Volume & Area — Mathslive.ie

Section 1 of 5

Cube & cuboid — the box

Think of a box, or a room.

The base is a rectangle. Each side is a rectangle too.

$A$ here means surface area — the total area of all the faces.

Must learn

1.$V = \ell \times b \times h$

2.$A = 2(\ell b + \ell h + bh)$

(i) Worked example — a closed box

Length $20$ m, width $12$ m, height $15$ m. Find volume and surface area.

$\ell = 20, \quad b = 12, \quad h = 15$

$V = \ell \times b \times h$

$V = 20 \times 12 \times 15$

$V = 3600$ m³

$A = 2(\ell b + \ell h + bh)$

$A = 2\big(20(12) + 20(15) + 12(15)\big)$

$A = 1440$ m²

(ii) Worked example — an open-top box

An open-top box has length $12$ m, height $6$ m and width $8$ m. Find area and volume.

Open top means there is no lid — so leave that face out of the area.

$\ell = 12, \quad h = 6, \quad b = 8$

$V = \ell \times b \times h = 12 \times 6 \times 8$

$V = 576$ m³

For the area: one base $+$ four walls (no lid):

$A = \ell b + 2\ell h + 2bh$

$A = 12(8) + 2(12)(6) + 2(8)(6)$

$A = 432$ m²

YOU TRY · 1

A closed box has length $10$ m, width $4$ m and height $3$ m. Find the volume and the total surface area.

Write down $\ell$, $b$, $h$ first. Then $V = \ell b h$ and $A = 2(\ell b + \ell h + bh)$.

$V = 10 \times 4 \times 3 = 120$ m³

$A = 2\big(10(4) + 10(3) + 4(3)\big) = 2(82)$

$A = 164$ m²

$V = 120$ m³, $A = 164$ m²

YOU TRY · 2

An open-top box has length $5$ m, width $4$ m and height $2$ m. Find the area of material needed to make it.

No lid — one base and four walls: $A = \ell b + 2\ell h + 2bh$.

$A = 5(4) + 2(5)(2) + 2(4)(2)$

$A = 20 + 20 + 16$

$A = 56$ m²

$56$ m²

Section 2 of 5

Cylinder

A cylinder is a tube with a circle on the top and a circle on the bottom.

The two circles each have area $A = \pi r^2$. The curved part rolls out into a rectangle.

Must learn

1.Volume: $V = \pi r^2 h$

2.Curved area: $A = 2\pi r h$

3.Each end circle: $A = \pi r^2$

Why $2\pi r h$? Unroll the curved surface — it becomes a rectangle, $h$ tall and $2\pi r$ wide (the circumference):

(i) Curved area and volume

A cylinder has radius $5.3$ cm and height $4.3$ cm. Find the volume and the curved area, to one decimal place.

$r = 5.3, \quad h = 4.3$

$A = 2\pi r h = 2\pi (5.3)(4.3)$

$A = 143.2$ cm²

$V = \pi r^2 h = \pi (5.3)^2 (4.3)$

$V = 379.46\ldots$

$V = 379.5$ cm³

(ii) Closed tank — surface area and volume

A tank has radius $8$ m and height $15$ m. Find the surface area and volume, to the nearest metre.

$r = 8, \quad h = 15$

$A = 2\pi r h = 2\pi (8)(15)$

$A = 754$ m²

$V = \pi r^2 h = \pi (8)^2 (15)$

$V = 3016$ m³

(iii) Open-top cylinder (in terms of $\pi$)

A cylinder with a base but an open top has radius $6$ cm and height $12$ cm. Find area and volume in terms of $\pi$.

Open top — so the curved surface plus one circle (the base).

$r = 6, \quad h = 12$

$A = 2\pi r h + \pi r^2$

$A = 2\pi (6)(12) + \pi (6)^2$

$A = 144\pi + 36\pi$

$A = 180\pi$ cm²

$V = \pi r^2 h = \pi (6)^2 (12)$

$V = 432\pi$ cm³

(iv) Solid cylinder (in terms of $\pi$)

Radius $8$ m, height $10$ m. Find area and volume of a solid cylinder in terms of $\pi$.

Solid — so the curved surface plus two circles.

$r = 8, \quad h = 10$

$A = 2\pi r h + 2\pi r^2$

$A = 2\pi (8)(10) + 2\pi (8)^2$

$A = 160\pi + 128\pi$

$A = 288\pi$ m²

$V = \pi r^2 h = \pi (8)^2 (10)$

$V = 640\pi$ m³

(v) Volume, curved area, then total of a solid

A cylinder has radius $5.5$ m and height $2.8$ m. To one decimal place find (i) volume, (ii) curved area, (iii) total area of the solid.

$r = 5.5, \quad h = 2.8$

$V = \pi r^2 h = \pi (5.5)^2 (2.8)$

$V = 266.1$ m³

Curved: $A = 2\pi r h = 2\pi (5.5)(2.8)$

$A = 96.8$ m²

Solid — add on the $2$ circles:

$A = 2\pi r h + 2\pi r^2 = 2\pi(5.5)(2.8) + 2\pi(5.5)^2$

$A = 286.8$ m²

(vi) Working backwards — find the height

A cylinder of radius $5$ cm has a volume of $100\pi$ cm³. Find the height.

$r = 5, \quad h = \,?\,, \quad V = 100\pi$

Backwards — keep the unknown on the left.

$\pi r^2 h = V$

$\pi (5)^2 h = 100\pi$

$25h = 100$

$h = 4$ cm

(vii) Working backwards — find the radius

A cylinder of height $6$ m has volume $24\pi$ m³. Find the radius.

$h = 6, \quad V = 24\pi, \quad r = \,?$

$\pi r^2 h = V$

$\pi r^2 (6) = 24\pi$

$6 r^2 = 24$

$r^2 = 4$

$r = \sqrt{4} = 2$ m

Same idea with decimals. Height $2.3$ m, volume $256.4$ m³:

$\pi r^2 (2.3) = 256.4$

$2.3\,\pi\, r^2 = 256.4$

$r^2 = \dfrac{256.4}{2.3\,\pi} = 35.48$

$r = \sqrt{35.48} = 5.95$

$r = 6$ m

And once more. Height $3.9$ m, volume $678.3$ m³ (to one decimal place):

$\pi r^2 (3.9) = 678.3$

$r^2 = \dfrac{678.3}{3.9\,\pi} = 55.36$

$r = 7.4$ m

YOU TRY · 3

A solid cylinder has radius $4$ cm and height $10$ cm. Find the volume and total surface area, in terms of $\pi$.

Solid — curved plus two circles: $A = 2\pi r h + 2\pi r^2$.

$V = \pi (4)^2 (10) = 160\pi$ cm³

$A = 2\pi(4)(10) + 2\pi(4)^2 = 80\pi + 32\pi$

$A = 112\pi$ cm²

$V = 160\pi$ cm³, $A = 112\pi$ cm²

YOU TRY · 4

A cylinder of radius $7$ cm has a volume of $343\pi$ cm³. Find the height.

Backwards: $\pi r^2 h = V$, keep the unknown on the left.

$\pi (7)^2 h = 343\pi$

$49h = 343$

$h = 7$ cm

YOU TRY · 5

A cylinder of height $5$ m has a volume of $180\pi$ m³. Find the radius.

$\pi r^2 h = V$, then divide and take the square root.

$\pi r^2 (5) = 180\pi$

$5 r^2 = 180 \;\Rightarrow\; r^2 = 36$

$r = \sqrt{36} = 6$ m

$r = 6$ m

Section 3 of 5

Cone

A cone has a circular base and comes to a point.

The slant height $\ell$ is the distance up the side. It is not the vertical height $h$.

Must learn

1.Slant height: $\ell^2 = h^2 + r^2$

2.Volume: $V = \dfrac{1}{3}\pi r^2 h$

3.Curved area: $A = \pi r \ell$

(i) Worked example (in terms of $\pi$)

A cone has radius $3$ cm and height $4$ cm. Find (i) volume, (ii) curved area, (iii) total area when solid.

$r = 3, \quad h = 4$

$V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi (3)^2 (4)$

$V = 12\pi$ cm³

Find the slant height first:

$\ell^2 = h^2 + r^2 = 4^2 + 3^2 = 25$

$\ell = \sqrt{25} = 5$ cm

Curved: $A = \pi r \ell = \pi (3)(5)$

$A = 15\pi$ cm²

Total when solid — add the base circle:

$A = \pi r \ell + \pi r^2 = 15\pi + \pi (3)^2$

$A = 15\pi + 9\pi$

$A = 24\pi$ cm²

(ii) Worked example (in terms of $\pi$)

Radius $12$ m, height $5$ m. Find (i) volume, (ii) total area when solid.

$r = 12, \quad h = 5$

$V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi (12)^2 (5)$

$V = 240\pi$ m³

$\ell^2 = h^2 + r^2 = 5^2 + 12^2 = 169$

$\ell = \sqrt{169} = 13$ m

$A = \pi r \ell + \pi r^2 = \pi (12)(13) + \pi (12)^2$

$A = 300\pi$ m²

(iii) Worked example (one decimal place)

A solid cone has radius $3.5$ m and height $2.1$ m. Find the volume and area, to one decimal place.

$r = 3.5, \quad h = 2.1$

$V = \dfrac{1}{3}\pi (3.5)^2 (2.1)$

$V = 26.9$ m³

$\ell^2 = (2.1)^2 + (3.5)^2 = 16.66$

$\ell = \sqrt{16.66} = 4.1$ m

$A = \pi r \ell + \pi r^2 = \pi (3.5)(4.1) + \pi (3.5)^2$

$A = 83.4$ m²

YOU TRY · 6

A solid cone has radius $6$ cm and height $8$ cm. Find the volume, the slant height, and the total surface area, in terms of $\pi$.

$V = \tfrac13 \pi r^2 h$, then $\ell^2 = h^2 + r^2$, then $A = \pi r \ell + \pi r^2$.

$V = \dfrac{1}{3}\pi (6)^2 (8) = 96\pi$ cm³

$\ell^2 = 8^2 + 6^2 = 100 \;\Rightarrow\; \ell = 10$ cm

$A = \pi(6)(10) + \pi(6)^2 = 60\pi + 36\pi$

$A = 96\pi$ cm²

$V = 96\pi$ cm³, $\ell = 10$ cm, $A = 96\pi$ cm²

YOU TRY · 7

A cone has radius $5$ m and height $12$ m. Find the slant height and the curved surface area, in terms of $\pi$.

Slant first: $\ell^2 = h^2 + r^2$. Curved area is $\pi r \ell$ only.

$\ell^2 = 12^2 + 5^2 = 169 \;\Rightarrow\; \ell = 13$ m

$A = \pi r \ell = \pi (5)(13)$

$A = 65\pi$ m²

$\ell = 13$ m, curved $A = 65\pi$ m²

Section 4 of 5

Sphere

A sphere is a ball. One measurement only — the radius $r$.

Must learn

1.Volume: $V = \dfrac{4}{3}\pi r^3$

2.Area: $A = 4\pi r^2$

(i) Worked example (in terms of $\pi$)

Radius $12$ cm. Find the area and volume in terms of $\pi$.

$r = 12$

$A = 4\pi r^2 = 4\pi (12)^2$

$A = 576\pi$ cm²

$V = \dfrac{4}{3}\pi r^3 = \dfrac{4}{3}\pi (12)^3$

$V = 2304\pi$ cm³

(ii) Worked example (one decimal place)

Radius $6.3$ m. Find the area and volume, to one decimal place.

$r = 6.3$

$A = 4\pi (6.3)^2$

$A = 498.8$ m²

$V = \dfrac{4}{3}\pi (6.3)^3$

$V = 1047.4$ m³

(iii) Working backwards — find the radius

The volume of a sphere is $36\pi$ m³. Find $r$.

$V = 36\pi$

$\dfrac{4}{3}\pi r^3 = 36\pi$

$r^3 = 36 \div \dfrac{4}{3}$

$r^3 = 27$

$r = \sqrt[3]{27} = 3$ m

Same method with decimals. Volume $673.8$ m³ (to one decimal place):

$\dfrac{4}{3}\pi r^3 = 673.8$

$r^3 = 673.8 \div \dfrac{4}{3}\pi = 160.85$

$r = \sqrt[3]{160.85} = 5.43$

$r = 5.4$ m

YOU TRY · 8

A sphere has radius $3$ cm. Find the area and the volume, in terms of $\pi$.

$A = 4\pi r^2$ and $V = \tfrac{4}{3}\pi r^3$.

$A = 4\pi (3)^2 = 36\pi$ cm²

$V = \dfrac{4}{3}\pi (3)^3 = 36\pi$ cm³

$A = 36\pi$ cm², $V = 36\pi$ cm³

YOU TRY · 9

A sphere has a volume of $288\pi$ cm³. Find the radius.

$\tfrac{4}{3}\pi r^3 = V$. Divide out, then cube root.

$\dfrac{4}{3}\pi r^3 = 288\pi$

$r^3 = 288 \div \dfrac{4}{3} = 216$

$r = \sqrt[3]{216} = 6$ cm

$r = 6$ cm

Section 5 of 5

Hemisphere

A hemisphere is half a sphere.

Halve the sphere formulas. The curved area is half of $4\pi r^2$:

Must learn

1.Volume: $V = \dfrac{2}{3}\pi r^3$

2.Curved area: $A = 2\pi r^2$

3.Solid total: $A = 2\pi r^2 + \pi r^2 = 3\pi r^2$

When it is solid, add the flat circle on top ($\pi r^2$) to the curved part ($2\pi r^2$) — giving $3\pi r^2$.

(i) Worked example (in terms of $\pi$)

A solid hemisphere has radius $6$ cm. Find the area and volume in terms of $\pi$.

$r = 6$

$V = \dfrac{2}{3}\pi r^3 = \dfrac{2}{3}\pi (6)^3$

$V = 144\pi$ cm³

Area of a solid hemisphere — curved part plus the top circle:

$A = 2\pi r^2 + \pi r^2 = 3\pi r^2$

$A = 3\pi (6)^2$

$A = 108\pi$ cm²

YOU TRY · 10

A solid hemisphere has radius $3$ cm. Find the volume and the total surface area, in terms of $\pi$.

$V = \tfrac{2}{3}\pi r^3$. Solid total area $= 3\pi r^2$.

$V = \dfrac{2}{3}\pi (3)^3 = 18\pi$ cm³

$A = 3\pi (3)^2 = 27\pi$ cm²

$V = 18\pi$ cm³, $A = 27\pi$ cm²

Summary

Volume & Area toolkit

1.Cuboid: $V = \ell b h$, $A = 2(\ell b + \ell h + bh)$

2.Cylinder: $V = \pi r^2 h$, curved $2\pi r h$, each end $\pi r^2$

3.Cone: $V = \dfrac{1}{3}\pi r^2 h$, curved $\pi r \ell$, $\ell^2 = h^2 + r^2$

4.Sphere: $V = \dfrac{4}{3}\pi r^3$, $A = 4\pi r^2$

5.Hemisphere: $V = \dfrac{2}{3}\pi r^3$, solid total $3\pi r^2$

6.Open top: drop one circle. Solid: keep both.

7.Backwards: keep the unknown on the left, then divide and root.

End of lesson

Volume & Area — HL · Mathslive.ie