IB DP Maths: AI HL复习笔记3.2.2 Volume & Surface Area
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
-
- 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:
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
How do I find the volume of a sphere?
- To calculate the volume of a sphere use the formula
-
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
Exam 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 10 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.
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
Exam 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.
i)Use the Pythagorean Theorem to find the distance VN.
ii)Calculate the area of the triangle ABV.
iii)Find the surface area of the right pyramid.
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