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Geometry formula solvers

Pythagorean Theorem

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

GeometryRelates the three sides of a right triangle.

Area of a Circle

A=πr2A = \pi r^{2}

GeometryArea enclosed by a circle of radius r.

Circumference of a Circle

C=2πrC = 2 \pi r

GeometryDistance around a circle of radius r.

Area of a Triangle

A=12bhA = \tfrac{1}{2} b h

GeometryArea from base and perpendicular height.

Rectangle Area

A=l⋅wA = l \cdot w

GeometryArea of a rectangle as length times width.

Parallelogram Area

A=b⋅hA = b \cdot h

GeometryArea of a parallelogram from its base and perpendicular height.

Trapezoid Area

A=a+b2⋅hA = \frac{a + b}{2} \cdot h

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

Ellipse Area

A=πabA = \pi a b

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

Circular Sector Area

A=12r2θA = \frac{1}{2} r^{2} \theta

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

Arc Length

s=rθs = r \theta

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

Distance Formula (2D)

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

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

Sphere Volume

V=43πr3V = \frac{4}{3} \pi r^{3}

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

Sphere Surface Area

S=4πr2S = 4 \pi r^{2}

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

Cylinder Volume

V=πr2hV = \pi r^{2} h

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

Cylinder Surface Area

S=2πr2+2πrhS = 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.

Cone Volume

V=13πr2hV = \frac{1}{3} \pi r^{2} h

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

Pyramid Volume

V=13BhV = \frac{1}{3} B h

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

Rectangular Prism Volume

V=l⋅w⋅hV = l \cdot w \cdot h

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

Cube Volume

V=a3V = a^{3}

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

Cube Surface Area

S=6a2S = 6 a^{2}

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

Cube Space Diagonal

d=a3d = a \sqrt{3}

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

Right-Triangle Sine Ratio (SOH)

sin⁡θ=oh\sin\theta = \frac{o}{h}

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

Right-Triangle Cosine Ratio (CAH)

cos⁡θ=ah\cos\theta = \frac{a}{h}

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

Right-Triangle Tangent Ratio (TOA)

tan⁡θ=oa\tan\theta = \frac{o}{a}

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

Law of Cosines

c2=a2+b2−2abcos⁡Cc^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

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

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

Triangle Area (Two Sides and Included Angle)

A=12 absin⁡CA = \tfrac{1}{2}\,ab\sin C

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

Slope Between Two Points

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

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

Slope-Intercept Form of a Line

y=mx+by = mx + b

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

Vertex x-Coordinate of a Parabola

h=−b2ah = -\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−b24ak = 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.

Midpoint Formula

xm=x1+x22x_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.

Circle Radius from Centre and a Point

r=(x−h)2+(y−k)2r = \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.

Distance from a Point to a Line

d=∣Ax0+By0+C∣A2+B2d = \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.

Point-Slope Form of a Line

y=y1+m(x−x1)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.

Slope from Standard Form of a Line

m=−ABm = -\frac{A}{B}

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

x-Intercept of a Line

xint=−bmx_{\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.

Perpendicular Slope Relation

m2=−1m1m_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.

Geometric Mean of Two Numbers

G=abG = \sqrt{ab}

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

Magnitude of a 2D Vector

∣v⃗∣=vx2+vy2|\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

∣v⃗∣=vx2+vy2+vz2|\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.

Cross Product z-Component of Two 2D Vectors

(a⃗×b⃗)z=axby−aybx(\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.

Parallelogram Area from Two Vectors

A=∣axby−aybx∣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.

Triangle Area from Two Vectors

A=12∣axby−aybx∣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.

Scalar Triple Product (Parallelepiped Volume)

V=∣a⃗⋅(b⃗×c⃗)∣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.

Area Moment of Inertia — Rectangle

I=bh312I = \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=πd464I = \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.

Elastic Section Modulus (S = I/c)

S=IcS = \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.

Polar Moment of Inertia — Solid Shaft

J=πd432J = \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.

Radius of Gyration (r = √(I/A))

r=IAr = \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.

Partially Filled Horizontal Cylindrical Tank

V=L[r2cos⁡−1 ⁣(r−hr)−(r−h)2rh−h2]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=100 ΔhLG = \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.

Elevation from Grade and Distance

E2=E1+G L100E_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.

Earthwork Volume by Average End Area

V=L (A1+A2)2V = \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=L (A1+4Am+A2)6V = \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.

Stockpile Volume (Truncated Pyramid)

V=h3(A1+A2+A1A2)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.

Concrete Volume with Waste Allowance

V=L W T(1+w100)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.

Radius from Degree of Curve (Arc Definition)

R=5729.578DR = \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.

Horizontal Curve Length from Degree of Curve

L=100 ΔDL = \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.

Elevation on a Parabolic Vertical Curve

E=EBVC+g1x100+A x2200 LE = 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.

Crest Vertical Curve Length for Sight Distance

L=A S2200(h1+h2)2L = \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.

Traverse Closure Error

Ec=(ΣLat)2+(ΣDep)2E_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.

Area of a Three-Sided Parcel by Coordinates

A=12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]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.