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Home › Chart guides › Geometry Formulas Chart
The area and perimeter of flat shapes from the square to the regular polygon, the volume and surface area of solids from the cube to the sphere, angle facts, and the theorems worth knowing, such as Pythagoras.
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Area, perimeter, volume and angles, every formula checked by computation.
| Shape | Given | Area | Perimeter |
|---|---|---|---|
| Square | side s | A = s² | P = 4 s |
| Rectangle | width w, height h | A = w h | P = 2 (w + h) |
| Triangle | base b, height h | A = b h / 2 | P = a + b + c |
| Parallelogram | base b, height h | A = b h | P = 2 (a + b) |
| Trapezoid | sides a and b, height h | A = (a + b) h / 2 | P = a + b + c + d |
| Circle | radius r | A = π r² | C = 2 π r |
| Ellipse | semi-axes a and b | A = π a b | no simple formula |
| Regular n-gon | n sides of length s | A = n s² / (4 tan(π/n)) | P = n s |
There is no simple formula for the distance round an ellipse. It needs an elliptic integral, so none is printed here rather than an approximation dressed up as a rule.
| Solid | Given | Volume | Surface area |
|---|---|---|---|
| Cube | edge s | V = s³ | S = 6 s² |
| Box | l by w by h | V = l w h | S = 2 (lw + lh + wh) |
| Cylinder | radius r, height h | V = π r² h | S = 2 π r (r + h) |
| Cone | radius r, height h | V = π r² h / 3 | S = π r (r + √(r² + h²)) |
| Sphere | radius r | V = 4 π r³ / 3 | S = 4 π r² |
| Pyramid | base area B, height h | V = B h / 3 | depends on the base |
The surface area of a pyramid depends on the shape of its base and on the slant of each face, so no single formula covers it and none is given.
| Angle | Measure | Shape | Angles add to |
|---|---|---|---|
| Right angle | 90° | Triangle | 180° |
| Straight angle | 180° | Quadrilateral | 360° |
| Full turn | 360° | Pentagon | 540° |
| Angles on a line | add to 180° | Hexagon | 720° |
| Angles round a point | add to 360° | Octagon | 1080° |
| Polygon of n sides | 180 (n − 2) | Decagon | 1440° |
| Pythagoras | in a right triangle a² + b² = c², c the longest side |
| Double every length | area becomes 4 times, volume 8 times, not 2 |
| Pi | 3.14159265, any circumference divided by its diameter |
| Degrees to radians | multiply by π, divide by 180 |
| Radians to degrees | multiply by 180, divide by π |
| Regular means | every side the same and every angle the same |
| Surface area | the area of the skin; volume is what fits inside |
A dimension and its formula have to use the same units. Areas come out in square units and volumes in cubic units, which is why doubling a length multiplies a volume by eight.
Height means the perpendicular height, measured square to the base, not the length of a sloping side.
Formulas as given in the matching English Wikipedia articles.