Mathematics notes, unit 6

Space

Textbook Mathematics Year 4 (DLP), pages 199–214

What this chapter covers

You can recognise right, acute and obtuse angles, and name the vertices and angles of a shape (6.1.1)
You can name equilateral, isosceles, scalene and right-angled triangles by their sides and angles (6.1.1)
You can identify and draw parallel lines and perpendicular lines (6.2.1, 6.2.2)
You can find the perimeter of squares, rectangles, triangles and regular polygons (6.3.1)
You can find the area of squares, rectangles and triangles in square units (6.3.2)
You can find the volume of cubes and cuboids in cubic units (6.4.1)
You can solve problems about perimeter, area and volume (6.5.1)

6.1Angles and triangles

An angle is the space between two straight lines that meet at a point. The point is called the vertex. A shape has as many angles as it has vertices: a square has 4 vertices and 4 angles.

Right angle: the corner of a square (a quarter turn).
Acute angle: smaller than a right angle.
Obtuse angle: bigger than a right angle.

Three angles: an acute angle, a right angle marked with a small square, and an obtuse angle — Acute, right and obtuse angles.

Paper folding helps. Fold a sheet of paper twice to make a right angle, then place its corner on any angle to see if that angle is smaller or bigger.

Types of triangle

We name triangles by their sides and their angles. The red marks on the sides show which sides are equal.

Equilateral, isosceles, scalene and right-angled triangles with equal sides marked — Equal sides have the same number of red marks.

Triangle	Sides	Angles
Equilateral	3 equal sides	3 acute angles
Isosceles	2 equal sides	3 acute angles, or 1 obtuse and 2 acute
Scalene	3 unequal sides	1 obtuse and 2 acute, or 3 acute
Right-angled	sides may differ	1 right angle and 2 acute angles

Do not say a triangle is "right" when you only see an angle that looks square. Check for the small square mark. A triangle can have only one right or obtuse angle.

A triangle has 3 sides of different lengths and one obtuse angle. What is its name?

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A scalene triangle: the three sides are unequal.

6.2Parallel lines and perpendicular lines

Parallel lines are always the same distance apart and never meet, however far they go.
Perpendicular lines cross each other at a right angle.

Two horizontal parallel lines, and two lines crossing at a right angle — Rails on a ladder are parallel. A window frame has perpendicular lines.

Drawing the lines

Parallel: place a ruler, draw a line along one edge, slide the ruler along the same direction, and draw a second line along the edge again.
Perpendicular: draw a horizontal line along a ruler. Hold a second ruler across it, or use the corner of a set square, and draw a line so that it makes a right angle.

Two lines that are not parallel may still not touch on the page. If you extend them far enough they will meet. Only parallel lines never meet. Also, crossing lines are perpendicular only when the angle is a right angle.

The top and bottom of a door are parallel. Which line is perpendicular to both?

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The side of the door (a vertical edge), because it meets the top and bottom at a right angle.

6.3Perimeter

Perimeter is the total length of all the sides of a shape, the distance all the way round its outline.

A rectangle 5 squares by 3 squares on a grid with the outline in red — 5 + 3 + 5 + 3 = 16 units. The length of one square is 1 unit.

Shape	Perimeter	Example
Square	4 × side	4 × 7 cm = 28 cm
Rectangle	2 × (length + width)	2 × (18 m + 10 m) = 56 m
Regular polygon with n sides	n × side	regular octagon, side 6 m: 8 × 6 m = 48 m
Other triangles and polygons	add all the sides	8 cm + 8 cm + 6 cm = 22 cm

Worked example: a regular pentagon with sides of 8 cm

Step 1. A regular pentagon has 5 equal sides.
Step 2. Perimeter = 5 × 8 cm = 40 cm.

The perimeter is a length, so its unit is cm or m, not cm² or m².

Find the perimeter of a rectangle 9 cm long and 4 cm wide.

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9 + 4 + 9 + 4 = 26 cm.

6.4Area

Area is the size of a surface. It is measured in square units: 1 square unit is a square with sides of 1 unit. We write it as unit², cm² or m².

A rectangle made of 12 unit squares in 4 columns and 3 rows — 4 units × 3 units = 12 square units.

Area of a rectangle = length × width
Area of a square = side × side
Area of a triangle = ½ × base × height

A rectangle can be cut into two equal triangles, so a triangle is half the area of the rectangle that has the same base and height.

A rectangle 8 cm by 6 cm cut by a diagonal; the lower triangle is shaded — Rectangle 8 cm × 6 cm = 48 cm². The triangle is half of it: 24 cm².

Worked example: a blue wall 6 m long and 3 m wide

Area = 6 m × 3 m = 18 m².
Two ways for a triangle with base 8 cm and height 6 cm: ½ × 8 × 6 = 24 cm², or 48 cm² ÷ 2 = 24 cm².

The height of a triangle is the straight distance at a right angle to the base, not the slanted side. Use cm × cm = cm², and m × m = m².

Find the area of a triangle with base 10 cm and height 7 cm.

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½ × 10 × 7 = 70 ÷ 2 = 35 cm².

6.5Volume

Volume is the amount of space a solid takes up. It is measured in cubic units: 1 unit³ is a cube with edges of 1 unit.

A block of unit cubes 3 long, 2 wide and 2 high — 3 × 2 × 2 = 12 cubes, so the volume is 12 unit³.

Volume of a cuboid = length × width × height
Volume of a cube = side × side × side

A cuboid 6 metres long, 2 metres wide and 3 metres high — 6 m × 2 m × 3 m = 36 m³.

Volume = base area × height. If the base area is 12 m² and the height is 2 m, the volume is 12 × 2 = 24 m³.

Worked example: how many cubes fit in a box?

A box is 16 cm × 12 cm × 8 cm. Each cube has sides of 4 cm.
Step 1. Volume of box = 16 × 12 × 8 = 1 536 cm³.
Step 2. Volume of one cube = 4 × 4 × 4 = 64 cm³.
Step 3. Number of cubes = 1 536 ÷ 64 = 24.

Volume uses three lengths multiplied together and ends in cm³ or m³. Do not confuse it with area (cm²) or perimeter (cm).

Find the volume of a cube with sides of 5 cm.

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5 × 5 × 5 = 125 cm³.

6.6Solving problems

Use the four steps: understand the problem, plan a strategy (draw a diagram), solve, then check.

Worked example: a square cow farm

Each side of a square farm is 16 m. Find the length of fence needed and the area of the farm.
Perimeter = 4 × 16 m = 64 m.
Area = 16 m × 16 m = 256 m².
Check: 256 ÷ 16 = 16. Correct.

Asked for	Think	Unit
Fence, ribbon, edging	perimeter: add the sides	cm, m
Paint, tiles, cloth	area: length × width	cm², m²
Space inside a box	volume: length × width × height	cm³, m³

The perimeter of a square card is 40 cm. Find its area.

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Side = 40 ÷ 4 = 10 cm. Area = 10 × 10 = 100 cm².

Notes follow Mathematics Year 4 (DLP), Unit 6, pp.199–214; figures redrawn.