Grade 7-12 Math Review · Grade 11-12 · Solid Mensuration
Surface Area and Volume of a Prism
One idea covers every prism there is: identical layers, stacked. The shape of the base never changes the method.
How much will it hold?
A rectangular water tank measures 1.2 m by 0.8 m by 0.5 m. How many litres does it hold when full?
V = 1.2 × 0.8 × 0.5 = 0.48 m³, and one cubic metre is 1 000 litres, so the tank holds 480 L. A stops at 0.48 and reads the cubic metres as litres. B and D slip one place in the conversion. The useful anchor: 1 m³ = 1 000 L and 1 000 cm³ = 1 L.
1
Volume is the base area, stacked h deep.
V = B h
B is the area of one base, not both, and h is the perpendicular distance between
the two bases. Because every cross-section of a prism is the same shape, the solid is
just that shape repeated — so multiply.
Rectangular prismV = ℓwhB = ℓw
Triangular prismV = (½bh₃)HB = ½bh₃, the triangle’s own height
CubeV = s³B = s² and h = s
2
Find B first, always, and write it down.
B = ½(6)(8) = 24 cm²
the triangle, on its own
V = B h = 24(15)
stack it 15 cm deep
V = 360 cm³
cm² × cm gives cm³
Two different heights are in play and they are easy to confuse: the triangle’s own
height inside B, and the prism’s length between the two bases. Naming them
differently on your page — h₃ for the triangle, h for the prism — costs
one second and saves the question.
3
Lateral area is one long rectangle.
LSA = p h TSA = p h + 2B
Unroll the sides of a prism and they form a single rectangle: its height is h, and its
width is the whole way round the base, which is the perimeter p. Then cap it with the two
bases. That is the entire derivation.
A triangular base: a prism 5 cm long stands on a right triangle with legs 7 cm and 24 cm. Find the TSA.
hypotenuse = √7² + 24² = √625 = 25 cm
the perimeter needs all three sides
p = 7 + 24 + 25 = 56, so LSA = 56 × 5 = 280 cm²
p h, the three rectangles
B = ½(7)(24) = 84, so 2B = 168 cm²
the two triangular ends
TSA = 280 + 168 = 448 cm²
p h + 2B
4
A worked total surface area, both ways.
p = 2(20 + 12) = 64 cm
round the 20 × 12 base
LSA = p h = 64(9) = 576 cm²
the four upright faces
2B = 2(20)(12) = 480 cm²
top and bottom
TSA = 576 + 480 = 1 056 cm²
check: 2(240 + 180 + 108) = 1 056 ✓
The check on the last line is the familiar 2(ℓw + ℓh + wh) for a box. Two
routes agreeing is worth the extra line, and for a box it takes one. A cube is the box
with all three edges equal to s: six square faces, so TSA = 6s².
5
Units: squared for surface, cubed for volume.
From
To
Multiply by
1 m
cm
100
1 m²
cm²
100² = 10 000
1 m³
cm³
100³ = 1 000 000
1 m³
litres
1 000
1 000 cm³
litres
1
Converting the length and then cubing is not the same as cubing and then converting
A cube of side 2 m has volume 8 m³. Convert first: 2 m = 200 cm, and
200³ = 8 000 000 cm³. Convert last: 8 m³ × 1 000 000 =
8 000 000 cm³ ✓. Multiplying 8 by only 100 gives 800 cm³, which is smaller than
the cube’s own face — a sanity check that catches it every time.
6
Backwards: the volume is given, a length is asked.
V = B h still holds when V is the number you know. Substitute everything you have, then
divide — and if the volume is in litres, turn it into cm³ first.
A rectangular container holds 6 L. Its base is 25 cm by 16 cm. How deep is it?
6 L = 6 000 cm³
1 L = 1 000 cm³
B = 25 × 16 = 400 cm²
the base on its own
6 000 = 400 h
substitute into V = B h
h = 15 cm
check: 25 × 16 × 15 = 6 000 ✓
Scaling: every edge of a 2 cm by 3 cm by 4 cm box is tripled. What happens to the volume?
Old V = 2 × 3 × 4 = 24 cm³
before
New V = 6 × 9 × 12 = 648 cm³
every length times 3
648 ÷ 24 = 27 = 3³
three lengths, each tripled
Scale every length by k and areas grow by k², volumes by k³. The box above gains 3² = 9 times the surface area.
Tripling the sides does not triple the volume
It is tempting to answer “3 times as big, 72 cm³”. But the box got three times
longer, three times wider and three times taller, so the volume is multiplied by
3 × 3 × 3 = 27. Multiplying the new edges out, 6 × 9 × 12 = 648, settles it.
7
Variations: a trapezoid base, an irrational area, a box with no lid.
Same B first, same p h; only the base changes.
A trapezoid base: parallel sides 5 cm and 9 cm, 4 cm apart; prism 10 cm long. Find V.
B = ½(5 + 9)(4) = 28 cm²
trapezoid area
V = 28 × 10 = 280 cm³
stacked 10 cm deep
The 4 cm is the trapezoid’s own height, not the prism’s length.
An equilateral triangle base of side 6 cm; prism 10 cm long. Find V.
B = (√34)(6²) = 9√3 cm²
equilateral triangle area
V = 9√3 × 10 = 90√3 cm³
about 155.9 cm³
Keep the √3 to the end; round only if a decimal is asked.
An open box (no lid) 30 cm by 20 cm by 10 cm tall. How much cardboard?
LSA = 2(30 + 20)(10) = 1 000 cm²
p h, the four sides
one base = 30 × 20 = 600 cm²
a bottom, but no top
total = 1 600 cm²
p h + B, not p h + 2B
Read the object: an open tank, a box without a lid or a tent with no floor drops a face.
On the exam
Exams ask for a volume or surface area from given edges, then turn it round: a capacity in litres, a missing depth from a known volume, or what happens when every length is doubled. The fastest route is B first, then V = B h or TSA = p h + 2B. The 10-second check is the units: cm² for surface, cm³ for volume, and 1 000 cm³ to the litre.
Now you try
Practice problem
Find the total surface area
A prism is 10 cm long. Its base is a right triangle with legs 3 cm and 4 cm.
The hypotenuse is 5, so p = 3 + 4 + 5 = 12 cm and LSA = 12(10) = 120 cm². One triangle is ½(3)(4) = 6 cm², so 2B = 12 cm². TSA = 120 + 12 = 132 cm². A stops at the lateral area and never caps the ends. C adds only one triangle. D is the volume, 6 × 10, which is a different question and carries cm³.
Hint: the perimeter needs all three sides, so find the hypotenuse before
you start. Then write LSA and 2B on separate lines — almost every wrong answer here
is one of those two left out.
