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Volume & Surface Area (DP IB Maths: AI HL)
Revision Note
Volume of 3D Shapes
What is volume?
- The volume of a 3D shape is a measure of how much 3D space it takes up
- A 3D shape is also called a solid
- You need to be able to calculate the volume of a number of common shapes
How do I find the volume of cuboids, prisms and cylinders?
- A prism is a 3-D shape that has two identical base shapes connected by parallel edges
- A prism has the same base shape all the way through
- A prism takes its name from its base
- To find the volume of any prism use the formula:
Volume of a prism = Ah
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- Where A is the area of the cross section and h is the base height
- h could also be the length of the prism, depending on how it is oriented
- This is in the formula booklet in the prior learning section at the beginning
- The base could be any shape so as long as you know its area and length you can calculate the volume of any prism
- Where A is the area of the cross section and h is the base height
- Note two special cases:
- To find the volume of a cuboid use the formula:
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- The volume of a cylinder can be found in the same way as a prism using the formula:
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- where is the radius, is the height (or length, depending on the orientation
- Note that a cylinder is technically not a prism as its base is not a polygon, however the method for finding its volume is the same
- Both of these are in the formula booklet in the prior learning section
How do I find the volume of pyramids and cones?
- In a right-pyramid the apex (the joining point of the triangular faces) is vertically above the centre of the base
- The base can be any shape but is usually a square, rectangle or triangle
- To calculate the volume of a right-pyramid use the formula
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- Where A is the area of the base, h is the height
- Note that the height must be vertical to the base
- A right cone is a circular-based pyramid with the vertical height joining the apex to the centre of the circular base
- To calculate the volume of a right-cone use the formula
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- Where is the radius, is the height
- These formulae are both given in the formula booklet
How do I find the volume of a sphere?
- To calculate the volume of a sphere use the formula
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- Where r is the radius
- the line segment from the centre of the sphere to the surface
- This formula is given in the formula booklet
- Where r is the radius
Examiner Tip
- Remember to make use of the formula booklet in the exam as all the volume formulae you need will be here
- Formulae for basic 3D objects (cuboid, cylinder and prism) are in the prior learning section
- Formulae for other 3D objects (pyramid, cone and sphere) are in the Topic 3: Geometry section
Worked example
A dessert can be modelled as a right-cone of radius 3 cm and height 12 cm and a scoop of ice-cream in the shape of a sphere of radius 3 cm. Find the total volume of the ice-cream and cone.
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Surface Area of 3D Shapes
What is surface area?
- The surface area of a 3D shape is the sum of the areas of all the faces that make up a shape
- A face is one of the flat or curved surfaces that make up a 3D shape
- It often helps to consider a 3D shape in the form of its 2D net
How do I find the surface area of cuboids, pyramids and prisms?
- Any prisms and pyramids that have polygons as their bases have only flat faces
- The surface area is simply found by adding up the areas of these flat faces
- Drawing a 2D net will help to see which faces the 3D shape is made up of
How do I find the surface area of cylinders, cones and spheres?
- Cones, cylinders and spheres all have curved faces so it is not always as easy to see their shape
- The net of a cylinder is made up of two identical circles and a rectangle
- The rectangle is the curved surface area and is harder to identify
- The length of the rectangle is the same as the circumference of the circle
- The area of the curved surface area is
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- where r is the radius, h is the height
- This is given in the formula book in the prior learning section
- The area of the total surface area of a cylinder is
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- This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given
- The net of a cone consists of the circular base along with the curved surface area
- The area of the curved surface area is
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- Where r is the radius and l is the slant height
- This is given in the formula book
- Be careful not to confuse the slant height, l, with the vertical height, h
- Note that r, h and l will create a right-triangle with l as the hypotenuse
- The area of the total surface area of a cone is
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- This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given
- To find the surface area of a sphere use the formula
-
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- where r is the radius (line segment from the centre to the surface)
- This is given in the formula booklet, you do not have to remember it
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Examiner Tip
- Remember to make use of the formula booklet in the exam as all the area formulae you need will be here
- Formulae for basic 2D shapes (parallelogram, triangle, trapezoid, circle, curved surface of a cylinder) are in the prior learning section
- Formulae for other 2D shapes (curved surface area of a cone and surface area of a sphere ) are in the Topic 3: Geometry section
Worked example
In the diagram below ABCD is the square base of a right pyramid with vertex V . The centre of the base is M. The sides of the square base are 3.6 cm and the vertical height is 8.2 cm.
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