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Katı cisimlerin hacim ve yüzey alanı hesaplayıcısı

Silindir, koni, küre, piramit, prizma, torus ve elipsoit dahil 14 cismin hacmini, yüzey alanını, eğik yüksekliğini ve litre cinsinden kapasitesini hesaplayın.

Güncellendi Doğrulanan örnekler: 13

Dene
Hacim
cm³
Hacim: 282.7433 cm³
En fazla ondalık basamak: 4; En yakına; eşit uzaklıkta sıfırdan uzağa
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)

Drag to rotate
Nasıl hesaplanır S

  1. Hacim

    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.

Katı cisimlerin hacim ve yüzey alanı hesaplayıcısı hakkında

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.

Çözümlü örnekler

Cylinder r = 3 cm, h = 10 cm

Solid
Cylinder
Yarıçap
3 cm
Yükseklik
10 cm
Sonuç birimi
Santimetre (cm)
Hacim
282.743339 cm³
Total surface area
245.044227 cm²
Lateral (curved) surface area
188.495559 cm²
Capacity
0.282743 L

Doğrulama kaynağı: Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L

Box 6 × 4 × 10 m

Solid
Cuboid (box)
Uzunluk
6 m
Genişlik
4 m
Yükseklik
10 m
Sonuç birimi
Metre (m)
Hacim
240 m³
Total surface area
248 m²
Space diagonal
12.328828 m
Capacity
240,000 L

Doğrulama kaynağı: 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
Yarıçap
3 cm
Yükseklik
4 cm
Sonuç birimi
Santimetre (cm)
Slant height
5 cm
Hacim
37.699112 cm³
Lateral (curved) surface area
47.12389 cm²
Total surface area
75.398224 cm²

Doğrulama kaynağı: 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
Yükseklik
4 cm
Sonuç birimi
Santimetre (cm)
Slant height
4.472136 cm
Hacim
205.25072 cm³
Total surface area
219.211186 cm²

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

Sorular

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.

“Katı cisimlerin hacim ve yüzey alanı hesaplayıcısı” ne kadar doğru sonuç verir?

Doğruluk, girdilerinize ve yöntemin varsayımlarına bağlıdır. Ondalık aritmetik 50 anlamlı basamak kullanır; ancak tahminler, sayısal yöntemler ve kaynak veriler daha az hassas olabilir. Gösterilen değerin yuvarlanması bu sınırları ortadan kaldırmaz. Bağımsız kaynaklarla doğrulanan çözümlü örnek sayısı: 13. Örneğin “Cylinder r = 3 cm, h = 10 cm”, Python 3.8 math: 90*pi, 78*pi, 60*pi; 282.743… cm³ ÷ 1000 = L ile karşılaştırılarak doğrulanır.

Yöntemin kaynağı nedir?

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.

Bu hesaplayıcı hakkında

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}

Kaynaklar

  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

Kaynaklarla doğrulandı

Bu hesaplayıcı, yanıtları bağımsız kaynaklardan alınan 13 çözümlü örnek içerir. Bunlar test paketinde çalıştırılır; burada da çalıştırabilirsiniz.

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