 5.06 Surface area and volume of composite solids

Worksheet
Surface area of composite solids
1

Find the surface area of the following composite solids:

a
b
c
d
e
2

Find the surface area of the following figures. Round your answers to two decimal places.

a
b
c
3

The solid shown is constructed by cutting out a quarter of a sphere from a cube. Find its surface area if the side length is 14.2 \text{ cm} and the radius of the sphere is half the side length.

4

Consider the hollow cylinder shown:

a

What is the external surface area of the curved surface to two decimal places?

b

What is the total surface area of the front and back rings to two decimal places?

c

What is the internal surface area of the curved surface to two decimal places?

d

Hence, what is the total surface area of the solid to two decimal places?

5

A solid consists of a cylinder (with radius of 5 \text{ mm}) attached to a rectangular prism on one of its faces.

a

Find the exposed surface area of the rectangular prism. Round your answer to two decimal places.

Note that an area is called 'exposed' if it is not covered by the other object.

b

Find the exposed surface area of the cylindrical top piece. Round your answer to two decimal places.

c

Hence, find the total surface area. Round your answer to two decimal places.

6

Consider the following shape with both boxes that are identical in size:

a

Find the surface area of both boxes if they had all faces exposed.

b

Find the surface area of the two circular faces of the cylinder. Round your answer to two decimal places.

c

Find the area of the curved face of the cylinder. Round your answer to two decimal places.

d

Hence, find the surface area of the solid.

Volume of composite solids
7

Find the volume of the following composite solids:

a
b
c
d
e
f
8

Find the volume of the following solids. All measurements are in metres. Round your answers to two decimal places where necessary.

a
b
9

Find the volume of the following composite figures. Round your answers to two decimal places.

a
b
10

This solid consists of a rectangular prism with a smaller rectangular prism cut out of it.

Find the volume of the solid.

11

A triangular tunnel is made through a rectangular prism as shown in the figure:

Find the volume of the solid formed.

Applications
12

The lid of this treasure chest is found to be exactly one half of a cylindrical barrel. Find the surface area of the chest, correct to two decimal places.

13

The outline of a paddy field used to grow rice is pictured on the right:

a

Find the area of the rice paddy.

b

Throughout the monsoon season, the rice paddy receives 16 \text{ cm} of rain. What volume of water, in cubic metres, has fallen on the paddy?

14

The image on the right shows the outline of a construction site for the foundations of a large office building:

a

Find the area of the construction site.

b

Prolonged rainstorms bring 0.21 \text{ m} of rain to the region and the foundations of the building are flooded. Determine the volume of water in the construction site.

15

A cylinder is pushed through a cube. The off-cut piece is indicated in red:

Find the total surface area of the off-cut. Round your answer to two decimal places.

16

A steel shed is to be constructed, with dimensions as shown below. The shed is to include a rectangular cut-out at the front for the entrance.

a

Determine the surface area of the shed. Round your answer to one decimal place.

b

Construction of the shed requires an additional 0.1\text{ m}^2 of sheet metal for each 1\text{ m}^2 of surface area, due to overlaps and wastage.

How much sheet metal is required to construct this shed? Round your answer up to the nearest square metre.

c

If the steel sheets cost \$18 per square metre, calculate the total cost of the steel required to build this shed. 17 The given diagram shows the design for a marquee (tent). The roof of the marquee has a height of 3\text{ m}. The material for the marquee costs \$44/\text{m}^{2}.

a

Find the total surface area of the marquee. Do NOT include the floor.

b

Find the total cost of the marquee material.

18

A wedding cake consists of three cylinders stacked on top of each other. The dimensions are as follows:

• The top layer has a radius and height of 20\text{ cm}.
• The middle layer has the same height as the top layer and a radius that is double that of the top layer.
• The bottom layer has a height that is double that of the top layer, and a radius that is triple that of the top layer.

All the sides and top surfaces are to be covered in icing, but not the base. Find the surface area of the cake that needs to be iced. Round your answer to the nearest square centimetre.