Geometry formula solvers

Categories — 73 of them, 1787 solvers in all

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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.

Centroid of a Composite Area (ȳ = ΣAȳ / ΣA)

\bar{y} = \frac{A_1 y_1 + A_2 y_2}{A_1 + A_2}

Strength of MaterialsCivil & SurveyingGeometryCentroid of an area built from two parts: the area-weighted average of the parts' own centroids, measured from any convenient datum. This is the step that must come before the parallel axis theorem, because it fixes the axis everything else is measured from.

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.

Related Rates: Inflating Sphere

\frac{dV}{dt} = 4\pi r^{2}\,\frac{dr}{dt}

CalculusGeometryThe chain rule on a sphere: how fast its volume grows from how fast its radius does, and back the other way. The balloon, the raindrop and the spreading oil drop are all this one relation.

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 Form of a Quadratic

y = a(x - h)^2 + k

AlgebraGeometryA parabola written around its own vertex (h, k), so the turning point, the vertical stretch and which way it opens are all readable without any algebra.

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.