Checkpoint 6 Maths · Geometry and Measure · 6Gg.05 · 6Gg.07

Capacity, Volume and Surface Area

Understand the difference between volume and capacity, link cm³ to ml, find volume by counting cubes, and calculate surface area from the areas of the faces.

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In the curriculum

Understand the difference between capacity and volume. (6Gg.05)
Understand how the area of 2D shapes is linked to the surface area of 3D shapes. (6Gg.07)
This means: know that volume is the space a 3D object takes up and capacity is how much a container can hold, use the right units (cm³, m³, ml, l), find volume by counting cubes, and find the surface area of cubes, cuboids and prisms by adding the areas of their faces (the area of the net).

1. Key words

Word / phrase	Meaning / use
volume	الحجم — the amount of space a 3D object takes up
capacity	السعة — the amount a container can hold when it is full
cubic centimetre (cm³)	سنتيمتر مكعب — the volume of a cube 1 cm by 1 cm by 1 cm
cubic metre (m³)	متر مكعب — the volume of a cube 1 m by 1 m by 1 m
millilitre (ml) / litre (l)	مليلتر / لتر — units of capacity; 1000 ml = 1 l
container	وعاء — something that holds things (a jug, a bottle, a box, a tank)
surface area	مساحة السطح الكلية — the total area of all the faces of a 3D shape
net	شبكة المجسم — the flat shape that folds to make the 3D shape
layer	طبقة — one level of cubes in a cuboid

2. Volume: how much space

The volume of an object is the amount of space it takes up. A brick, a box, a stone and even you have a volume. We measure volume in cubic units: a cube with edges of 1 cm has a volume of 1 cm³ (one cubic centimetre). Big volumes, like a room, are measured in m³.

To find the volume of a shape made of centimetre cubes, count the cubes. For a cuboid, count in layers: count the cubes in one layer, then multiply by the number of layers.

One layer of the cuboid has 4 × 3 = 12 cubes. There are 2 layers, so the volume is 12 × 2 = 24 cm³.

Count how many cubes are along the length and the width of the bottom layer, and multiply (4 × 3 = 12).
Count the layers (2).
Multiply: 12 × 2 = 24 cubes, so the volume is 24 cm³.
Remember the cubes you cannot see: a cuboid is solid, so every layer is full.

Notice that 4 × 3 × 2 = 24: for a cuboid, multiplying the length, width and height gives the number of centimetre cubes.

3. Capacity: how much a container holds

The capacity of a container is the greatest amount it can hold when it is full: water in a bottle, juice in a carton, rice in a jar. Capacity is usually measured in millilitres (ml) and litres (l). 1 litre = 1000 ml.

This jug has a capacity of 1000 ml (1 litre). The labels go up in 200 ml and each small mark is 50 ml. The water level is 650 ml.

Find the difference between two labelled numbers (400 − 200 = 200 ml).
Count the spaces between them (4 spaces).
Divide: 200 ÷ 4 = 50 ml for each small mark.
Count up from the labelled number just below the level: 600 + 50 = 650 ml. Read the level at the flat surface of the liquid, with your eye level with it.

	Volume	Capacity
What it means	the space an object takes up	how much a container can hold
Which objects	any 3D object, solid or hollow	only containers (they are hollow and can be filled)
Usual units	cm³, m³	ml, l
Example	the volume of a brick is 1500 cm³	the capacity of a bottle is 1.5 l
Changes?	a stone's volume stays the same	a jug's capacity stays the same, but the volume of water in it can change

So a full 1 litre bottle has a capacity of 1 litre. If you pour some out, the capacity is still 1 litre, but the volume of water inside is now less. The volume of liquid in a container can never be more than its capacity (or it overflows).

الفرق بين الحجم والسعة

الحجم = المساحة اللي الجسم واخدها في الفراغ (cm³). أي جسم ليه حجم، حتى الطوبة والحجر.
السعة = الوعاء يشيل قد إيه لما يتملي (ml و l). الكوباية والزجاجة والخزان ليهم سعة.
افتكر: 1 cm³ = 1 ml و 1000 ml = 1 لتر.

4. Linking cm³ and ml

Volume and capacity units are linked: 1 cm³ of space holds exactly 1 ml of liquid. A hollow cube 10 cm by 10 cm by 10 cm has 10 × 10 × 10 = 1000 centimetre cubes of space, so it holds 1000 ml = 1 litre.

A 10 cm cube holds exactly 1 litre.

Volume	Capacity
1 cm³	1 ml
250 cm³	250 ml
1000 cm³	1000 ml = 1 l
2500 cm³	2500 ml = 2.5 l

Finding the volume of a stone (displacement). When you put a solid object into water, the water level goes up by the volume of the object, because the object pushes the water out of the way.

The water level rises from 300 ml to 380 ml, so the stone has a volume of 380 − 300 = 80 cm³.

5. Surface area: adding the areas of the faces

The surface area of a 3D shape is the total area of all its faces. Imagine wrapping the shape in paper with no overlaps: the amount of paper is the surface area. If you unfold the shape into its net, the area of the net = the surface area. Surface area is an area, so the units are cm² or m², not cm³.

Sketch the net, or list all the faces (do not forget the ones at the back and underneath).
Find the area of each face (rectangle: length × width; right-angled triangle: base × height ÷ 2).
Add all the areas. For a cuboid, opposite faces are identical, so find three different faces and double the total.
Write the answer in square units (cm²).

The net of a cuboid 5 cm by 3 cm by 2 cm. Faces: two of 15 cm², two of 10 cm², two of 6 cm². Surface area = 2 × (15 + 10 + 6) = 62 cm².

Cube: all 6 faces are identical squares, so surface area = 6 × area of one face. A cube with edges of 3 cm: one face = 3 × 3 = 9 cm², surface area = 6 × 9 = 54 cm².

Triangular prism: 2 triangle ends + 3 rectangles. If the ends are right-angled triangles, you can use base × height ÷ 2 for each end.

A triangular prism 10 cm long. Its ends are right-angled triangles with sides 3 cm, 4 cm and 5 cm.

	Area	Surface area	Volume	Capacity
What	space inside a flat 2D shape	total area of all faces of a 3D shape	space a 3D object takes up	amount a container holds
Units	cm², m²	cm², m²	cm³, m³	ml, l

Same volume, different surface area

8 centimetre cubes can be put together as a 2 × 2 × 2 cube (surface area 6 × 4 = 24 cm²) or as a long 8 × 1 × 1 rod (surface area 2 × (8 + 8 + 1) = 34 cm²). Both have a volume of 8 cm³, but the long rod has more surface showing. The more spread out the cubes are, the bigger the surface area.

6. Worked examples

Worked example 1 A cuboid is made of centimetre cubes. It is 4 cubes long, 3 cubes wide and 2 cubes high (see section 2). What is its volume? [1]

One layer: 4 × 3 = 12 cubes. Two layers: 12 × 2 = 24 cubes.
Volume = 24 cm³.
Common mistake: counting only the cubes you can see in the picture (some are hidden inside and at the back).

Worked example 2 Write volume or capacity for each sentence. [2] (a) The amount of water a fish tank can hold. (b) The amount of space a wooden block takes up. (c) A carton holds 250 ml of juice. (d) A stone takes up 80 cm³ of space.

(a) capacity (a container, how much it can hold)
(b) volume (a solid object, space it takes up)
(c) capacity (container, ml)
(d) volume (solid object, cm³)
Mark-scheme style: 2 marks for all four correct, 1 mark for two or three correct.

Worked example 3 Look at the jug in section 3. (a) How much water is in the jug? [1] (b) How much more water is needed to fill the jug to its capacity of 1 litre? [1]

(a) Each small mark is 200 ÷ 4 = 50 ml. The level is one mark above 600: 650 ml.
(b) 1 litre = 1000 ml. 1000 − 650 = 350 ml.
Common mistake: thinking each small mark is 10 ml or 100 ml without working it out.

Worked example 4 A container has 300 ml of water. Layla puts a stone in it and the level rises to 380 ml (see section 4). What is the volume of the stone? Give the units. [1]

The rise is 380 − 300 = 80 ml. The stone pushed out 80 ml of water, and 1 ml = 1 cm³.
Volume of the stone = 80 cm³.

Worked example 5 A cuboid is 5 cm long, 3 cm wide and 2 cm high. Calculate its surface area. [2]

Top and bottom: 5 × 3 = 15 cm² each. Front and back: 5 × 2 = 10 cm² each. Two ends: 3 × 2 = 6 cm² each.
Total: 15 + 15 + 10 + 10 + 6 + 6 = 62 cm² (or 2 × (15 + 10 + 6) = 62).
Mark-scheme style: 1 mark for at least two correct face areas; 1 mark for 62.
Common mistakes: adding only three faces (31 cm²), or working out the volume (5 × 3 × 2 = 30 cm³) instead.

Worked example 6 (calculator allowed) Calculate the surface area of the triangular prism in section 5. Its ends are right-angled triangles with sides 3 cm, 4 cm and 5 cm, and it is 10 cm long. [3]

Each triangle end: 4 × 3 ÷ 2 = 6 cm². Two ends: 12 cm².
Rectangles (one for each side of the triangle, each 10 cm long): 4 × 10 = 40, 3 × 10 = 30, 5 × 10 = 50 cm². Total 120 cm².
Surface area = 12 + 120 = 132 cm².
Common mistake: using the 5 cm side for the triangle's area. For the rectangles all three sides are used, because each side of the triangle has its own rectangle.

Common mistakes

Mixing up volume and capacity: capacity is only for containers; volume is for any 3D object.
Wrong units: volume in cm³ or m³, capacity in ml or l, surface area in cm² or m².
Counting only the visible cubes in a drawing of a cuboid (hidden cubes count too).
Reading scales without working out what one small mark is worth.
Forgetting that 1 l = 1000 ml (not 100 ml).
Leaving out faces when finding surface area: a cuboid always has 6 faces, a triangular prism 5.
Confusing surface area (cm², add face areas) with volume (cm³, count cubes).

How Checkpoint asks this

"Draw a ring around the best unit" or "Write the missing unit" (cm³, ml, l, cm²).
Sort statements or objects into volume / capacity, sometimes in a table or with matching lines.
Read the level of liquid on a measuring jug or cylinder; then "How much more is needed to fill it?"
"This shape is made of centimetre cubes. What is its volume?" Count layers; remember hidden cubes.
"Here is the net of a cuboid. Work out the surface area." Show the area of each face.
"Explain the difference between volume and capacity" or "Is Ahmed correct?" Use the words space and holds.
Converting between ml and l (2.5 l = 2500 ml) and using 1 cm³ = 1 ml.

7. Practice

Remember

Volume = space an object takes up (cm³, m³). Capacity = how much a container holds (ml, l).
1 litre = 1000 ml and 1 cm³ = 1 ml, so a 10 cm cube holds 1 litre.
Volume of a cuboid of cubes: cubes in one layer × number of layers.
Read a scale by working out what each small mark is worth first.
Surface area = total area of all the faces = area of the net (cm²). Cube: 6 × one face. Cuboid: 2 × (three different faces).
A rise in water level gives the volume of the object put in.

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