The surface area of a 3D object is the total area of the shapes which form the surface of the object. For an object whose surface is composed of flat polygonal faces, such as the prisms described below, we can find the surface area by adding up the areas of all the faces. For objects with curved surfaces, such as cylinders and spheres, finding the surface area is a little more difficult.

Surface area of basic 3D shapes

In the previous lessons, we looked at how to find the surface area of prisms, cylinders, pyramids, cones, and spheres.

3D shape	Surface area formula
Prism	$SA=2\times\text{area of base }+\text{area of rectangles }$`SA`=2×area of base +area of rectangles
Cylinder	$SA=2\pi r^2+2\pi rh$`SA`=2π`r`2+2π`rh`
Pyramid	$SA=\text{area of base }+\text{area of triangles }$`SA`=area of base +area of triangles
Cone	$SA=\pi r^2+\pi rs$`SA`=π`r`2+π`rs`
Sphere	$SA=4\pi r^2$`SA`=4π`r`2

Using these formulas, we can solve real-world problems by modelling objects (or parts of objects) as 3D shapes.

Surface area of composite shapes

In practical problems we often encounter 3D shapes that consist of two or more of the simple shapes discussed above that have been joined together in some way. These are called composite shapes. To find the surface area of a composite shape we just add together the surface areas of these simple shapes as appropriate, being careful not to count the faces where shapes join together.

Practice questions

question 1

Find the surface area of the composite figure shown, which consists of a cone and a hemisphere joined at their bases.

Round your answer to two decimal places.

question 2

This is the design for a marquee (tent). The roof of the marquee has a height of $3$3 metres. The material for the marquee costs $\$44$$44 per m².

A tent composed of a rectangular prism which represents and a triangular prism on top. The bottom rectangular lateral face of the triangular prism is the top rectangular lateral face of the rectangular prism. The tent is oriented such that on its front is a triangular face of the triangular prism, and a rectangular face of the rectangular prism. The left and right side of the triangle face at the front extends to the triangle face at the back, forming two rectangular lateral faces of the triangular prism. These two rectangular lateral faces of the triangular prism are labeled as "Roof," indicating that these two rectangles form the roof of the tent. The triangle face at the front of the tent has its left and right sides both labeled to measure $5$ m, and height that measures $3$ m. The base side of the triangle is the top side of the rectangle face at the front, and it is drawn in dashed lines. The rectangle face at the front of the tent has length that measures $8$ m and height that measures $7$ m. The width of the whole tent measures $10$ m as labeled. The rectangle face at the front is labeled "Front," and the rectangular lateral face of the rectangular prism at the right is labeled "Side."

What is the area of the front of the marquee?
What is the surface area of one of the side walls (not including the roof)?
What is the surface area of the entire roof?
What is the total surface area? Do NOT include the floor of the marquee.
What is the total cost of the marquee material?

Outcomes

ACMGM020

calculate the surface areas of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the surface area of a cylindrical food container

5.06 Applications of surface area