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Geometry formulas


Symbols

name symbol
area $$A$$
lateral area $$A_{L}$$
right section area $$A_{R}$$
total area $$A_{T}$$
apothem, catheti, height, side $$a$$
base area $$B$$
base perimeter $$B_{P}$$
base, catheti, side $$b$$
circunference $$C$$
chord length, hypotenuse, side $$c$$
diagonal, diameter $$D$$
diagonal, side $$d$$
edge $$E$$
lateral edge $$E_{L}$$
face $$F$$
generatrix $$g$$
height, segment height $$h$$
arc length $$L$$
line $$l$$
perimeter $$P$$
radius $$r$$
semiperimeter $$S$$
sides $$s$$
vertex, volume $$V$$
angle, inclination angle $$\theta$$

Polygons

name notation perimeter area
Triangle $$a,b,c:\ \mathsf{sides}\\b:\ \mathsf{base}\\h:\ \mathsf{height}\\S:\ \mathsf{semiperimeter}\\P:\ \mathsf{perimeter}$$ $$P = a + b + c\\S=\frac{a + b + c}{2}$$ $$A = \frac{bh}{2}\\A = \sqrt{S \left(S - a\right)\left(S - b\right)\left(S - c\right)}$$
Equilateral triangle $$a:\ \mathsf{side}$$ $$P = 3a$$ $$A = \frac{\sqrt{3}}{4}a^{2}$$
Right triangle $$a,b:\ \mathsf{legs, cathetus}\\c:\ \mathsf{hypotenuse}$$ $$P = a + b + c$$ $$A = \frac{ab}{2}$$
Square $$a:\ \mathsf{side}$$ $$P = 4a$$ $$A = a^{2}$$
Rectangle $$b:\ \mathsf{base}\\h:\ \mathsf{height}$$ $$P = 2\left(b + h\right)$$ $$A = bh$$
Rhombus $$a:\ \mathsf{side}\\D:\ \mathsf{long\ diagonal}\\d:\ \mathsf{short\ diagonal}$$ $$P = 4a$$ $$A = \frac{Dd}{2}$$
Rhomboid $$a:\ \mathsf{side}\\b:\ \mathsf{base, side}\\h:\ \mathsf{height}$$ $$P = 2\left(a + b\right)$$ $$A = bh$$
Trapezoid $$a,b,c,d:\ \mathsf{sides}\\l:\ \mathsf{sides}\\h:\ \mathsf{sides\ number}$$ $$P = a+b+c+d$$ $$A = \left(\frac{a+c}{2}\right)h$$
Convex quadrilateral $$a:\ \mathsf{apothem}\\d_{1},d_{2}:\ \mathsf{diagonals}$$ $$P = a+b+c+d$$ $$A = \frac{1}{4}\sqrt{4\left( d_{1}d_{2} \right)\left( a^{2} - b^{2} + c^{2} - d^{2} \right)}$$
Regular polygons $$a:\ \mathsf{apothem}\\n:\ \mathsf{number\ of\ sides}\\s:\ \mathsf{side}$$ $$P = ns$$ $$A = \frac{Pa}{2}$$
Circle $$D:\ \mathsf{diameter}\\r:\ \mathsf{radius}\\ \pi:\ \mathsf{3.1416}$$ $$P = \pi D \\ P = 2 \pi r$$ $$A = \frac{\pi D^{2}}{4} \\ A = \pi r^{2}$$
Annulus $$D:\ \mathsf{outer\ diameter}\\d:\ \mathsf{inner\ diameter}\\R:\ \mathsf{outer\ radius}\\r:\ \mathsf{inner\ radius}\\ \pi:\ \mathsf{3.1416}$$ $$P_{o} = \pi D = 2\pi R \\ P_{i} = \pi d = 2\pi r \\ P_{t} = \pi \left( D + d \right) \\ \\ P_{t} = 2\pi \left( R + r \right)$$ $$A = \frac{\pi}{4}\left(D^{2} - d^{2}\right) \\ A = \pi \left(R^{2} - r^{2}\right)$$
Circular sector $$L:\ \mathsf{arc\ length}\\r:\ \mathsf{radius}\\ \theta:\ \mathsf{angle}\\ \pi:\ \mathsf{3.1416}$$ $$L = r \theta \\ P = L + 2r$$ $$A = \frac{r^{2}\theta}{2} = \frac{\theta\pi r^{2}}{360^{\circ}} $$
Circular segment $$c:\ \mathsf{chord\ length}\\ h:\ \mathsf{segment\ height}\\ r:\ \mathsf{radius}\\ \pi:\ \mathsf{3.1416}\\ \theta:\ \mathsf{angle}$$ $$P = r \theta + c$$ $$A = \frac{ r^{2}\theta}{2} - \frac{c \left( r - h \right)}{2} \\ A = \frac{ \pi r^{2}\theta}{360^{\circ}} - \frac{c \left( r - h \right)}{2} $$
Ellipse $$a:\ \mathsf{semi-major\ axis}\\ b:\ \mathsf{semi-minor\ axis}$$ $$P = \pi \left( a + b \right)$$ $$A = \pi ab $$

Polyhedra

Regular polyhedra

Euler's theorem for regular polyhedra: \( C + V = A + 2 \).

name notation perimeter area volume
Tetrahedron $$E:\ \mathsf{edge}$$ $$P = 6E$$ $$A = 4\sqrt{3}E$$ $$V = \frac{\sqrt{2}}{12}E$$
Cube $$E:\ \mathsf{edge}$$ $$P = 12E$$ $$A = 6E^{2}$$ $$V = A^{3}$$
Octahedron $$E:\ \mathsf{edge}$$ $$P = 12E$$ $$A = 2\sqrt{3}E^{2}$$ $$V = \frac{\sqrt{2}}{3}E^{3}$$
Dodecahedron $$E:\ \mathsf{edge}$$ $$P = 30E$$ $$A = 15 \sqrt{\frac{5 + 2 \sqrt{5}}{5}} E^{2}$$ $$V = \frac{15 + 7 \sqrt{5}}{4}E^{3}$$
Icosahedron $$E:\ \mathsf{edge}$$ $$P = 30E$$ $$A = 5\sqrt{3}E^{2}$$ $$V = \frac{5}{6} \sqrt{\frac{7 + 3\sqrt{5}}{2}}E^{2}$$

Irregular polyhedra

name notation perimeter area volume
Oblique prism $$A_{L}:\ \mathsf{lateral\ area}\\A_{R}:\ \mathsf{right\ section\ area}\\A_{T}:\ \mathsf{total\ area}\\B:\ \mathsf{base\ area}\\B_{P}:\ \mathsf{base\ perimeter}\\E_{L}:\ \mathsf{lateral\ edge}\\h:\ \mathsf{height}\\P_{R}:\ \mathsf{right\ section\ perimeter}\\ \theta:\ \mathsf{inclination\ angle}$$ $$P = 2B_{P} + E_{L}(F - 2)\\P = 2B_{P} + \frac{h}{\sin\theta}(F - 2)$$ $$A_{L} = P_{R}E_{L}\\A_{T} = A_{L} + 2B\\A_{T} = P_{R}E_{L} + 2B$$ $$V = Bh$$
Right prism $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\B:\ \mathsf{base\ area}\\B_{P}:\ \mathsf{base\ perimeter}\\h:\ \mathsf{height}$$ $$P = 2B_{P} + h(F - 2)$$ $$A_{L} = B_{P}h\\A_{T} = A_{L} + 2B\\A_{T} = B_{P}h + 2B$$ $$V = Bh$$
Parallelepiped $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\a:\ \mathsf{length}\\b:\ \mathsf{width}\\c:\ \mathsf{height}$$ $$P = 4 \left( a + b+ c \right)$$ $$A_{L} = 2 \left( a + b \right) c\\A_{T} = 2 \left( a + b \right) c + 2ab$$ $$V = abc$$
Pyramid $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\a:\ \mathsf{apothem}\\B:\ \mathsf{base\ area}\\B_{P}:\ \mathsf{base\ perimeter}\\h:\ \mathsf{height}$$ $$$$ $$A_{L} = \frac{1}{2}B_{P}a\\A_{T} = \frac{1}{2}B_{P}a + B$$ $$V = \frac{1}{3}Bh$$
Truncated pyramid $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\a:\ \mathsf{apothem}\\B:\ \mathsf{base\ area\ (bottom)}\\B^{\prime}:\ \mathsf{base\ area (top)}\\B_{P}:\ \mathsf{base\ perimeter (bottom)}\\B^{\prime}_{P}:\ \mathsf{base\ perimeter (top)}\\h:\ \mathsf{height}$$ $$$$ $$A_{L} = \left( \frac{B_{P} + B_{P}^{\prime}}{2} \right) a \\A_{T} = \left( \frac{B_{P} + B_{P}^{\prime}}{2} \right) a + B + B^{\prime}$$ $$V = h\left( B + B^{\prime} + \sqrt{BB^{\prime}} \right)$$
Oblique cylinder $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\B:\ \mathsf{base\ area}\\g:\ \mathsf{generatrix}\\h:\ \mathsf{height}\\P_{R}:\ \mathsf{right\ section\ perimeter}$$ $$$$ $$A_{L} = P_{R} g \\A_{T} = P_{R} g + 2B$$ $$V = Bh$$
Right Cylinder $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\h:\ \mathsf{height}\\r:\ \mathsf{base\ radius}$$ $$$$ $$A_{L} = 2 \pi rh \\A_{T} = 2 \pi rh + 2 \pi r^{2}$$ $$V = \pi r^{2} h$$
Cone $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\g:\ \mathsf{generatrix}\\h:\ \mathsf{height}\\r:\ \mathsf{base\ radius}$$ $$$$ $$A_{L} = 2 \pi rg \\A_{T} = 2 \pi rg + 2 \pi r^{2}$$ $$V = \frac{1}{3}\pi r^{2} h$$
Cone $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\g:\ \mathsf{generatrix}\\h:\ \mathsf{height}\\r:\ \mathsf{base\ radius(bottom)}\\r:\ \mathsf{base\ radius(top)}$$ $$$$ $$A_{L} = 2 \pi g \left(r + r^{\prime}\right) \\A_{T} = 2 \pi g \left(r + r^{\prime}\right) + \pi \left(r^{2} + r^{\prime 2}\right)$$ $$V = \frac{1}{3}\pi h\left( r^{2} + r^{\prime 2} + rr^{\prime} \right)$$
Sphere $$A:\ \mathsf{area}\\r:\ \mathsf{sphere\ radius}$$ $$$$ $$A = 4 \pi r^{2}$$ $$V = \frac{4}{3}\pi r^{3}$$
Spherical cap $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\h:\ \mathsf{height}\\R:\ \mathsf{sphere\ radius}\\r:\ \mathsf{circle\ radius}$$ $$$$ $$A = 2 \pi Rh \\ A = \pi \left( r^{2} + h^{2} \right) \\ A_{T} = A + \pi r^{2}$$ $$V = \frac{1}{3} \pi h^{2} \left( 3R - h \right) \\ V = \frac{1}{2} \pi h \left( r + \frac{h^{2}}{3} \right)$$
Spheric zone $$A_{L}:\ \mathsf{lateral\ area}\\A_{T}:\ \mathsf{total\ area}\\h:\ \mathsf{height}\\R:\ \mathsf{sphere\ radius}\\r:\ \mathsf{lower\ base\ radius}\\r^{\prime}:\ \mathsf{upper\ base\ radius}$$ $$$$ $$A_{L} = 2 \pi Rh \\ A_{T} = 2 \pi Rh + \pi \left( r^{2} + r^{\prime 2} \right)$$ $$V = \frac{1}{2} \pi h \left( r^{2} + r^{\prime 2} + \frac{h^{2}}{3} \right)$$
spherical lune, biangle $$A:\ \mathsf{area}\\AB:\ \mathsf{maximum\ arc}\\R:\ \mathsf{sphere\ radius}$$ $$$$ $$A = 2R AB$$ $$$$