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Volume of 3D shapes: Formulae

Volume of 3D shapes: Formulae

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Summary

Volume of 3D shapes: Formulae

​​In a nutshell

There are different formulae to learn for the volumes of 3D shapes.



More 3D shapes

There are two more 3D shapes that you need to be familiar with: prisms and frustums.


Prism

A prism is a 3D shape that is a 2D shape stretched in the third dimension. The 2D shape is called it's cross-section.


Examples


Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

This is a prism with a cross-section that is a hexagon. Therefore, this shape is called a hexagonal prism.

This is a prism with a cross-section that is a triangle. Therefore, this shape is called a triangular prism.

A cylinder is also a type of prism, it has a circular cross-section.


Frustum

A frustum is a smaller cone taken away from a bigger cone, or a cone with it's top part cut off.


Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae



Formulae for volumes of 3D shapes

Here are a list of 3D shapes and the formulae to calculate their volumes.


SHAPE

FORMULA

DIAGRAM

Cube

V=x×x×x=x3V=x\times x\times x=x^3


Where xx​ is the length of a single side.

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

​​Cuboid

V=length×width×heightV=\text{length}\times \text{width}\times \text{height}​​


V=lwhV=lwh​​

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Cylinder

V=π×radius2×heightV=\pi\times\text{radius}^2\times \text{height}​​


V=πr2hV=\pi r^2h​​

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Prism

V=A×lV=A\times l​​


Where AA​ is the area of the cross-section, and ll​ is the length.

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Sphere

​​​V=43π×radius3V=\dfrac{4}{3}\pi\times\text{radius}^3​​


V=43πr3V=\dfrac{4}{3}\pi r^3​​

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Cone

V=13π×radius2×heightV=\dfrac{1}{3}\pi\times\text{radius}^2\times \text{height}​​


V=13πr2hV=\dfrac{1}{3}\pi r^2h​​

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Pyramid

V=13×A×hV=\dfrac{1}{3}\times A\times h​​


Where AA​ is the base area, and hh​ is the height.

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

Frustum

V=volume of bigger conevolume of smaller coneV=\text{volume of bigger cone} -\text{volume of smaller cone}​​


Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae



==​​

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae

-​​

Maths; Shapes and area; KS4 Year 10; Volume of 3D shapes: Formulae


Example 1

What is the exact volume of a cone that has a radius of 5cm5cm and a height of 10cm10cm?


Substitute r=5r=5 and h=10h=10 into the formula for the volume of a cone:

V=13πr2hV=\dfrac{1}{3}\pi r^2h​​


V=13π(5)2(10)V=\dfrac{1}{3}\pi (5)^2(10)​​


V=250π3cm3V=\dfrac{250\pi}{3}cm^3​​


The exact volume of the cone is 250π3 cm3\underline{\dfrac{250\pi}{3} \space cm^3}.


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FAQs - Frequently Asked Questions

What is a frustum?

What is a prism?

How do you find the volume of a 3D shape?

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