Solid figures

Lateral area of a rectangular (right) prism

$$S_{son} = P\cdot h$$

S_lat - lateral area
P - the perimeter of the base
h - prism height

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Calculate 'S_lat'

Surface area of a rectangular (right) prism

$$S = S_{son}+2\cdot S_{pagr}$$

S - surface area
S_lat - lateral area
S_base - area of base

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Volume of a rectangular (right) prism

$$V = S_{pagr}\cdot h$$

V - volume
S_base - area of base
h - prism height

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Diagonal of a rectangular parallelepiped

$$d^{2} = a^{2}+b^{2}+c^{2}$$

d - diagonal of a rectangular parallelepiped
a, b, c - sides

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Lateral area of a rectangular parallelepiped

$$S_{son} = 2\cdot (a\cdot c+b\cdot c)$$

S_lat - lateral area
a, b, c - sides

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Calculate 'S_lat'

Surface area of a rectangular parallelepiped

$$S_{son} = 2\cdot (a\cdot b+b\cdot c+a\cdot c)$$

S - surface area
a, b, c - sides

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Calculate 'S_lat'

Volume of a rectangular parallelepiped

$$V = S_{pagr}\cdot h$$

S_base - area of base
h - height

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Volume of a rectangular parallelepiped

$$V = a\cdot b\cdot c$$

V - volume
a, b, c - sides

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Lateral area of a cube

$$S_{son} = 4\cdot a^{2}$$

S_lat - lateral area
a - side

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Calculate 'S_lat'

Surface area of a cube

$$S = 6\cdot a^{2}$$

S - surface area
a - side

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Volume of a cube

V - volume
a - side

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Lateral area of a regular pyramid

$$S_{son} = \frac{1}{2}\cdot P\cdot h_{s}$$

S_lat - lateral area
P - the perimeter of the base
h_s - apothem (slant height)

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Calculate 'S_lat'

Lateral area of a regular pyramid

$$S_{son} = \frac{S_{pagr}}{cos(\phi)}$$

S_lat - lateral area
S_base - area of base
φ - angle between the slant edge and the base

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Calculate 'S_lat'

Volume of a regular pyramid

$$V = \frac{1}{3}\cdot S_{pagr}\cdot h$$

V - volume
S_base - area of base
h - pyramid height

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Lateral area of a frustum of a regular pyramid

$$S_{son} = \frac{1}{2}\cdot (P1+P2)\cdot h_{s}$$

S_lat - lateral area
P1, P2 - perimeters of lower and upper bases
h_s - apothem (slant height)

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Calculate 'S_lat'

Lateral area of a frustum of a regular pyramid

$$S_{son} = \frac{S1-S2}{cos(\phi)}$$

S_lat - lateral area
S1, S2 - areas of lower and upper bases
φ - angle between the slant edge and the lower base

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Calculate 'S_lat'

Surface area of a frustum of a pyramid

$$S = S_{son}+S1+S2$$

S - surface area
S_lat - lateral area
S1, S2 - areas of lower and upper bases

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Volume of frustum of a pyramid

$$V = \frac{1}{3}\cdot h\cdot (S1+S2+\sqrt {S1\cdot S2})$$

V - volume
h - height of frustum of a pyramid
S1, S2 - areas of lower and upper bases

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Lateral area of a cylinder

$$S_{son} = 2\cdot \pi\cdot r\cdot h$$

S_lat - lateral area
r - radius
h - cylinder height

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Calculate 'S_lat'

Area of a base of a cylinder

$$S_{pagr} = \pi\cdot r^{2}$$

S_base - area of base
r - radius

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Calculate 'S_base'

Surface area of a cylinder

$$S = 2\cdot \pi\cdot r\cdot (r+h)$$

S - surface area
r - radius
h - cylinder height

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Volume of a cylinder

$$V = \pi\cdot r^{2}\cdot h$$

V - volume
r - radius
h - cylinder height

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Lateral area of a cone

$$S_{son} = \pi\cdot r\cdot l$$

S_lat - lateral area
r - radius
l - ruling of a cone

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Calculate 'S_lat'

Surface area of a cone

$$S = \pi\cdot r\cdot (r+l)$$

S - surface area
r - radius
l - ruling of a cone

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Lateral area of a cone (net)

$$S = \frac{\pi\cdot l^{2}\cdot \alpha}{360}$$

S_lat - lateral area
l - ruling of a cone
α - angle of a net

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Volume of a cone

$$V = \frac{1}{3}\cdot \pi\cdot r^{2}\cdot h$$

V - volume
r - radius
h - cone height

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Lateral area of a frustum of a cone

$$S_{son} = \pi\cdot (R+r)\cdot l$$

S_lat - lateral area
R - radius of lower base
r - radius of upper base
l - slant height

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Calculate 'S_lat'

Surface area of a frustum of a cone

$$S = \pi\cdot (R+r)\cdot l+\pi\cdot R^{2}+\pi\cdot r^{2}$$

S - surface area
R - radius of lower base
r - radius of upper base
l - slant height

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Volume of a frustum of a cone

$$V = \frac{1}{3}\cdot \pi\cdot h\cdot (R^{2}+r^{2}+R\cdot r)$$

V - volume
R - radius of lower base
r - radius of upper base
h - height of a frustum of a cone

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Surface area of a sphere (ball)

$$S = 4\cdot \pi\cdot R^{2}$$

S - surface area
R - radius

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Volume of a sphere (ball)

$$V = \frac{4}{3}\cdot \pi\cdot R^{3}$$

V - volume
R - radius

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Surface area of a spherical cap (segment)

$$S = 2\cdot \pi\cdot R\cdot h$$

S - surface area
R - radius of a sphere
h - height of a cap (segment)

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Volume of a spherical cap (segment)

$$V = \pi\cdot h^{2}\cdot (R-\frac{h}{3})$$

V - volume
R - radius of a sphere
h - height of a cap (segment)

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Volume of a spherical cap (segment) using radius of base

$$V = \frac{1}{6}\cdot \pi\cdot h\cdot (h^{2}+3\cdot r^{2})$$

V - volume
h - height of a cap (segment)
r - radius of a cap (segment) base

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Surface area of a spherical segment

$$S = 2\cdot \pi\cdot R\cdot h$$

S - surface area
R - radius of a sphere
h - height of a spherical segment

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Volume of a spherical segment

$$V = \frac{1}{6}\cdot \pi\cdot h^{3}+\frac{1}{2}\cdot \pi\cdot (r1^{2}+r2^{2})\cdot h$$

V - volume
h - height of a spherical segment
r1, r2 - radii of bases of a spherical segment

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Surface area of a spherical sector (cone)

$$S = \pi\cdot R\cdot (2\cdot h+r)$$

S - surface area
R - radius of a sphere
h - height of a spherical cap belonging to spherical sector
r - radius of a base of a spherical cap belonging to spherical sector

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Volume of a spherical sector (cone)

$$V = \frac{2}{3}\cdot \pi\cdot R^{2}\cdot h$$

V - volume
R - radius of a sphere
h - height of a spherical cap belonging to spherical sector

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