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Volume e área da superfície de sólidos

Volume, área da superfície, altura inclinada e capacidade em litros de 14 sólidos: cilindros, cones, esferas, pirâmides, prismas, toros e elipsoides.

Atualizado Exemplos verificados: 13

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Volume
cm³
Volume: 282.7433 cm³
Máximo de casas decimais: 4; Ao mais próximo; empates afastando-se de zero
Total surface area
245.0442cm²
Lateral (curved) surface area
188.4956cm²
Base area
28.2743cm²
Capacity
0.282743L

The cylinder (r = 3 cm, h = 10 cm) holds 282.7433 cm³ — 0.2827 litres — and its outside surface is 245.0442 cm².

Cylinder (r = 3 cm, h = 10 cm)

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Como é calculado S

  1. Volume

    V=πr2h=π×32×10=282.743339 cm3V = \pi r^2 h = \pi \times 3^2 \times 10 = 282.743339\,\text{cm}^{3}

  2. Curved surface

    Slateral=2πrh=188.495559 cm2S_\text{lateral} = 2\pi r h = 188.495559\,\text{cm}^{2}

  3. Total surface area (with both ends)

    S=2πr(r+h)=245.044227 cm2S = 2\pi r(r + h) = 245.044227\,\text{cm}^{2}

  4. Capacity

    282.743339 cm3=0.28274334 L282.743339\,\text{cm}^{3} = 0.28274334\,\text{L}

    1 L = 1000 cm³ = 0.001 m³.

Sobre Volume e área da superfície de sólidos

Choose one of 14 solids and enter its dimensions to get the volume and total surface area, plus the curved (lateral) surface, base area, slant height and space diagonal where the solid has them. The volume is also given as a capacity in litres, using 1 L = 1,000 cm³. The formulas are the standard ones, for example V = πr²h for a cylinder, V = ⅓πh(R² + Rr + r²) for a frustum and V = 2π²Rr² for a torus.

The default cylinder, 3 cm in radius and 10 cm tall, holds 282.743 cm³ (0.283 L) and has 245.044 cm² of surface, of which 188.496 cm² is the curved side. Volume sizes tanks, containers and concrete pours; surface area sizes paint, insulation and sheet material.

Every result is exact except the surface of a general ellipsoid, which uses Knud Thomsen's approximation, within about 1.06%. The 3D model is drawn to the proportions entered; drag it to rotate.

Exemplos resolvidos

Cylinder r = 3 cm, h = 10 cm

Solid
Cylinder
Raio
3 cm
Altura
10 cm
Unidade dos resultados
Centímetros (cm)
Volume
282.743339 cm³
Total surface area
245.044227 cm²
Lateral (curved) surface area
188.495559 cm²
Capacity
0.282743 L

Fonte de verificação: Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L

Box 6 × 4 × 10 m

Solid
Cuboid (box)
Comprimento
6 m
Largura
4 m
Altura
10 m
Unidade dos resultados
Metros (m)
Volume
240 m³
Total surface area
248 m²
Space diagonal
12.328828 m
Capacity
240,000 L

Fonte de verificação: 6·4·10; 2(24 + 60 + 40); Python 3.8 math.sqrt(152); 1 m³ = 1000 L

Cone r = 3, h = 4 (slant 5)

Solid
Cone
Raio
3 cm
Altura
4 cm
Unidade dos resultados
Centímetros (cm)
Slant height
5 cm
Volume
37.699112 cm³
Lateral (curved) surface area
47.12389 cm²
Total surface area
75.398224 cm²

Fonte de verificação: 3-4-5 slant; Python 3.8 math: 12*pi, 15*pi, 24*pi

Frustum R = 5, r = 3, h = 4

Solid
Frustum of a cone
Bottom radius (R)
5 cm
Top radius (r)
3 cm
Altura
4 cm
Unidade dos resultados
Centímetros (cm)
Slant height
4.472136 cm
Volume
205.25072 cm³
Total surface area
219.211186 cm²

Fonte de verificação: Python 3.8 math: sqrt(20), pi*4*(25+15+9)/3, pi*8*sqrt(20) + 34*pi

Perguntas

How do you calculate the volume of a cylinder?

Multiply the area of the circular end by the height: V = πr²h. A radius of 3 cm and a height of 10 cm give 90π ≈ 282.74 cm³, or 0.283 L. Use the radius, not the diameter: a tank 1 m across and 1.5 m tall has r = 0.5 m and holds π × 0.25 × 1.5 ≈ 1.178 m³, which is 1,178 L.

How do you find the surface area of a cylinder?

Add the curved side, 2πrh, to the two circular ends, 2πr²: S = 2πr(r + h). For r = 3 cm and h = 10 cm the curved side is 60π ≈ 188.50 cm² and the total is 78π ≈ 245.04 cm². An open-topped tank has only one end, so subtract πr², here 28.27 cm², to get 216.77 cm².

How many litres are in a cubic metre?

1,000. A litre is exactly one cubic decimetre, 1,000 cm³, a definition fixed by the General Conference on Weights and Measures (CGPM) in 1964. Divide cubic centimetres by 1,000 to get litres. In US units, one gallon is exactly 231 in³, or 3.785411784 L, and one cubic foot is 28.316846592 L.

What is the formula for the volume of a sphere?

V = ⁴⁄₃πr³, which is two thirds of the cylinder that just encloses the sphere, as Archimedes showed. A sphere of radius 3 cm holds 36π ≈ 113.10 cm³, and a ball 1 m across holds 0.5236 m³, or 523.6 L. Volume grows with the cube of the radius, so doubling the diameter multiplies the volume by 8.

How do you calculate the volume of a cone or a pyramid?

Take one third of the base area times the height: V = ⅓Bh. A cone of radius 3 and height 4 has V = ⅓π × 9 × 4 = 12π ≈ 37.70, and a square pyramid with a 6 × 6 base and height 4 has V = ⅓ × 36 × 4 = 48. A cone holds exactly one third of the cylinder with the same base and height.

Qual é a precisão de “Volume e área da superfície de sólidos”?

A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 13. Por exemplo, “Cylinder r = 3 cm, h = 10 cm” é verificado com Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L.

De onde vem o método?

Zwillinger, D. (ed.) CRC Standard Mathematical Tables and Formulas, 33rd ed., §4.6 (solids); Weisstein, E. W. “Torus”, “Spherical Cap”, “Ellipsoid” — MathWorld; Thomsen, K. — ellipsoid surface approximation, as quoted in “Ellipsoid § Surface area”, Wikipedia.

Sobre esta calculadora

Vsphere=43πr3,Vcone=13πr2hVfrustum=13πh(R2+Rr+r2),Vtorus=2π2Rr2\begin{gathered} V_\text{sphere} = \tfrac{4}{3}\pi r^3,\quad V_\text{cone} = \tfrac{1}{3}\pi r^2 h \\ V_\text{frustum} = \tfrac{1}{3}\pi h(R^2 + Rr + r^2),\quad V_\text{torus} = 2\pi^2 R r^2 \end{gathered}

Fontes

  1. Zwillinger, D. (ed.) CRC Standard Mathematical Tables and Formulas, 33rd ed., §4.6 (solids)
  2. Weisstein, E. W. “Torus”, “Spherical Cap”, “Ellipsoid” — MathWorld
  3. Thomsen, K. — ellipsoid surface approximation, as quoted in “Ellipsoid § Surface area”, Wikipedia

Verificado com as referências

Esta calculadora inclui 13 exemplos resolvidos com respostas de fontes independentes. Eles fazem parte do conjunto de testes e você também pode executá-los aqui.

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