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.

A composite figure consisting of a hemisphere and a cone that are conjoined at their bases. The flat base of the hemisphere serves as the circular base of the cone. The composite figure is oriented vertically, with the apex of the cone directed upward.
A vertical dashed line extending from the cone’s apex to the center of its base measures $10$10 cm. A horizontal dashed line is also drawn from the center of the cone's base to its circumference and measures $4$4 cm. These two dashed lines intersect at a right angle, indicated by a small square at the intersection point.

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 three-dimensional diagram of a composite figure. The structure has a rectangular prism base with dimensions measured $8$8 m in length, $10$10 m in width, and height of $7$7 m. Attached to the top of the rectangular prism base is a triangular prism, which adds an additional $\text{ 3}$ 3 m to the structure’s height. A dashed line indicates the height of the triangle, perpendicular to the base of the triangle. Each of the slanting sides of the triangular base measures $5$5 m.

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

1.2.2.3

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

5.06 Applications of surface area