You may be asked to calculate the volume of a prism for GCSE Maths. Put simply, a prism is a 3D shape, and the space it occupies is the volume. It's relevant to real life, such as working out how much material is needed to build a solid object. Exams will test if you understand the formula and can apply it to different types of prisms, such as finding the volume of a triangular prism, a hexagonal prism, or a pentagonal prism.
This article will explore what a prism is, the different types of prisms, and how to calculate the volume of the prism. Clear examples are included, which calculate the volume step by step. Illustrations are given for each prism type. It is suitable for GCSE Maths revision across all major exam boards.
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What Is a Prism?
A prism is a three-dimensional (3D) shape with the same cross-section running across its length. This means that if we were to split the shape in half parallel to each side, each slice looks the same. The two matching ends are known as the cross-section - they are always identical and parallel.
The sides joining the cross-sections are normally rectangular faces. All prisms have a regular structure, due to these sides connecting identical shapes. The upshot is that the volume for all prisms can be calculated with a single method.
A prism is titled after the shape of its cross-section. For example:
- Rectangular prism - this has a rectangular cross-section. It's also known as a cuboid.
- Triangular prism - this has a triangular cross-section.
- Pentagonal prism - the cross-section is pentagon-shaped.
A shape is not a prism when its cross-section changes size across its length. This includes pyramids and cones.
Types of Prisms
There are various types of prisms. They are named after the shape of their cross-section. Despite the different shapes a cross-section can take, the method for finding the volume stays the same.
The prisms you will commonly see in GCSE Maths are listed below.
Rectangular Prism (Cuboid)
A rectangular prism is also called a cuboid. It has a rectangular cross-section. Examples from everyday life include boxes, books, and bricks. This is the most common prism, because all faces are rectangles.
Triangular Prism
A triangular prism has a triangular cross-section. The triangle shape is maintained throughout the prism, while the sides that connect the triangles are rectangles. It is common in GCSE Maths, as the student needs to find the area of a triangle, as well as the prism volume.
Other Prisms
The following prisms can also be encountered in GCSE Maths. They are less common, but should still be revised, and students are advised to tackle questions focused on these 3D shapes:
- Pentagonal prisms
- Hexagonal prisms
- Trapezium-based prisms
Remember that a shape is a prism when the cross-section stays the same across its length, regardless of its appearance. The focus in GCSE Maths is normally on rectangular and triangular prisms.
What is the volume of a prism?
Volume measures the space taken up by a 3D shape. It says how much space the shape can hold inside it. It is not the amount of space the shape takes up on the outside. Cubic units are normally used to measure volume, e.g. cm³ or m³.
For example, imagine filling a shape with small cubes. Each cube is 1 cm³, which means the total cubes needed to fill the shape represents its volume. This effectively is how we calculate the volume of prisms.
Volume and surface area are two distinct measurements. The area is space on a flat, 2D surface. It's important to remember this because calculating the volume of a shape begins by finding the area of the cross-section, which is extended through its length.
The formula to calculate the volume
The formula for finding the volume of a prism is multiplying the area of its cross-section by its length. The cross-section stays the same throughout the shape, so we can simply calculate this area against the length of the prism:
Volume = area of cross-section x length of the prism
The cross-section is the flat shape at the front and back of the prism, which are identical to each other. Their appearance depends on the prism type: for example, it could be a rectangle or a triangle. The length is the distance between the cross-sections.
All measurements need to use the same units before you find the volume, which must be written in cubic units. Remember to always begin your calculations by finding the area of the cross-section. Finding the area depends on the type of cross-section. The calculation for simple shapes is below:
- Circle - Measure the diameter or radius
- Rectangle - Measure the width and depth
- Pipe/tube - Measure the outer radius and inner radius
Worked Prism examples
The method to find the volume remains the same: get the area of the cross-section, then multiply by the length of the shape. Learn how to calculate the volume with the worked examples below.
Example 1: Rectangular Prism (Cuboid)
A rectangular prism has a rectangular cross-section. Multiply the width by the height to find the cross-sectional area.
Let's say the cross-section is 5 cm by 3 cm, while the length is 10 cm:
- The cross-section area is 15 cm² (5 x 3)
- The volume is 150 cm³ (15 x 10)
Example 2: Triangular Prism
The cross-section for a triangular prism is a triangle. This means we need to use the triangle area formula to find the area.
Let's say the triangular cross-section has a base of 6 cm and a height of 4 cm. The length is 8 cm:
- The area of the triangle is 12 cm² (6 x 4 / 2)
- The volume is 96 cm³ (12 x 8)
Example 3: Any other Prism
We can apply the formula for the volume to all prisms. As we have stressed in this guide, multiply the area of the cross-section by the length to return the volume.
GCSE exam questions may feature a trapezium prism or a pentagonal prism. While this shape has a more complex cross-section, the method stays the same: cross-section area x length.
Practice questions
1
What defines a prism?
2
Which shape is a prism?
3
What does volume measure in a 3D shape?
4
What's the volume of a prism with a cross-section area of 10 cm² and a length of 4 cm?
5
What is the volume of the prism?
6
Which image is a prism?
Final thoughts - Volume of prism
Finding the volume of a prism links ideas about 2D areas and 3D shapes. The method is straightforward: find the area of the cross-section, multiply it by the length, and write the answer in cubic units. We can apply this formula to all types of prisms, including triangular prisms and rectangular prisms.
The only caveat is that students need to be able to find the area of a surface. If you need a refresher, read this BBC article to calculate the area of a surface. You can also read this post by ThirdSpaceLearning, which looks more widely at the types and properties of 3D shapes.
If you need help preparing for this topic, TeachTutti has experienced GCSE Maths tutors. Every tutor has an enhanced DBS check and will tailor lessons to your specific needs, such as a wider look at the properties of 3D shapes.
