Geometry formulas

The sides are x, x√3, and 2x.

The hypotenuse is leg × √2.

cos θ = (u·v) / (|u||v|).

Arc length is the fraction of the circumference.

Multiply length by width.

Area of the disk.

Circumference of the rim.

Diameter.

From area.

Central angle in radians is arc length divided by radius.

The same opening in degrees.

Average the x-coordinates of the three vertices.

Average the y-coordinates of the three vertices.

G also sits two-thirds of the way along each median, from the vertex toward the opposite midpoint.

The chord subtended by θ is 2 r sin(θ/2).

Compare sides for equality; compare a²+b² with c² for the angle type.

Each minute moves the minute hand 6° and the hour hand 0.5°.

Complementary angles sum to 90°.

Compute slant height, then surface and volume.

Add or subtract full turns until the angle lies in that interval.

V = a³; S = 6a²; face diagonal a√2; space diagonal a√3.

Enter Radius (r) and Height (h) for the Cylinder Calculator; use finite values in the indicated domain. The calculator reports Surface and Volume: Cylinder volume πr²h and total surface 2πr(r+h). Radius and Height must share a length unit; area and volume use its square and cube.

Apply the Pythagorean theorem to the coordinate differences.

Enter Semi-axis (a) and Semi-axis (b) for the Ellipse Area Calculator; use finite values in the indicated domain. The calculator reports Area: Ellipse area is π a b. Semi-major a and Semi-minor b are radii, not full axis lengths.

Volume is (4/3)πabc.

Radius is the distance from the center (h, k) to the given point (x, y).

Standard form of the circle equation.

Height is (√3/2)s; area is (√3/4)s².

Convex polyhedra satisfy χ = 2.

The aspect ratio is φ = (1+√5)/2.

Volume is (2/3)π r³; total surface is 3π r².

s is the semiperimeter; area is √[s(s−a)(s−b)(s−c)].

Height to the base is √(a² − (b/2)²).

Kite diagonals are perpendicular; area is half the product.

Area is base times perpendicular height.

Mode 0: rectangle 2(ℓ+w). Mode 1: regular n-gon n·s.

Add the three side lengths.

Enter Number of sides (n) for the Polygon Angles Calculator; use finite values in the indicated domain. The calculator reports Interior sum (°), Each interior (°), and Each exterior (°): Interior sum (n−2)180°; exterior 360°/n. Number of sides must be a whole number of at least 3.

Pyramid volume is one-third base area times height.

A triple satisfies a² + b² = c² after sorting so c is largest.

Take the square root of a² + b².

Close the loop and apply the shoelace formula.

Reduce to [0, 360), then take the acute distance to the x-axis.

Area uses side length and apothem ap = s / (2 tan(π/n)).

The diagonals of a rhombus are perpendicular bisectors; area is half their product.

Sector area is the matching fraction of π r².

Segment area is the sector minus the triangular portion.

Area of the half-disk is half of π r².

Perimeter is the curved arc plus the diameter.

Corresponding sides of similar triangles are proportional.

The slant is the hypotenuse of r and h.

Enter Radius (r) for the Sphere Calculator; use finite values in the indicated domain. The calculator reports Surface area and Volume: Surface 4πr² and volume 4/3 πr³. Radius must be nonnegative; Surface area uses squared units and Volume uses cubed units.

Supplementary angles sum to 180°.

Sum the three unique face areas and double.

Pick the solid. A sphere uses radius r, a cube uses side a, and a rectangular box uses length, width, and height. Keep every length in the same unit.

Surface area S is the outside. Sphere 4πr², cube 6a², box twice the three pairwise faces.

Volume V is the interior. Sphere (4/3)πr³, cube a³, box ℓwh.

Divide. The surface area to volume ratio is S/V. The unit is 1 over length: 1/cm if you typed centimetres. Closed forms 3/r and 6/a skip the long way for a sphere and a cube.

If the point is at distance d from the center, tangent length is √(d² − r²).

V = a³ √2 / 12.

V = 2π² R r²; S = 4π² R r, with R > r > 0.

Average the parallel sides and multiply by height.

Interior angles of a Euclidean triangle sum to 180°.

Base-height is (1/2)bh; otherwise Heron’s formula from three sides.

h_a = 2A/a. If area is 0, A = (1/2)bc sin α.

Each side must be shorter than the sum of the other two.

Volume is (Heron area of the triangular face) times length.

Multiply length, width, and height.