Mathematics notes, unit 6

Space

Textbook Mathematics Year 6 SK (DLP, Semakan 2017), pages 165-186

What this chapter covers

You can draw and measure angles up to 180° with a protractor and a ruler (6.1.2)
You can draw regular polygons with up to eight sides and measure their interior angles (6.1.1)
You can name the centre, circumference, radius and diameter of a circle and relate the radius to the diameter (6.2.1)
You can draw circles of a given radius with a modified ruler, a pair of pencils or a round object (6.2.2)
You can solve problems about regular polygons and circles in daily life (6.3.1)

6.1Angles and regular polygons

Forming angles

An angle is formed when two straight lines meet at a point. In Year 6 you form angles up to 180° with a protractor and a ruler.

Four angles: 50 degrees acute, 90 degrees right, 108 degrees obtuse, 180 degrees straight — An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, and a straight angle is 180°.

Worked example: form a 50° angle at A

Step 1. Draw a straight line and mark two points, A and B.
Step 2. Place the baseline of the protractor exactly on AB and its centre exactly on A.
Step 3. Start at 0° on the scale that begins on line AB. Count up to 50°. Mark the point C.
Step 4. Remove the protractor. Join A and C with a ruler. Label the angle 50°.

A protractor has two scales. Start reading from the 0° that lies on your baseline. A 50° angle is acute, so it must look small. If your angle looks like 130°, you read the wrong scale.

Steps follow the Mathematics Year 6 (DLP) textbook, pp. 165-167.

What type of angle is 108°? Which scale of the protractor would you read to form it?

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It is an obtuse angle (more than 90°, less than 180°). Read the scale whose 0° is on the baseline.

Regular polygons

A regular polygon has all sides equal and all interior angles equal. An interior angle is an angle inside the shape at a corner.

Regular polygon	Sides	Each interior angle
Equilateral triangle	3	60°
Square	4	90°
Pentagon	5	108°
Hexagon	6	120°
Heptagon	7	about 128.5°
Octagon	8	135°

Ways to draw them

Triangle grid paper: join three points the same number of units apart. Every angle is 60°.
Ruler and protractor: draw a side, then measure the interior angle at each end (for a pentagon, 108° at A and at B) and mark the next side the same length.
Hexagon: join six equilateral triangles at one point, or mark six points the same distance apart.
Dotted or square grid paper: good for a square.

A regular hexagon split into six equilateral triangles — Six equilateral triangles make a regular hexagon.

Worked example: the photo frame

Anding makes a six-sided photo frame with equal sides. The perimeter is 120 cm.
Step 1. Six equal sides, so divide the perimeter by 6.
Step 2. 120 ÷ 6 = 20 cm for each side.

Perimeter of a regular polygon = number of sides × length of one side. A regular pentagon with 5 cm sides has perimeter 5 × 5 = 25 cm.

"Equal sides" is not enough. A slanted four-sided shape can have four equal sides, but its angles are not all 90°, so it is not a regular polygon.

A regular octagon has sides of 5 m. Find its perimeter and one interior angle.

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Perimeter = 8 × 5 = 40 m. Each interior angle is 135°.

6.2Circles

Circle with centre O, one radius, one diameter and the circumference labelled — The parts of a circle.

Centre: the point in the middle, usually called O.
Circumference: the curved line that forms the boundary of the circle.
Radius: a straight line from the centre to the circumference.
Diameter: a straight line between two points on the circumference that passes through the centre.

All the radii of one circle are equal. The diameter is twice the radius, so radius = diameter ÷ 2.

Worked example

The diameter of a circle is 7 cm. The radius is 7 ÷ 2 = 3.5 cm. A circle with radius 4 cm has a diameter of 4 × 2 = 8 cm.

Do not mix up radius and diameter. A diameter must pass through the centre. A line between two points on the circumference that misses the centre is not a diameter.

Drawing a circle

Modified ruler: put one pencil at the 0 cm hole (the centre) and the other at the radius hole, for example 4 cm. Keep the centre still and move the other pencil round once.
Round object: trace a cup. To find its centre, cut the circle out and fold it in four. The folds cross at the centre.
Label the centre and draw and label one radius.

A compass is not used in this chapter. The two-pencil trick works the same way: the pencil at the centre must not move.

A circle has a diameter of 18 cm. What is its radius? What radius does a circle of diameter 9.4 cm have?

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18 ÷ 2 = 9 cm. 9.4 ÷ 2 = 4.7 cm.

6.3Solving problems

Understand the problem, plan a strategy, solve, then check. Draw a plan when the question asks for a shape in a space.

Worked example: a circle cut from a square plank

Chee Cheong has a 25 cm × 25 cm plank. He wants a circle of radius 8 cm.
Step 1. Diameter = 8 × 2 = 16 cm.
Step 2. 16 cm is less than 25 cm, so the circle fits. The space left across the plank is 25 − 16 = 9 cm.

Think smart: two polygons

Charles's regular polygon has 2 times as many sides as Raymond's. With up to eight sides, the pairs are: Raymond triangle (3) and Charles hexagon (6); Raymond square (4) and Charles octagon (8).

A cake is baked in a tin with a radius of 10 cm. What is the smallest square paper side that covers the base?

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Diameter = 20 cm, so the paper must be at least 20 cm × 20 cm.

Summary

Form an angle with a protractor: baseline on the line, centre on the vertex, read from 0°. Angles go up to 180°.
Regular polygons have equal sides and equal angles: 60°, 90°, 108°, 120°, about 128.5° and 135° for 3 to 8 sides.
Perimeter of a regular polygon = sides × side length.
A circle has a centre, a circumference, radii and diameters. Diameter = 2 × radius.
Draw a circle with a modified ruler or two pencils; the centre pencil stays still.