Course

Cambridge O Level Mathematics: Geometry and Mensuration (4024)

Name: Cambridge O Level Mathematics: Geometry and Mensuration (4024)
Price: 1699 BDT

Sections 4 and 5 of 4024 in plain steps: geometry words, constructions and nets, angle facts with reasons, symmetry, bearings, similarity, the circle theorems, then units, areas, circles, and the surface area and volume of solids.

100 lessons18h 4mEnglish

What you'll learn

Use and explain the geometry words the exam uses, from a reflex angle to a frustum
Measure and draw lines and angles, construct a triangle from three sides, and draw and use nets
Find unknown angles in several steps and give the correct reason for each one
Draw and read scale drawings and give three-figure bearings, including back bearings
Show that two triangles are similar, and use a scale factor on a length, an area or a volume
Use every circle theorem and circle symmetry property in the syllabus, with its reason
Change between metric units of length, mass, area, volume and capacity
Find perimeters, areas, arc lengths and sector areas, with exact answers in terms of π when asked
Find the surface area and volume of any solid in the syllabus, including compound solids and a frustum, and work back to a missing length

Requirements

Number and Algebra at O Level standard, or our courses on them: substituting into a formula and solving a simple equation
Pythagoras' theorem, as taught in our Trigonometry course, for the chord and cone lessons
A ruler, a protractor, a pair of compasses and a calculator

Course content

100 lessons · 18h 4m

4.1 · Geometrical terms

The words every geometry question uses: points, lines, planes and angles, then triangles, special quadrilaterals, polygons, similar and congruent shapes, solids and the parts of a circle.

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Points, lines and planes

A point marks a place, a line goes on for ever, a plane is a flat surface. Parallel lines never meet; perpendicular lines meet at 90°.

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Acute, obtuse, reflex and right angles

Angles are named by size, from acute to reflex, and by three letters such as angle ABC, with the vertex in the middle.

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The perpendicular bisector

The line that cuts a segment in half at right angles. Every point on it is the same distance from both ends.

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Triangles: equilateral, isosceles, scalene and right-angled

Triangles are named by their sides and their angles. Equal sides are marked with dashes, and equal sides face equal angles.

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Special quadrilaterals

Square, rectangle, kite, rhombus, parallelogram and trapezium, each named by its sides, angles and diagonals.

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Polygons: regular and irregular

A polygon is a flat shape with straight sides: pentagon, hexagon, octagon, decagon. Regular means all sides and all angles equal.

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Similar and congruent shapes, and scale factor

Congruent shapes are the same shape and size. Similar shapes are the same shape, and the scale factor says how many times bigger.

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Solids: faces, edges and vertices

Cube, cuboid, prism, cylinder, pyramid, cone, sphere, hemisphere and frustum, and how to count faces, edges and vertices.

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The vocabulary of a circle

Centre, radius, diameter, circumference, chord, tangent, arc, sector and segment, drawn on one round flower bed.

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10.

Geometrical terms: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

11.

Geometrical terms: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

12.

Geometrical terms: summary

The key facts, formulas and common mistakes of the chapter on one page.

4.2 · Geometrical constructions

Measure and draw lines and angles, construct a triangle from three sides with a ruler and compasses, and draw, read and use nets of solids.

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Measuring and drawing lines

Line up zero, read to the nearest millimetre, and draw every straight edge with a ruler.

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Measuring and drawing angles

Put the protractor centre on the vertex, start from zero on one arm, and read the right scale. Reflex angles are 360° minus the small angle.

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Constructing a triangle given three sides

Draw one side, swing two arcs with the other two lengths, and join to where they cross. Leave the arcs showing.

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Constructing other shapes from triangles

A rhombus is two triangles side by side, and a triangle can only be drawn when the two short sides add to more than the long one.

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Nets of cubes and cuboids

A net is the flat shape that folds into a solid. A cuboid net has six rectangles in three matching pairs.

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Nets of prisms and pyramids

A prism net is a rectangle strip with two end faces; a pyramid net is the base with a triangle on every edge.

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Using nets to find surface area and volume

Read the lengths off the net, add the face areas for the surface area, and use the same lengths for the volume.

12m

Geometrical constructions: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

Geometrical constructions: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

10.

Geometrical constructions: summary

The key facts, formulas and common mistakes of the chapter on one page.

4.6 · Angles

Angles at a point and on a line, vertically opposite angles, the three parallel-line pairs, angle sums of triangles, quadrilaterals and polygons, and a reason on every line.

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Angles at a point and on a straight line

Angles around a point add to 360°, and angles on a straight line add to 180°.

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Vertically opposite angles

Two crossing lines make two pairs of equal angles, because each pair makes 180° with the same angle.

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Corresponding angles

A line crossing two parallel lines makes angles in matching positions, and matching angles are equal.

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Alternate angles

Alternate angles sit on opposite sides of the crossing line, between the parallels, in a Z shape. They are equal.

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Co-interior angles

Co-interior angles sit on the same side, between the parallels, in a C shape. They add to 180°.

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Angle sum of a triangle

Draw a line through one corner parallel to the opposite side, and two pairs of alternate angles show the three angles make 180°.

12m

Angles in isosceles and equilateral triangles

Equal sides face equal angles, so one angle of an isosceles triangle gives the other two. Every angle of an equilateral triangle is 60°.

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The exterior angle of a triangle

An exterior angle equals the two interior angles opposite it, because both make 180° with the same angle.

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Angle sum of a quadrilateral

One diagonal cuts any quadrilateral into two triangles, so its angles add to 360°.

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10.

Interior angles of a polygon

Diagonals from one corner cut an n-sided polygon into n − 2 triangles, so the angles add to (n − 2) × 180°.

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11.

Exterior angles of a polygon

Walk round any polygon and you turn once, so the exterior angles add to 360°. For a regular polygon, 360° ÷ n gives each one.

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12.

Calculating angles and giving reasons

Most questions need two or three steps. Write each angle with its reason in proper words, as the mark scheme asks.

12m

13.

Angles: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

14.

Angles: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

15.

Angles: summary

The key facts, formulas and common mistakes of the chapter on one page.

4.5 · Symmetry

Line symmetry and the order of rotational symmetry in flat shapes, what they tell you about triangles, quadrilaterals and polygons, and the planes and axes of symmetry of solids.

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Line symmetry

A line of symmetry folds a shape onto itself. Each point has a matching point the same distance away on the other side.

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Rotational symmetry

The order of rotational symmetry is how many times a shape fits onto itself in one full turn.

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Symmetry of triangles and quadrilaterals

An isosceles triangle has one line of symmetry and a rhombus two. Each line explains an equal pair of sides or angles.

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Symmetry of regular polygons

A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.

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Planes of symmetry of solids

A plane of symmetry cuts a solid into two mirror halves. Count them for prisms, cylinders, pyramids and cones.

12m

Axes of rotational symmetry of solids

An axis of symmetry is a line a solid turns about and still looks the same. A cone and a cylinder have one special axis.

12m

Symmetry: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

Symmetry: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

Symmetry: summary

The key facts, formulas and common mistakes of the chapter on one page.

4.3 · Scale drawings and bearings

Draw and read scale drawings and map scales, then use compass directions and three-figure bearings, find a back bearing, and plan a route on a scale drawing.

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Scale drawings

A scale such as 1 cm to 2 m means every centimetre on paper stands for 2 metres on the ground.

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Map scales such as 1 : 50 000

A ratio scale has no units, so 1 cm on the map is 50 000 cm on the ground. Then change centimetres into metres or kilometres.

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Drawing a scale drawing

Choose a scale, work out each length on paper, then draw with a ruler and protractor and label the scale.

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North, east, south and west

The four compass points and the four between them. "Due east" means exactly east.

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Three-figure bearings

A bearing is measured from north, turning clockwise, and always written with three figures, such as 065°.

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Finding a back bearing

To turn round and go back, add 180° to the bearing, or take 180° away if it is 180° or more.

12m

Bearings on scale drawings

Draw each leg of a route to scale on its bearing, then measure the distance and bearing back to the start.

12m

Scale drawings and bearings: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

Scale drawings and bearings: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

10.

Scale drawings and bearings: summary

The key facts, formulas and common mistakes of the chapter on one page.

4.4 · Similarity

Calculate lengths in similar shapes, show that two triangles are similar, and use the scale factor, its square and its cube for lengths, areas and volumes.

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Calculating lengths of similar shapes

Find the scale factor from one pair of matching sides, then multiply to go bigger or divide to go smaller.

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Showing that two triangles are similar

Two pairs of equal angles are enough. Give a reason for each angle, then write the letters in matching order.

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Similar triangles that overlap

When a small triangle sits inside a big one, draw them apart first, then match the sides. The whole side is not the part.

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Areas of similar shapes

Area uses two lengths, so if lengths are multiplied by k, areas are multiplied by k².

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Surface areas of similar solids

Surface area is an area too, so similar solids have surface areas in the ratio k².

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Volumes of similar solids

Volume uses three lengths, so similar solids have volumes in the ratio k³.

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From an area or volume ratio back to lengths

Square root an area ratio or cube root a volume ratio to get the length scale factor, then use it on the other quantity.

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Solving problems involving similarity

The flag pole and its shadow make a triangle similar to a stick and its shadow, so one short measurement finds the height.

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10.

Similarity: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

11.

Similarity: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

12.

Similarity: summary

The key facts, formulas and common mistakes of the chapter on one page.

4.7 · Circle theorems I

The six angle theorems of 4.7: semicircle, tangent and radius, angle at the centre, same segment, cyclic quadrilateral and the alternate segment theorem, each used with its reason.

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Angle in a semicircle

An angle drawn from the two ends of a diameter to the circle is always 90°.

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Angle between tangent and radius

A tangent touches the circle at one point, and it makes 90° with the radius to that point.

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Angle at the centre is twice the angle at the circumference

Two radii make two isosceles triangles, and together they show the angle at the centre is double.

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Angles in the same segment are equal

Angles standing on the same arc, on the same side, are each half the same angle at the centre, so they are equal.

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Opposite angles of a cyclic quadrilateral

A cyclic quadrilateral has all four corners on a circle. Its opposite angles add to 180°.

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The alternate segment theorem

The angle between a tangent and a chord equals the angle in the segment on the other side of the chord.

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Calculating angles using circle theorems

Spot the feature first (a diameter, a tangent, four points on the circle), name the theorem, and give a reason on every line.

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Circle theorems I: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

Circle theorems I: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

10.

Circle theorems I: summary

The key facts, formulas and common mistakes of the chapter on one page.

4.8 · Circle theorems II

The three symmetry properties of 4.8: equal chords are equidistant from the centre, the perpendicular bisector of a chord passes through the centre, and tangents from a point are equal.

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The perpendicular bisector of a chord

The perpendicular bisector of any chord passes through the centre, so two chords are enough to find the centre.

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Chords, radius and distance from the centre

Half the chord, the distance from the centre and the radius make a right-angled triangle, so Pythagoras' theorem links them.

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Equal chords are equidistant from the centre

Equal chords make congruent right-angled triangles with the radius, so they are the same distance from the centre.

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Tangents from an external point are equal

Two tangents from one point outside a circle have equal lengths, and the line to the centre cuts the angle between them in half.

12m

Using the symmetry properties of circles

Mix the three properties with the 4.7 theorems, and name the property as the reason each time.

12m

Circle theorems II: mixed practice

A timed drill of mixed questions from the whole chapter, from easy to exam level, with a worked answer for each.

12m

Circle theorems II: exam practice

Past-paper style questions with command words, marks, full worked solutions and the mark scheme.

15m

Circle theorems II: summary

The key facts, formulas and common mistakes of the chapter on one page.

5.1 · Units of measure

Metric units of length, mass, area, volume and capacity, and how to change each one into a larger or smaller unit.

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Metric units of length

mm, cm, m and km: multiply to go to a smaller unit, divide to go to a larger one.

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Metric units of mass

There are 1000 grams in a kilogram. Change the units before you add or compare.

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Converting units of area

One metre is 100 cm, so one square metre is 100 × 100 square centimetres. Area units use the length factor twice.

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Converting units of volume

A cubic metre holds 100 × 100 × 100 cubic centimetres. Volume units use the length factor three times.

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Volume and capacity

Capacity is how much a container holds. 1 cm³ holds 1 ml, and 1 m³ holds 1000 litres.

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About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.