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Year 8AC9M8M02

Volume and Surface Area of Prisms

Solve problems involving the volume and capacity of right prisms using appropriate units.

Volume fills the inside. Surface area wraps the outside. Two different questions about the same box.

Builds onYr 4 · Area and perimeter

Method

Volume of a prism
V=area of cross-section×lengthV = \text{area of cross-section} \times \text{length}
For a box, that is $l \times w \times h$.
Surface area of a box
SA=2(lw+lh+wh)SA = 2(lw + lh + wh)
Six faces, in three matching pairs.
Units
V in cm3,SA in cm2V \text{ in cm}^{3}, \quad SA \text{ in cm}^{2}
Volume is cubed, surface area is squared.

Worked example

e.g. a box 5 cm×3 cm×4 cm\text{a box } 5 \text{ cm} \times 3 \text{ cm} \times 4 \text{ cm}
  1. Label the three edgesl=5, w=3, h=4l = 5,\ w = 3,\ h = 4
    A rectangular box with edges five, three and four centimetres5 cm4 cm3 cm
  2. For volume, multiply all threeV=5×3×4V = 5 \times 3 \times 4
  3. Work it out. Units are cubedV=60 cm3V = 60 \text{ cm}^{3}
  4. For surface area, find the three different faceslw=15, lh=20, wh=12lw = 15,\ lh = 20,\ wh = 12
  5. There are two of each, so add them and doubleSA=2(15+20+12)SA = 2(15 + 20 + 12)
  6. Work it out. Units are squaredSA=94 cm2SA = 94 \text{ cm}^{2}

Common mistakes

WrongSA=lw+lh+whSA = lw + lh + wh
RightSA=2(lw+lh+wh)SA = 2(lw + lh + wh)
A box has six faces, not three. Every face has a matching one on the opposite side.
WrongV in cm2V \text{ in cm}^{2}
RightV in cm3V \text{ in cm}^{3}
Volume fills space in three directions, so the units are cubed.
WrongAdding the three edges for volume\text{Adding the three edges for volume}
RightMultiply them\text{Multiply them}
5+3+4=125 + 3 + 4 = 12 is not a volume. Volume counts unit cubes, so you multiply.

Practice

Find the volume and the surface area:

1
2×3×42 \times 3 \times 4
AnswerV=24, SA=52V = 24,\ SA = 52
2
10×5×210 \times 5 \times 2
AnswerV=100, SA=160V = 100,\ SA = 160
3
6×6×66 \times 6 \times 6
AnswerV=216, SA=216V = 216,\ SA = 216
4
8×3×58 \times 3 \times 5
AnswerV=120, SA=158V = 120,\ SA = 158
5
12×4×312 \times 4 \times 3
AnswerV=144, SA=192V = 144,\ SA = 192
6
7×7×27 \times 7 \times 2
AnswerV=98, SA=154V = 98,\ SA = 154
7
A cube has V=125. Find its edge\text{A cube has } V = 125. \text{ Find its edge}
Answer55
8
A box is 4×5×h with V=100. Find h\text{A box is } 4 \times 5 \times h \text{ with } V = 100. \text{ Find } h
Answer55
Next step
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