In this section we calculate the volume and also surface area the 3-D shapes such together *cubes*, *cuboids*, *prisms* and also *cylinders*.

You are watching: Surface area of three dimensional figures

Cube | Volume = x³Surface area = 6x² | |

Cuboid | Volume = xyzSurface area = 2xy + 2xz + 2yz | |

Cylinder | Volume = π r²hArea of curved surface = 2π rhArea that each finish = π r²Total surface ar area = 2π rh + 2π r² | |

Prism | A prism has a uniform cross-sectionVolume = area of cross ar × size = A l |

Calculate the *volume* the the cuboid shown.

Calculate the *surface area* that the cuboid shown.

## Example 2

Calculate the *volume* and also total *surface area* of the cylinder shown.

Volume | = | π r²h = π × 4² × 6 = 96 π |

= | 301.5928947 cm³ | |

= | 302 cm³ (to 3 far-ranging figures) |

Area of curved surface | = | 2π rh = 2 × π × 4 × 6 |

= | 48π | |

= | 150.7964474 cm² |

Area of each end | = | π r² = π × 4² |

= | 16π | |

= | 50.26548246 cm² |

Total surface ar area | = | 150.7964474 + (2 × 50.26548246) |

= | 251.3274123 cm² | |

= | 251 cm² (to 3 significant figures) |

Note: native the functioning we deserve to see that the area of the curved surface ar is 48*π*, and also that the area of each finish is 16*π*. The total surface area is therefore

48π + (2 × 16π) | = 80π = 251.3274123 cm² |

= 251 cm² (to 3 far-reaching figures) |

concern 2

Giving her answers exactly to 3 far-ranging figures, calculation the *volume* and *total surface area* of every of the following cylinders:

inquiry 5

The diagram shows a wood block that has had a hole drilled in it. The diameter of the feet is 2 cm.Calculate the *volume* that this solid, giving your answer correct to 2 decimal places.

Volume = block – hole = 4 × 6 × 6 – 1² ×

*π*× 6 = 144 – 6

*π*= 125.15 (to 2 d.p.)

inquiry 6

A concrete beam is to remainder on two concrete pillars. The beam is a cuboid v sides of length 0.5m, 3m and 0.4m.The pillars have diameter 0.4m and also height 2m.Calculate the *total volume* the concrete needed to do the beam and the pillars. Round her answer to a judicious level that accuracy.

question 7

The diagram reflects the cross-section of a pipeline of length 50 cm.The inside diameter that the pipeline is 20 cm and also the external diameter is 30 cm.

Calculate the *volume* of metal needed to do the pipe. Round your answer to a judicious level of accuracy.

Calculate the *total surface area* the the pipe, including the within surface.Round your answer come 3 significant figures.

Total surface area | = 2 × (15² – 10²) × π + 30π × 50 + 20π × 50 |

= 250π + 1500π + 1000π = 2750π | |

= 8639.379797 cm² = 8640 cm² (to 3 s.f.) |

concern 8

The diagram reflects a prism.The cross-section of the prism is composed of a rectangle and a semicircle.

concern 10

A cylinder has a diameter that 12 cm and a curved surface ar area of 132*π* or 415 cm² (to 3 far-ranging figures).

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These cuboids room made from small cubes. Write exactly how many little cubes there space in every cuboid. The first is done for you.

(i) | 12 tiny cubes | |

(ii) | little cubes | |

(iii) | tiny cubes | |

(iv) | tiny cubes |

Area the cross-section = × basic × height = × 6 × 8 = 24 cm²