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2D shapes

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Area of a rectangle Substituting into formulaeThis topic is relevant for:

Here we will learn about cuboids, including nets of cuboids and volume and surface area of cuboids.

There are also volume and surface area of a cuboid* *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

A** cuboid **(or a rectangular cuboid)is a polyhedron ( 3D shape) with 6 rectangular faces. They can also be called rectangular prisms.

Cubes are a special type of cuboid where the length, width and height are all equal.

Examples of cuboids:

A **face **of a 3 D shape is a flat surface.

Cuboids have a total of \bf{6} ** faces** – all of which are rectangular (or square) and contain four right angles. They have three pairs of identical faces:

An **edge **of a 3 D shape is a straight line between two faces.

Cuboids have \bf{12} **edges**:

A **vertex **is a point where two or more edges meet.

Cuboids have \bf{8} **vertices**:

A **net **of a three-dimensional shape is the 2 dimensional shape it would make if it were unfolded and laid flat. Nets can be folded up to make the 3 D shapes.

We can label the vertices (corners) of a cuboid to help us identify certain edges or faces.

E.g.

Using this labelling we can identify lengths, e.g. the length AB:

We can also identify faces, e.g. the face ABCD:

The **volume **of a 3 D shape is how much space there is inside of the shape.

The formula for the volume of a cuboid is:

Volume = length \times width \times heightE.g.

This cuboid is made from 12 cubes.

Its volume is

\begin{aligned} &Volume = length \times width \times height \\\\ &Volume = 2 \times 2 \times 3 \\\\ &Volume = 12cm^3 \end{aligned}Volume is measured in cubic units, e.g. mm^3, \ cm^3 or m^3.

**Step-by-step guide:** Volume of a cuboid

Get your free cuboid worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free cuboid worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEWork out the volume of this cuboid

**Substitute the values into the formula.**

Here the length is 7cm , the width is 12cm and the height is 2cm .

\begin{aligned} &Volume = length \times width \times height \\\\ &Volume = 7 \times 12 \times 2 \end{aligned}2**Do the calculation.**

3**Include the units.**

The measurements are in cm therefore the volume will be in cm^3 .

Volume=168cm^3The **surface area** of a 3 D shape is the total area of all of the faces of the shape

To work out the surface area of a cuboid, we need to work out the area of each rectangular face and add them all together.

Cuboids have three pairs of equal faces – top and bottom, front and back, and left and right.

Since it is an area, surface area is measured in square units (e.g. mm^2, cm^2, m^2 etc).

Face | Area |

Bottom | 5 × 6 = 30 |

Top | 30 |

Front | 6 × 2 = 12 |

Back | 12 |

Right side | 2 × 5 = 10 |

Left side | 10 |

**Step-by-step guide: **Surface area of a cuboid

Work out the surface area of the cuboid

**Work out the area of each face.**

We know that the areas of the top and bottom are equal, the front and back are equal and the two sides are equal.

Face | Area |

Bottom | 7 × 12 = 84 |

Top | 84 |

Front | 7 × 2 = 14 |

Back | 14 |

Right side | 2 × 12 = 24 |

Left side | 24 |

2**Add the six areas together.**

3**Include the units.**

The measurements on the cuboid are in cm therefore the total surface area of the cuboid = 244cm^2.

**Missing/incorrect units**

You should always include units in your answer.

Volume is measured in units cubed (e.g. mm^3, cm^3, m^3 etc)

Surface area is measured in units squared (e.g. mm^2, cm^2, m^2 etc)

**Calculating with different units**

You need to make sure all measurements are in the same units before calculating volume or surface area. (E.g. you can’t have some in cm and some in m )

**Dividing by three rather than cube rooting**

If you are given the volume of a cube and you need to find the side length, remember the inverse of cubed is cube root, not divide by 3.

E.g.

If the volume of a cube is 8cm^3 , the side length is \sqrt[3]{8}=2cm (not 8\div3 )

**Calculating volume instead of surface area**

Volume and surface area are different things – volume tells us the space within the shape whereas surface area is the total area of the faces. To find surface area, work out the area of each face and add them together.

**Equal faces**

A common mistake is to think that four of the faces are equal.

E.g.

The pair of blue faces are equal to each other.

The pair of green faces are equal to each other.

The pair of oranges faces are equal to each other.

**Cuboid vs parallelepiped**

A common error is to confuse a rectangular cuboid with a rectangular parallelepiped. Cuboids have 6 rectangular faces whereas parallelepipeds have 6 faces that are parallelograms.

1. Work out the volume of the cuboid

90\mathrm{mm}^{3}

332\mathrm{mm}^{3}

360\mathrm{mm}^{3}

160\mathrm{mm}^{3}

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
\text{Volume }&= 4 \times 10 \times 9\\\\
\text{Volume }&=360\mathrm{mm}^{3}
\end{aligned}

2. Work out the volume of the cuboid

6\mathrm{cm}^{3}

600\mathrm{cm}^{3}

60\mathrm{cm}^{3}

598\mathrm{cm}^{3}

Notice that some of the units are in cm and some in m so the first thing we need to do is convert 0.25m to cm. \ 0.25m=25cm.

\begin{aligned} \text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\ \text{Volume }&= 3 \times 25 \times 8\\\\ \text{Volume }&=600\mathrm{cm}^{3} \end{aligned}

3. The volume of this cuboid is 300cm^{3} . Work out the height of the cuboid.

2cm

150cm

50cm

12cm

\begin{aligned}
\text{Volume }&= \text{ length }\times \text{ width }\times \text{ height }\\\\
300&= 25 \times 6 \times h\\\\
300&=150h\\\\
h&=2\mathrm{cm}
\end{aligned}

4. Work out the surface area of this cuboid

990\mathrm{m}^{2}

792\mathrm{m}^{2}

492\mathrm{m}^{2}

642\mathrm{m}^{2}

Work out the area of each of the six faces:

Face | Area |

Bottom | 6 × 15 = 90 |

Top | 90 |

Front | 6 × 11 = 66 |

Back | 66 |

Right side | 11 × 15 = 165 |

Left side | 165 |

\text{Total surface area: }90+90+66+66+165+165=642\mathrm{m}^{2}

5. Work out the surface area of this cuboid

621\mathrm{cm}^{2}

9000\mathrm{cm}^{2}

2700\mathrm{cm}^{2}

3000\mathrm{cm}^{2}

First we need to make all the units the same. 0.3m=30cm.

Then work out the area of each of the six faces:

Face | Area |

Bottom | 20 × 30 = 600 |

Top | 600 |

Front | 20 × 15 = 300 |

Back | 300 |

Right side | 30 × 15 = 450 |

Left side | 450 |

\text{Total surface area: } 600+600+300+300+450+450=2700\mathrm{cm}^{2}

6. Work out the surface area of this cuboid

(140+38a) \ \mathrm{cm}^{2}

(140+56a) \ \mathrm{cm}^{2}

70a \ \mathrm{cm}^{2}

140a \ \mathrm{cm}^{2}

Work out the area of each of the six faces:

Face | Area |

Bottom | 14 × a = 14a |

Top | 14a |

Front | 14 × 5 = 70 |

Back | 70 |

Right side | 5 × a = 5a |

Left side | 5a |

\text{Total surface area: }14a+14a+70+70+5a+5a=(140+38a) \ \mathrm{cm}^{2}

1. Calculate the surface area of the cuboid. Give your answer in m^2.

**(3 marks)**

Show answer

240cm=2.4m

**(1)**

\text{Surface area } = 2.8+2.8+9.6+9.6+1.68+1.68

**(1)**

\text{Surface area }=28.16\mathrm{m}^{2}

**(1)**

2. (a) Each of these diagrams is made from 6 identical squares with side length 7cm . Which diagram is the net of a cube?

(b) What would the volume of the cube made from this net be?

**(3 marks)**

Show answer

(a)

D – cao

**(1)**

(b)

\text{Volume }=7 \times 7 \times 7

**(1)**

\text{Volume }=343

**(1)**

3. The volume of a cube is 125\mathrm{cm}^{3}.

What is the surface area of the cube?

**(4 marks)**

Show answer

\text{Side length of cube }=\sqrt[3]{125}

**(1)**

\text{Side length of cube }= 5cm

**(1)**

\text{Area of each face }=5 \times 5 = 25

**(1)**

\text{Total surface area }=6 \times 25 = 150\mathrm{cm}^{2}

**(1)**

You have now learned how to:

- Recognise a cuboid or a cube
- Calculate the volume of a cuboid
- Calculate the surface area of a cuboid
- Apply formulae to calculate and solve problems involving cuboids (including cubes)
- Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3-D

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Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

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