12.16 Surface area of cones, pyramids and spheres
Total surface area
Surface area is the total area of all the faces on a 3D object.
$\text{Surface area of a prism }=\text{Sum of areas of all faces}$Surface area of a prism =Sum of areas of all faces
Pyramids
A pyramid can be made in the following way. Use any polygon as a base. There can be square bases, triangular bases or even hexagonal bases. Then, connect every vertex of the base to an apex point above the base, and you have a pyramid.
Square and rectangular based pyramids, are the most common you will come across in mathematics, but also in the real world.
In the interactive below, notice that the slope height corresponds to the height of the 2D triangle, which is used in calculating surface area.
$\text{Surface area of right pyramid }=\text{Area of base }+\text{Area of triangles }$Surface area of right pyramid =Area of base +Area of triangles
Practice question
Question 1
Find the surface area of the square pyramid shown. Include all faces in your calculations.
Spheres
A solid three-dimensional circular object is a sphere.
Archimedes showed that the surface area of the circular component of the cylinder wrapping the circle, has area $2\pi r\times2r=4\pi r^2$2πr×2r=4πr2 and that this area is the same value for the surface area of a sphere.
$\text{Surface area of a sphere }=4\pi r^2$Surface area of a sphere =4πr2
Practice question
Question 2
Find the surface area of the sphere shown.
Round your answer to two decimal places.
Cones
The surface area of a cone is related to cylinders in the same way that a pyramid is related to the prism.
$\text{Surface area of right cone}=\text{Area of base }+\text{Area of sector }$Surface area of right cone=Area of base +Area of sector
$SA=\pi r^2+\pi rs$SA=πr2+πrs
Practice question
Question 3
Find the surface area of the cone shown.
Round your answer to two decimal places.
Compound solids
Sometimes, the shape is a compound solid (also known as a composite solid), it is made up of a combination of other solids. These are some examples of compound solids.
To find the surface area of compound solids, it is necessary to be able to visualise the different shapes that make up the various surfaces. Once these are identified (the different faces and shapes), then calculate the areas of each face and add them up separately.
Don't forget to subtract faces which are not on the surface, like the circle where the cylinder sits on the rectangular prism in the middle image above.
Practice questions
Question 4
In the diagram, the roof has a height of $3$3 metres. Find the surface area of the figure shown,
Round your answer to two decimal places.
Question 5
We wish to find the surface area of the given solid.
What is the surface area of the faces as seen from the top view?
What is the surface area of the faces as seen from the left side view?
What is the surface area of the faces as seen from the front view?
Therefore, what is the total surface area, including all faces of the solid?