GM Lesson 035 Capacity and Mixed Measurement Problems

Learning Intentions

By the end of this lesson, students will be able to:

• Calculate volumes and capacities of standard three-dimensional objects.
• Connect volume and capacity in practical contexts.
• Solve practical problems involving shape and measurement.

Prerequisites

Students should already be able to:

• Calculate areas of rectangles, triangles, circles, parallelograms and trapeziums.
• Calculate volumes of prisms, cylinders, pyramids, cones and spheres.
• Substitute values into measurement formulae.
• Convert between common metric units of volume and capacity.

Key Idea Summary

Volume measures the amount of space inside a three-dimensional object.

Capacity measures how much a container can hold.

The key conversions are:

Useful syllabus formulae:

Prism:

Cylinder:

Pyramid:

Cone:

Sphere:

A mixed measurement problem often requires more than one step:

1. Identify the solid or combination of solids.
2. Select the correct formula.
3. Substitute the measurements.
4. Calculate the volume.
5. Convert volume to capacity if required.
6. Interpret the answer in context.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Worked Example 1: Capacity of a Rectangular Tank

A rectangular water tank has length , width and height . Find its capacity in litres.

The tank is a rectangular prism, so:

The base area is:

Therefore:

Since:

the capacity is:

Therefore, the tank has a capacity of .

Worked Example 2: Capacity of a Cylindrical Container

A cylindrical container has radius and height . Find its capacity in litres, correct to one decimal place.

For a cylinder:

Substitute and :

Convert to litres:

Therefore, the capacity is approximately .

Worked Example 3: Mixed Solid Problem

A decorative container is made from a cylinder with a cone on top. The cylinder has radius and height . The cone has the same radius and perpendicular height . Find the total volume of the solid.

Cylinder volume:

Cone volume:

Total volume:

Therefore, the total volume is approximately .

Since:

the capacity would be approximately , or .

Worked Example 4: Practical Capacity Problem

A spherical fish bowl has radius . It is filled to of its total capacity. Estimate the amount of water in the bowl in litres, correct to one decimal place.

For a sphere:

Substitute :

The bowl is filled to :

Convert to litres:

Therefore, the bowl contains approximately of water.

Understanding Checks

Check 1

A container has volume . What is its capacity in litres?

Check 2

A prism has base area and height . Which formula should be used to find its volume?

Check 3

A cylinder has diameter and height . What value should be substituted for in the formula?

Check 4

Explain why is the same capacity as .

Check 5

A cone and a cylinder have the same radius and perpendicular height. How does the cone’s volume compare to the cylinder’s volume?

Exercises

Time allocation: approximately minutes

Simple Familiar Exercises

Exercise 1

Convert each volume to capacity.

a. to litres b. to litres c. to cubic centimetres d. to cubic centimetres

Exercise 2

A rectangular prism has length , width and height . Find its volume in cubic centimetres and its capacity in litres.

Exercise 3

A cylinder has radius and height . Find its volume in cubic centimetres, correct to one decimal place.

Exercise 4

A triangular prism has triangular base area and length . Find its volume.

Complex Familiar Exercises

Exercise 5

A cylindrical drink bottle has radius and height . Find its capacity in millilitres, correct to the nearest millilitre.

Exercise 6

A cone has radius and perpendicular height . Find its volume in cubic centimetres, correct to one decimal place.

Exercise 7

A square-based pyramid has base side length and perpendicular height . Find its volume.

Exercise 8

A spherical container has radius . Find its total capacity in litres, correct to two decimal places.

Homework Problems

Homework should take no more than minutes.

Homework 1

Convert each measurement.

a. to litres b. to cubic centimetres c. to litres d. to cubic centimetres

Homework 2

A rectangular storage box has length , width and height . Find its volume in cubic centimetres and capacity in litres.

Homework 3

A cylinder has radius and height . Find its capacity in litres, correct to one decimal place.

Homework 4

A cone has diameter and perpendicular height . Find its volume in cubic centimetres, correct to one decimal place.

Homework 5

A sphere has radius . Find its volume, correct to one decimal place.

Homework 6

A composite solid is made from a cylinder with a cone on top. Both parts have radius . The cylinder has height and the cone has perpendicular height .

Find the total volume of the solid in cubic centimetres.

Homework 7

A swimming pool is shaped like a rectangular prism. It is long, wide and deep.

a. Find the volume of the pool in cubic metres. b. Find the capacity of the pool in litres.

Homework 8

A cylindrical rainwater tank has diameter and height .

a. Find the volume of the tank in cubic metres. b. Convert the volume to litres. c. If the tank is currently full, estimate how many litres of water it contains.

Next: GM Lesson 036 Conditions for Similarity