingGeometryProjects an elevation along a uniform grade, adding the rise over a measured horizontal distance to the known starting elevation.

Elevation on a Parabolic Vertical Curve

E = E_{BVC} + \frac{g_1 x}{100} + \frac{A\,x^{2}}{200\,L}

Civil & SurveyingGeometryElevation at any station on an equal-tangent parabolic vertical curve, measured from the beginning of vertical curve.

Ellipse Area

A = \pi a b

GeometryArea of an ellipse from its semi-major and semi-minor axes, with π ≈ 3.14159265.

Ellipse Perimeter (Ramanujan Approximation)

P \approx \pi \left[ 3(a+b) - \sqrt{(3a+b)(a+3b)} \right]

GeometryPerimeter of an ellipse by Ramanujan's second approximation — accurate to better than one part in a billion for ordinary shapes.

Ellipsoid Volume

V = \frac{4}{3}\pi a b c

GeometryVolume of an ellipsoid from its three semi-axes; setting a = b = c returns the sphere.

Euclid's Pythagorean Triple

c = m^2 + n^2

GeometryEuclid's generator turns any two whole numbers into a right triangle: legs m² − n² and 2mn, hypotenuse m² + n². Every primitive triple arises exactly once this way.

Geometric Mean of Two Numbers

G = \sqrt{ab}

AlgebraGeometryThe square root of the product of two numbers, the average that suits growth rates, ratios and scale factors of any kind.

Hemisphere Total Surface Area

A = 3\pi r^2

GeometryTotal surface of a solid hemisphere: the curved half-sphere 2*pi*r^2 plus the flat circular base pi*r^2, giving 3*pi*r^2.

Hemisphere Volume

V = \frac{2}{3}\pi r^3

GeometryVolume of half a sphere, exactly half of the full sphere's (4/3)*pi*r^3.

Heron's Formula (Triangle Area from Three Sides)

A = \sqrt{s(s-a)(s-b)(s-c)}, \; s = \tfrac{a+b+c}{2}

GeometryArea of any triangle from the lengths of its three sides, using the semiperimeter.

Horizontal Curve Length from Degree of Curve

L = \frac{100\,\Delta}{D}

Civil & SurveyingGeometryLength of a circular curve in 100 ft stations, from the total deflection angle and the degree of curve, on the arc definition.

Kite Area

A = \frac{d_1 d_2}{2}

GeometryArea of a kite as half the product of its two perpendicular diagonals.

Ladder Setback and Angle (4-to-1 Rule)

\theta = \arctan\!\left(\frac{H}{B}\right)

Trades & ConstructionGeometryRelates the height a ladder reaches, the distance its feet sit out from the wall, and the resulting angle. The 4-to-1 rule sets the setback at a quarter of the height, which lands at about 75.5°.

Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

TrigonometryGeometryFinds the third side of any triangle from two sides and their included angle, or the angle from all three sides.

Law of Sines

\frac{a}{\sin A} = \frac{b}{\sin B}

TrigonometryGeometryIn any triangle, each side is proportional to the sine of its opposite angle.

Magnitude of a 2D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2}

Vectors & MatricesAlgebraGeometryFinds the length of a two-dimensional vector from its x- and y-components, the Pythagorean theorem written for arrows instead of triangles.

Magnitude of a 3D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Vectors & MatricesAlgebraGeometryFinds the length of a three-dimensional vector from its x-, y-, and z-components, extending Pythagoras into space.

Midpoint Formula

x_m = \frac{x_1 + x_2}{2}

AlgebraGeometryThe coordinate halfway between two endpoints along one axis; apply it to x and again to y for the midpoint of a segment.

Moment of Inertia — I-Beam or Built-Up Section

I = \frac{B H^{3} - (B - t_w)(H - 2t_f)^{3}}{12}

Strength of MaterialsCivil & SurveyingGeometryStrong-axis second moment of area of a doubly symmetric I-shape, taken as the full bounding rectangle minus the two rectangular voids beside the web.

Parallel Axis Theorem (I = I_c + Ad²)

I = I_c + A d^{2}

Strength of MaterialsCivil & SurveyingGeometrySecond moment of area of a shape about an axis parallel to its own centroidal axis: add A times the offset squared. The transfer term that builds every plate girder and flitch beam.

Parallelogram Area

A = b \cdot h

GeometryArea of a parallelogram from its base and perpendicular height.

Parallelogram Area from Two Vectors

A = \left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the parallelogram spanned by two plane vectors directly from their four components, no angle or height required.

Parallelogram Perimeter

P = 2(a + b)

GeometryPerimeter of a parallelogram from its two side lengths.

Partially Filled Horizontal Cylindrical Tank

V = L \left[ r^{2} \cos^{-1}\!\left(\frac{r-h}{r}\right) - (r-h)\sqrt{2rh - h^{2}} \right]

HVAC & HydronicsFluid MechanicsGeometryLiquid volume in a horizontal cylinder from the wetted depth, using the circular segment area times the tank length.

Percent Grade from Rise and Run

G = \frac{100\,\Delta h}{L}

Civil & SurveyingGeometryPercent grade is one hundred times the vertical rise divided by the horizontal run, measured level, never along the slope.

Perpendicular Slope Relation

m_2 = -\frac{1}{m_1}

AlgebraGeometrySlope of a line perpendicular to a given line — the negative reciprocal, so that the two slopes always multiply to −1.

Plastic Section Modulus — Rectangle

Z = \frac{b h^{2}}{4}

Strength of MaterialsCivil & SurveyingGeometryPlastic section modulus of a solid rectangle, the first moment of the two half-areas about the equal-area axis. Exactly 1.5 times the elastic section modulus bh²/6.

Point-Slope Form of a Line

y = y_1 + m(x - x_1)

AlgebraGeometryEquation of a straight line through one known point with a known slope, giving the y-value at any x you care to choose.

Polar Moment of Inertia — Solid Shaft

J = \frac{\pi d^{4}}{32}

Strength of MaterialsMechanicsGeometryPolar moment of inertia of a solid round shaft, J = πd⁴/32, in m⁴ — exactly twice the diametral moment of inertia.

Prism Volume (General Cross-Section)

V = B L

GeometryVolume of any prism or extrusion whose cross-section does not change along its length: cross-sectional area times length.

Pyramid Volume

V = \frac{1}{3} B h

GeometryVolume of any pyramid as one-third of its base area times its perpendicular height.

Pythagorean Theorem

a^{2} + b^{2} = c^{2}

GeometryRelates the three sides of a right triangle.

Radius from Degree of Curve (Arc Definition)

R = \frac{5729.578}{D}

Civil & SurveyingGeometryConverts degree of curve to radius using the arc definition, where D is the central angle subtending one 100 ft station of arc.

Radius of Gyration (r = √(I/A))

r = \sqrt{\frac{I}{A}}

Strength of MaterialsMechanicsGeometryRadius of gyration of a cross-section, r = √(I/A), the shape property that decides how slender a column really is.

Rectangle Area

A = l \cdot w

GeometryArea of a rectangle as length times width.

Rectangle Diagonal

d = \sqrt{l^2 + w^2}

GeometryDiagonal of a rectangle from its length and width.

Rectangle Perimeter

P = 2(l + w)

GeometryPerimeter of a rectangle from its length and width.

Rectangular Prism Space Diagonal

d = \sqrt{l^2 + w^2 + h^2}

GeometryThe corner-to-opposite-corner distance through the inside of a box — the longest straight object it can hold.

Rectangular Prism Surface Area

A = 2(lw + lh + wh)

GeometryTotal surface area of a box — three pairs of matching rectangular faces, from length, width and height.

Rectangular Prism Volume

V = l \cdot w \cdot h

GeometryVolume of a box as the product of its length, width, and height.

Regular Polygon Area (from Side Length)

A = \frac{n s^2}{4 \tan(\pi/n)}

GeometryArea of a regular polygon with n equal sides of length s — pentagon, hexagon, octagon and beyond.

Regular Polygon Perimeter

P = n s

GeometryPerimeter of a regular polygon: number of sides times side length.

Regular Tetrahedron Surface Area

A = \sqrt{3}\,s^2

GeometryTotal surface area of a regular tetrahedron: four equilateral triangles of edge s, which sum to exactly √3 s².

Regular Tetrahedron Volume

V = \frac{s^3}{6\sqrt{2}}

GeometryVolume of a regular tetrahedron — four identical equilateral triangles — from its single edge length.

Rhombus Area (from Diagonals)

A = \frac{d_1 d_2}{2}

GeometryArea of a rhombus as half the product of its two diagonals.

Rhombus Perimeter

P = 4s

GeometryPerimeter of a rhombus — four equal sides, like a leaning square.

Right-Triangle Cosine Ratio (CAH)

\cos\theta = \frac{a}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its adjacent side and the hypotenuse.

Right-Triangle Sine Ratio (SOH)

\sin\theta = \frac{o}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its opposite side and the hypotenuse.

Right-Triangle Tangent Ratio (TOA)

\tan\theta = \frac{o}{a}

TrigonometryGeometryRelates an acute angle of a right triangle to the ratio of its opposite and adjacent legs.

Roof Area from Footprint and Pitch

A_r = A_f\sqrt{1 + \left(\frac{p}{12}\right)^2}

Trades & ConstructionGeometryTurns the plan area a roof covers into the sloping area you actually have to shingle, using the pitch multiplier for a pitch of p in 12.

Roof Pitch to Slope Angle

\theta = \arctan\!\left(\frac{p}{12}\right)

Trades & ConstructionGeometryConverts a roof pitch quoted the framing way, p inches of rise per 12 inches of level run, into the slope angle from horizontal, and back.

Scalar Triple Product (Parallelepiped Volume)

V = \left|\vec{a}\cdot(\vec{b}\times\vec{c})\right|

Vectors & MatricesGeometryAlgebraFinds the volume of the parallelepiped spanned by three space vectors, from the nine components, as the size of their scalar triple product.

Slope Between Two Points

m = \frac{y_2 - y_1}{x_2 - x_1}

AlgebraGeometrySteepness of a line as rise over run between two points.

Slope from Standard Form of a Line

m = -\frac{A}{B}

AlgebraGeometrySlope of a line written in standard form Ax + By = C, read straight off the two coefficients without rearranging anything.

Slope-Intercept Form of a Line

y = mx + b

AlgebraGeometryEquation of a straight line from its slope and y-intercept.

Sphere Surface Area

S = 4 \pi r^{2}

GeometrySurface area of a sphere of radius r, using π ≈ 3.14159265.

Sphere Volume

V = \frac{4}{3} \pi r^{3}

GeometryVolume enclosed by a sphere of radius r, using π ≈ 3.14159265.

Spherical Cap Volume

V = \frac{\pi h^2}{3}(3R - h)

GeometryVolume of the piece a flat plane slices off a sphere, from the sphere radius R and the cap height h.

Square Area

A = s^2

GeometryArea of a square from its side length.

Square Diagonal

d = s\sqrt{2}

GeometryDiagonal of a square from its side length.

Square Perimeter

P = 4s

GeometryPerimeter of a square as four times its side length.

Square Pyramid Slant Height

l = \sqrt{h^2 + \left(\tfrac{s}{2}\right)^2}

GeometrySlant height of a square pyramid — apex down the middle of a face — from the vertical height and the base side.

Square Pyramid Surface Area

A = s^2 + 2sl

GeometryTotal surface area of a square pyramid: the square base plus four identical triangular faces, from the base side and the slant height.

Stair Stringer Length

L = \sqrt{H^2 + R_t^2}

Trades & ConstructionGeometryThe straight-line length of the stringer board, the hypotenuse of the total rise and the total run.

Stockpile Volume (Truncated Pyramid)

V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)

Civil & SurveyingGeometryVolume of a flat-topped stockpile or borrow pit from its base area, top area and height; the answer is reported in cubic yards.

Torus Surface Area

A = 4\pi^2 R r

GeometrySurface area of a ring torus: the tube's circumference swept once around the ring, by Pappus's centroid theorem.

Torus Volume

V = 2\pi^2 R r^2

GeometryVolume of a ring torus from the centre-to-tube radius R and the tube radius r, by Pappus's centroid theorem.

Trapezoid Area

A = \frac{a + b}{2} \cdot h

GeometryArea of a trapezoid as the average of its two parallel sides times the perpendicular height.

Traverse Closure Error

E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}

Civil & SurveyingGeometryLinear misclosure of a closed traverse, combining the residual sums of the latitudes and departures as a right triangle.

Triangle Area (Two Sides and Included Angle)

A = \tfrac{1}{2}\,ab\sin C

TrigonometryGeometryComputes a triangle's area from two sides and the angle between them, with no height needed.

Triangle Area from Two Vectors

A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the triangle formed by two plane vectors from a shared vertex, using only their components — the shoelace formula.

Triangle Perimeter

P = a + b + c

GeometryPerimeter of a triangle as the sum of its three sides.

Triangular Prism Volume

V = \tfrac{1}{2} b h_t L

GeometryVolume of a triangular prism — the triangular end area times the prism length, from the triangle's base and height.

Vertex x-Coordinate of a Parabola

h = -\frac{b}{2a}

AlgebraGeometryThe x-coordinate of a parabola's vertex, sitting midway between the two roots and marking the axis of symmetry.

Vertex y-Coordinate of a Parabola

k = c - \frac{b^2}{4a}

AlgebraGeometryThe y-coordinate of a parabola's vertex — its minimum value when a is positive, and its maximum when a is negative.

x-Intercept of a Line

x_{\text{int}} = -\frac{b}{m}

AlgebraGeometryWhere a line in slope-intercept form crosses the x-axis, found by setting y to zero and solving the remaining equation for x.