Area, Perimeter, and Volume

TSI Math Study Guide

Area, perimeter, and Volume

Geometry questions on the TSI often involve finding the perimeter, circumference, area, surface area, or volume of a figure. The calculations are usually straightforward once you identify the correct formula and which measurements are actually needed.

Before calculating, decide which one the question is asking for

Distance around a figure  ·  space inside a flat figure  ·  outside surface of a solid  ·  space inside a 3D object

Perimeter and circumference

Perimeter — the total distance around a polygon’s outside edge. Add the lengths of all sides. Measured in ordinary units (feet, meters, inches), not square units.
l w

Rectangle

P = 2l + 2w

A garden 14 ft by 9 ft:

P = 2(14) + 2(9)P = 46 feet

r

Circle — circumference

C = 2πr = πd

d = 2r

Radius 6 in:

C = 2π(6) = 12πC ≈ 37.68 inchesπ ≈ 3.14

On the test

If the answer choices contain π, leave your answer in terms of π.

Area

Area — the space inside a figure’s boundary. Always measured in square units.
l w

Rectangle

A = lw

A floor 12 ft by 8 ft:

A = 12(8) = 96 ft2

b h

Triangle

A = ½bh

Base 10 cm, height 7 cm:

A = ½(10)(7) = 35 cm2

b h

Parallelogram

A = bh

Base 11 m, height 6 m:

A = 11(6) = 66 m2

r

Circle — area

A = πr2

A patio with radius 5 ft:

A = π(5)2 = 25πA ≈ 78.5 ft2

⚠ Two things to watch

For triangles and parallelograms, the height must be perpendicular to the base — don’t use a slanted side.

Don’t confuse the two circle formulas: C = 2πr but A = πr2.

Composite figures

Composite figure — made from two or more simpler shapes. Divide it into shapes you know, find each area, then add or subtract.
10 ft 8 ft 4 × 3

An L-shaped floor: a 10 × 8 rectangle with a 4 × 3 corner removed.

10(8) = 80large rectangle4(3) = 12missing corner80 − 12 = 68 ft2subtract

Composite questions look more complicated than they are. The key is breaking the figure into familiar pieces.

Volume

Volume — the space inside a three-dimensional object. Measured in cubic units (cm3, ft3, m3). For prisms and cylinders, think V = (area of base)(height).
l w h

Rectangular prism

V = lwh

A box 8 × 5 × 3 ft:

V = 8(5)(3) = 120 ft3

r h

Cylinder

V = πr2h

Radius 4 cm, height 9 cm:

V = π(16)(9) = 144πV ≈ 452.16 cm3

h r

Cone

V = ⅓πr2h

One-third of the matching cylinder.

Radius 3 in, height 8 in:

V = ⅓π(9)(8) = 24π in3

r

Sphere

V = 43πr3

Radius 3 cm:

V = 43π(27) = 36π cm3

Surface area

Surface area — the total area covering the outside of a 3D object. Volume measures the space inside; surface area measures the outside covering.

Rectangular prism

SA = 2lw + 2lh + 2wh

Or add up the area of each of the six faces.

A box with l = 5, w = 3, h = 2:

SA = 2(5)(3) + 2(5)(2) + 2(3)(2)SA = 30 + 20 + 12SA = 62 square units

Cylinder

SA = 2πr2 + 2πrh

2πr2 is the two circular ends; 2πrh is the curved side.

Radius 3 cm, height 5 cm:

SA = 2π(9) + 2π(3)(5)SA = 18π + 30πSA = 48π cm2

Using dimensions correctly

Problems often provide extra measurements. Don’t assume every number must be used. A triangle’s area needs its base and perpendicular height — a third side length may be unnecessary. A cylinder’s volume needs r and h; you don’t need the circumference first unless the problem gives information that requires you to find the radius.

Units matter

Perimeter, circumference

ftmin

Regular units

Area, surface area

ft2m2in2

Square units

Volume

ft3m3in3

Cubic units

If a volume problem produces an answer in square feet, something has gone wrong.

TSI strategy: identify what is being measured

Before choosing a formula, ask

Am I finding a distance, an area, or a volume?

Fencing around a yard

Perimeter

Carpet covering a floor

Area

Liquid a tank can hold

Volume

Material to cover the outside of a box

Surface area

🔑 Key tip

Pay attention to both the shape and the units. A circle with radius r might need 2πr for circumference or πr2 for area. A cylinder might need πr2h for volume or 2πr2 + 2πrh for surface area.

The numbers may be the same, but the question determines which formula you need.

Area, Perimeter, and Volume Review Quiz

Next: Unit Conversions

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