Unit Conversions

TSI Math Study Guide

Unit Conversions

Geometry problems sometimes give measurements in different units. Before calculating a perimeter, area, surface area, or volume, you may need to convert the measurements into compatible units.

What’s tested: converting length, area, and volume measurements within the customary and metric systems.

Length conversions

Customary

12 inches= 1 foot3 feet= 1 yard5,280 feet= 1 mile

Metric — powers of 10

1 kilometer= 1,000 meters1 meter= 100 centimeters1 centimeter= 10 millimeters

Conversion factor — a fraction equal to 1, such as 12 in1 ft. Arrange it so the unit you don’t want cancels. That tells you whether to multiply or divide.

To a smaller unit

Convert 7 feet to inches.

7 ft × 12 in1 ftfeet cancel7(12) = 84 inthe number gets larger

To a larger unit

Convert 72 inches to feet.

72 in × 1 ft12 ininches cancel72 ÷ 12 = 6 ftthe number gets smaller

Metric

Convert 4.5 meters to centimeters.

4.5 m × 100 cm1 m450 cmoften just a decimal shift, but the factor prevents mistakes

Convert before using a formula

Measurements should be in the same unit before they go into a geometry formula together.

ProblemA rectangular rug measures 8 feet by 60 inches. What is its area in square feet?

Don’t

8(60) = 480

Different units multiplied together — meaningless.

Convert first, calculate second

60 in × 1 ft12 in = 5 ftconvertA = 8(5) = 40 ft2then calculate

Area and volume conversions

Area is measured in square units and volume in cubic units, so the conversion factor gets squared or cubed.

⚠ Watch out

It’s tempting to think 1 ft2 = 12 in2. But a square that measures 1 foot by 1 foot measures 12 inches by 12 inches — that’s 144 square inches.

LENGTH1 ft = 12 inAREA1 ft2 = (12 in)2 = 144 in2VOLUME1 ft3 = (12 in)3 = 1,728 in3

1 ft = 12 in 1 ft = 12 in 12 × 12 = 144 in²

Area example

Convert 3 ft2 to square inches.

3(144) = 432 in2

Volume example

Convert 2 ft3 to cubic inches.

2(1728) = 3,456 in3

Same idea in metric

1 m= 100 cm1 m2= 10,000 cm21 m3= 1,000,000 cm3

Using units to check your work

Units cancel

48 in × 1 ft12 in = 4 ft

Inches cancel, leaving feet. Correct setup.

Units don’t cancel

48 in × 12 in1 ft

You’d get in2/ft. The factor is upside down.

Estimation check

Converting to a smaller unit (meters → centimeters) should make the number larger: 3 m = 300 cm. An answer like 0.03 cm should immediately look wrong.

TSI strategy: make the units match first

When a problem gives several measurements, check the units before choosing a formula.

ProblemA rectangle measures 10 ft × 48 in. Find its area.
48 in = 4 ftdon’t multiply 10 by 48 — convert first A = 10(4) = 40 ft2

Getting all measurements into compatible units first makes the rest of the problem much easier.

🔑 Key tip

The conversion changes with the dimension being measured. If 1 ft = 12 in, then 1 ft2 = 122 = 144 in2 and 1 ft3 = 123 = 1,728 in3.

Use the ordinary factor for length, square it for area, cube it for volume.

Unit Conversions Review Quiz

Next: Geometry Concepts

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