UNIT 9 • GEOMETRY & TRIGONOMETRY

Volume & Surface Area
Cylinders, Cones, Spheres and More

Volume is the space a solid fills, in cubic units; surface area is the total area of its outside, in square units. A cylinder, for example, has $V = \pi r^2 h$ and $TSA = 2\pi r(r + h)$ .

THE MOST-USED FORMULAS9 SOLIDS

\begin{gathered} \text{Cylinder: } V = \pi r^2 h \\ \text{Cone: } V = \tfrac{1}{3}\pi r^2 h \\ \text{Sphere: } V = \tfrac{4}{3}\pi r^3 \end{gathered}

1. Name the SolidMatch it to its formulas; a cone or pyramid needs the slant height for its surface.

2. Radius, not DiameterMost formulas use the radius

r

. Halve the diameter first.

3. UnitsVolume in cm³ (or litres), surface area in cm². Keep every length in the same unit.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A closed cylindrical tank has radius 7 cm and height 10 cm. Taking $\pi = \tfrac{22}{7}$ , find its volume, curved surface area and total surface area.

STEP 01 • VOLUME

V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 10 = 22 \times 7 \times 10 = 1540 \text{ cm}^3

STEP 02 • CURVED SURFACE AREA

The side of the tank unrolls into a rectangle $2\pi r$ wide and $h$ tall:

CSA = 2\pi r h = 2 \times \frac{22}{7} \times 7 \times 10 = 440 \text{ cm}^2

STEP 03 • TOTAL SURFACE AREA

Add the top and bottom circles:

TSA = 2\pi r(r + h) = 2 \times \frac{22}{7} \times 7 \times (7 + 10) = 44 \times 17 = 748 \text{ cm}^2

FINAL CONCLUSION

V = 1540 cm³, CSA = 440 cm², TSA = 748 cm²

V = 1540 \text{ cm}^3 = 1.54 \text{ litres}

The calculator below opens on this tank. Change π to see the answer with π kept exact.

LABORATORY

Interactive Volume & Surface Area Calculator

LABORATORY

Volume & Surface Area Engine

ShapeUnit

Radius rHeight h

Value of π

Common Questions

What is the difference between curved and total surface area?

Curved (or lateral) surface area is only the side surface. Total surface area adds the flat ends too: for a cylinder,

TSA = CSA + 2\pi r^2 = 2\pi r(r + h)

What is the formula for the volume of a cone?

V = \tfrac{1}{3}\pi r^2 h

: one third of the cylinder with the same base and height. Its curved surface area uses the slant height

l = \sqrt{r^2 + h^2}

CSA = \pi r l

What is the volume and surface area of a sphere?

V = \tfrac{4}{3}\pi r^3

and

S = 4\pi r^2

. A hemisphere has half the volume,

\tfrac{2}{3}\pi r^3

, and total surface area

3\pi r^2

(curved part plus the flat circle).

How do you convert between cm³ and litres?

1 litre = 1000 cm³, and 1 m³ = 1000 litres. So a 1540 cm³ tank holds 1.54 litres.

PreviousArea & Perimeter

Unit 9•40 of 47

NextPythagoras Theorem

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 9 • GEOMETRY & TRIGONOMETRY

Volume & Surface Area
Cylinders, Cones, Spheres and More

Volume is the space a solid fills, in cubic units; surface area is the total area of its outside, in square units. A cylinder, for example, has $V = \pi r^2 h$ and $TSA = 2\pi r(r + h)$ .

THE MOST-USED FORMULAS9 SOLIDS

\begin{gathered} \text{Cylinder: } V = \pi r^2 h \\ \text{Cone: } V = \tfrac{1}{3}\pi r^2 h \\ \text{Sphere: } V = \tfrac{4}{3}\pi r^3 \end{gathered}

1. Name the SolidMatch it to its formulas; a cone or pyramid needs the slant height for its surface.

2. Radius, not DiameterMost formulas use the radius

r

. Halve the diameter first.

3. UnitsVolume in cm³ (or litres), surface area in cm². Keep every length in the same unit.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

A closed cylindrical tank has radius 7 cm and height 10 cm. Taking $\pi = \tfrac{22}{7}$ , find its volume, curved surface area and total surface area.

STEP 01 • VOLUME

V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 10 = 22 \times 7 \times 10 = 1540 \text{ cm}^3

STEP 02 • CURVED SURFACE AREA

The side of the tank unrolls into a rectangle $2\pi r$ wide and $h$ tall:

CSA = 2\pi r h = 2 \times \frac{22}{7} \times 7 \times 10 = 440 \text{ cm}^2

STEP 03 • TOTAL SURFACE AREA

Add the top and bottom circles:

TSA = 2\pi r(r + h) = 2 \times \frac{22}{7} \times 7 \times (7 + 10) = 44 \times 17 = 748 \text{ cm}^2

FINAL CONCLUSION

V = 1540 cm³, CSA = 440 cm², TSA = 748 cm²

V = 1540 \text{ cm}^3 = 1.54 \text{ litres}

The calculator below opens on this tank. Change π to see the answer with π kept exact.

LABORATORY

Interactive Volume & Surface Area Calculator

LABORATORY

Volume & Surface Area Engine

ShapeUnit

Radius rHeight h

Value of π

Common Questions

What is the difference between curved and total surface area?

Curved (or lateral) surface area is only the side surface. Total surface area adds the flat ends too: for a cylinder,

TSA = CSA + 2\pi r^2 = 2\pi r(r + h)

What is the formula for the volume of a cone?

V = \tfrac{1}{3}\pi r^2 h

: one third of the cylinder with the same base and height. Its curved surface area uses the slant height

l = \sqrt{r^2 + h^2}

CSA = \pi r l

What is the volume and surface area of a sphere?

V = \tfrac{4}{3}\pi r^3

and

S = 4\pi r^2

. A hemisphere has half the volume,

\tfrac{2}{3}\pi r^3

, and total surface area

3\pi r^2

(curved part plus the flat circle).

How do you convert between cm³ and litres?

1 litre = 1000 cm³, and 1 m³ = 1000 litres. So a 1540 cm³ tank holds 1.54 litres.

PreviousArea & Perimeter

Unit 9•40 of 47

NextPythagoras Theorem

Volume & Surface Area Cylinders, Cones, Spheres and More

Step-by-Step Worked Example

V = 1540 cm³, CSA = 440 cm², TSA = 748 cm²

Interactive Volume & Surface Area Calculator

Volume & Surface Area Engine

Common Questions

NETZ Engine

Volume & Surface Area Cylinders, Cones, Spheres and More

Step-by-Step Worked Example

V = 1540 cm³, CSA = 440 cm², TSA = 748 cm²

Interactive Volume & Surface Area Calculator

Volume & Surface Area Engine

Common Questions

Volume & Surface Area
Cylinders, Cones, Spheres and More

Volume & Surface Area
Cylinders, Cones, Spheres and More