Mathematics notes, unit 6

Space

Textbook Mathematics Year 5 (DLP), Part 1 and Part 2, pages Part 2, 217–234

What this chapter covers

You can name regular polygons and state their characteristics, including sides, corners, interior angles, symmetrical axes and diagonals (6.1.1)
You can measure the interior angles of regular polygons with a protractor (6.2.1)
You can find the perimeter of composite shapes made from two regular polygons (6.3.1)
You can find the area of composite shapes made from squares, rectangles and triangles (6.3.2)
You can find the volume of composite shapes made from cubes and cuboids (6.4.1)
You can solve everyday problems about the perimeter, area and volume of composite shapes (6.5.1)

Space is about shapes. In this unit you study regular polygons, measure their interior angles with a protractor, and then find the perimeter, area and volume of shapes that are made by joining two simpler shapes together (composite shapes).

Plan for a composite shape. Split it into shapes you already know (rectangle, square, triangle, cuboid, cube). Work out each part. Then add the parts together, or subtract the piece that was taken away.

6.1Regular polygons

A polygon is a closed shape made of straight sides. A regular polygon has all sides equal in length and all interior angles equal in size.

A corner (also called a vertex) is where two sides meet. An interior angle is the angle inside the shape at a corner. A diagonal is a straight line drawn from one corner to another corner that is not next to it.

Six regular polygons: equilateral triangle, square, pentagon, hexagon, heptagon and octagon, each labelled with its number of sides — Six regular polygons. The number of sides always equals the number of corners and the number of interior angles.

Regular or not regular?

Both conditions must be true. A rectangle that is not a square has equal angles but its sides are not all equal. A rhombus that is not a square has equal sides but its angles are not all equal. So neither is a regular polygon.

A regular hexagon is regular; a rectangle, a rhombus and an uneven pentagon are not regular — Only the hexagon has equal sides and equal angles.

A shape with equal sides is not always regular. Check the angles too. A square is regular, but a rectangle or a rhombus that is not a square is not.

Characteristics of regular polygons

Regular hexagon ABCDEF with all nine diagonals drawn — A hexagon has 6 corners and 9 diagonals.

Regular pentagon with its five dashed lines of symmetry — A pentagon has 5 symmetrical axes (lines of symmetry).

Regular polygon	Sides	Corners	Interior angles	Symmetrical axes	Diagonals
Equilateral triangle	3	3	3	3	0
Square	4	4	4	4	2
Pentagon	5	5	5	5	5
Hexagon	6	6	6	6	9
Heptagon	7	7	7	7	14
Octagon	8	8	8	8	20

Patterns to remember. Sides = corners = interior angles = symmetrical axes. The diagonals go up by 2, 3, 4, 5, 6 as the polygon gets one more side: 0, 2, 5, 9, 14, 20. To check a number of diagonals, each corner joins to (sides − 3) other corners, so diagonals = sides × (sides − 3) ÷ 2. For a hexagon: 6 × 3 ÷ 2 = 9.

Worked example: name the polygon

A regular polygon has 7 corners and 14 diagonals.
7 corners means 7 sides, so it is a regular heptagon. Check: 7 × 4 ÷ 2 = 14 diagonals.

a) How many symmetrical axes does a regular octagon have? b) A regular polygon has 5 diagonals. Name it. c) Is a rhombus that is not a square a regular polygon? Why?

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a) 8 (same as its number of sides).
b) Pentagon (5 sides, 5 diagonals).
c) No. Its sides are equal but its interior angles are not all equal.

6.2Measuring interior angles

A protractor measures angles in degrees (°). It has an inner scale and an outer scale, a baseline and a centre.

A protractor placed on one interior angle of a regular pentagon ABCDE: the centre is on corner A, the baseline lies along side AE and the reading is 108 degrees — Measuring angle a of the regular pentagon: centre on corner A, baseline along AE. The reading is 108°.

Figure: Mathematics Year 5 (DLP) textbook, p. 219.

Place the centre of the protractor on the corner of the angle.
Line up the baseline with one arm of the angle, so that arm points to a 0° mark.
Start reading from 0° on the scale that begins at that arm, and follow the same scale to the other arm.
Write the value with the degree sign.

Two scales. The two scales give two numbers that add up to 180°. Decide first if the angle is smaller or larger than a right angle (90°), then choose the reading that fits. The angle a in the pentagon is more than 90°, so the correct reading is 108°, not 72°.

Types of angle

Type	Size
Acute angle	more than 0° and less than 90°
Right angle	exactly 90°
Obtuse angle	more than 90° and less than 180°

Every interior angle of a regular polygon is the same size. Measure just one:

Regular polygon	Each interior angle	Type
Equilateral triangle	60°	acute
Square	90°	right angle
Pentagon	108°	obtuse
Hexagon	120°	obtuse
Octagon	135°	obtuse

The more sides a regular polygon has, the larger each interior angle becomes. Your own measurement may be 1° or 2° away from these values. That is fine.

The protractor reading on the outer scale for one arm of an angle is 40°, and the other arm is on the baseline at 0°. What is the angle, and what type is it?

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The angle is 40°. It is less than 90°, so it is an acute angle.

6.3Perimeter and area of composite shapes

A composite shape is a shape made by joining two or more simpler shapes. Its perimeter is the distance round the outside of the shape. Its area is the space inside, measured in square units (cm², m²).

Perimeter

A square with a regular pentagon on top, all sides 9 cm, joined along a dashed side — A square and a regular pentagon share one side (dashed). Only the outer sides are counted.

Worked example

The square has 4 sides and the pentagon has 5 sides. They share one side, which is inside the shape.
Outer sides = 4 + 5 − 2 = 7 sides (the shared side is removed from each polygon).
Perimeter = 7 × 9 cm = 63 cm

Do not count the joining side. Counting all 9 sides gives 81 cm, which is wrong. Only the outside edges are part of the perimeter.

For a shape with sides of different lengths, label every outer side, then add them one by one starting from a corner. Tick each side as you add it so none is missed or counted twice.

Area

A rectangle 6 m wide and 4 m high with a right-angled triangle of base 6 m and height 3 m on top; the left side is 7 m — Rectangle 6 m by 4 m, with a right-angled triangle on top. The dashed line separates the two parts.

Shape	Area
Rectangle	length × breadth
Square	side × side
Triangle	½ × base × height

Worked example: add the parts

Rectangle: 6 m × 4 m = 24 m²
Triangle: ½ × 6 m × 3 m = 9 m²
Total area = 24 m² + 9 m² = 33 m²

A 12 m by 9 m rectangle with a 4 m by 3 m rectangle removed from the top right corner — A 12 m by 9 m rectangle with a 4 m by 3 m corner cut away (dashed).

Worked example: subtract the missing piece

Whole rectangle: 12 m × 9 m = 108 m²
Piece removed: 4 m × 3 m = 12 m²
Shaded area = 108 m² − 12 m² = 96 m²

Height of a triangle. The height is the straight-up distance from the base, at a right angle to it. It is not the slanted side.

A square of side 5 cm has a right-angled triangle of base 5 cm and height 4 cm joined to one side. Find the area of the composite shape.

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Square: 5 × 5 = 25 cm²
Triangle: ½ × 5 × 4 = 10 cm²
Total: 25 + 10 = 35 cm²

6.4Volume of composite shapes

Volume is the amount of space a solid takes up. It is measured in cubic units (cm³, m³). Volume of a cuboid = length × breadth × height. A cube is a cuboid with all edges equal, so its volume is side × side × side.

A cuboid 9 cm long, 4 cm deep and 2 cm high with a 4 cm by 4 cm by 3 cm block on top — Two cuboids joined together.

Worked example: add the parts

Lower cuboid: 9 cm × 4 cm × 2 cm = 72 cm³
Upper cuboid: 4 cm × 4 cm × 3 cm = 48 cm³
Volume of the composite shape = 72 cm³ + 48 cm³ = 120 cm³

A block 8 cm long, 5 cm deep and 6 cm high with a piece 3 cm long and 2 cm high cut off the top right — An 8 cm by 5 cm by 6 cm block with a piece cut away. The piece that was cut away was 3 cm long and 2 cm high.

Worked example: subtract the piece removed

Whole block: 8 × 5 × 6 = 240 cm³
Piece cut away: 3 × 5 × 2 = 30 cm³
Remaining volume = 240 cm³ − 30 cm³ = 210 cm³
Check by adding the two parts that are left: 8 × 5 × 4 = 160 cm³ and 5 × 5 × 2 = 50 cm³; 160 + 50 = 210 cm³.

The volume of a composite shape stays the same however the parts are arranged, as long as no part is added or taken away.

Units. Perimeter is in cm or m. Area is in cm² or m². Volume is in cm³ or m³. Check the unit before you write the answer.

A cuboid measures 10 cm × 8 cm × 4 cm. A cube of side 4 cm is cut out of it. What is the volume left?

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Cuboid: 10 × 8 × 4 = 320 cm³
Cube: 4 × 4 × 4 = 64 cm³
Left: 320 − 64 = 256 cm³

6.5Solving problems

The textbook uses four steps. Understand the problem, plan the strategy, solve, then check.

Worked example: fence round a pond

A pond is made of a regular octagon with sides of 6 m joined to a rectangle 15 m long and 6 m wide. One 6 m side of the rectangle is also a side of the octagon. A fence goes round the outside of the pond. How long is the fence? (The textbook, p. 228, solves a similar problem.)

Understand: we need the length round the outside of the pond.
Plan: outer sides of the octagon + outer sides of the rectangle, leaving out the shared side.
Solve: octagon: 8 sides, 1 shared, so 7 × 6 m = 42 m. Rectangle: 15 m + 15 m + 6 m = 36 m (the other 6 m side is shared).
Fence = 42 m + 36 m = 78 m
Check: octagon 8 × 6 = 48 m, rectangle 2 × (15 + 6) = 42 m. 48 + 42 − (2 × 6) = 78 m. ✓

Worked example: is the money enough?

A rectangular path 7 m by 3 m and another rectangular path 5 m by 2 m are to be tiled. One square metre costs RM80. Mak has RM2 000.
Area = 7 × 3 + 5 × 2 = 21 + 10 = 31 m²
Cost = 31 × RM80 = RM2 480
RM2 480 is more than RM2 000, so the money is not enough. She needs RM480 more.

In a cost problem, always do three things: find the area (or length), multiply by the price for each unit, then compare with the money and write a sentence.

A garden 10 m by 6 m has a pond 3 m by 2 m inside it. Grass costs RM12 for each m². How much does the grass cost for the part without the pond?

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Garden: 10 × 6 = 60 m²
Pond: 3 × 2 = 6 m²
Grass area: 60 − 6 = 54 m²
Cost: 54 × RM12 = RM648

Notes follow Mathematics Year 5 (DLP), Unit 6, pp. 217-234. Worked examples use numbers made for these notes.