What is rotational symmetry ?

A shape is said to have rotational symmetry if, during a 360° rotation about its centre, it looks the same as it did in its original position .

What is meant by the order of rotational symmetry ?

The order of rotational symmetry refers to the number of times a shape looks the same as it is rotated 360° about its centre.

True or False? A shape can have order 0 rotational symmetry.

False. A shape can never have order 0 rotational symmetry. A shape is said to have no rotational symmetry when the order of rotational symmetry is 1 , i.e. it only looks like its original position when it has been rotated through the full 360°.

How can tracing paper be used to help work out the order of rotational symmetry of a shape?

You can use tracing paper to help you work out the order of symmetry for a particular shape by doing the following: Sketch the shape onto the tracing paper. Draw an arrow pointing upwards on the tracing paper. Place your pencil in the centre of the shape and rotate the tracing paper until the arrow is pointing upwards again (360º). Count how many times the shape on the tracing paper maps over the shape on the paper beneath exactly; this is the order of rotational symmetry .

What is line symmetry ?

A shape is said to have line symmetry when one half of the shape is a mirror image of the other half.

What is a line of symmetry ?

A line of symmetry is a line across which a shape can be folded or reflected so that the two halves match exactly.

True or False? A line of symmetry is also known as a line of reflection .

True. A line of symmetry is also known as a line of reflection . It can also be called a mirror line .

How can tracing paper be used to help sketch the reflection of a shape?

You can use tracing paper to help you sketch the reflection of a shape: Trace the shape and the line of symmetry on the tracing paper. Flip the paper over the line of symmetry. The traced image will now be in its reflected position.

What is a plane of symmetry in a 3D shape?

A plane of symmetry in a 3D shape is a plane (flat 2D shape) that splits a 3D shape into two congruent (identical) halves that are mirror images of each other.

True or False? A cylinder has infinite rotational symmetry .

True. A cylinder has infinite rotational symmetry about the axis that runs through the centre point of both circular faces.

How are equilateral, isosceles and scalene triangles told apart by their sides ?

An equilateral triangle has three equal sides, an isosceles triangle has two, and a scalene triangle has three sides all of different lengths. The fourth type, right-angled , is named by its 90° angle rather than by its sides.

What makes a square a special kind of rectangle?

A square is a rectangle in which all four sides have the same length . Everything that is true of a rectangle is therefore true of a square as well, but not the other way round.

What makes a rhombus a special kind of parallelogram?

A rhombus is a parallelogram in which all four sides have the same length . Both shapes have two pairs of parallel sides, and the rhombus adds the equal side lengths on top of that.

A kite is a quadrilateral with two pairs of equal adjacent sides. Unlike every other named quadrilateral it has no parallel sides at all.

In which quadrilaterals are the two diagonals perpendicular to each other?

The diagonals are perpendicular in a square , a rhombus and a kite . In a rectangle or a parallelogram that is not one of those, the diagonals cross at an angle that is not 90°.

What does an isosceles trapezium have that an ordinary trapezium does not?

Its two non-parallel sides are the same length, and its two diagonals are equal in length as well. Both kinds have exactly one pair of parallel sides, and in each case those parallel sides are of different lengths.

True or False? A parallelogram has two pairs of parallel sides, and so does a trapezium.

False. A parallelogram has two pairs of parallel sides, but a trapezium has exactly one pair. That single pair of parallel sides is the defining feature of a trapezium.

Define the circumference of a circle.

The circumference is the perimeter of a circle, the distance all the way round its edge. It is the circle's own name for what any other shape would call its perimeter.

Define a chord of a circle.

A chord is a straight line joining two points on the circumference of a circle. The longest chord that any circle has is its diameter.

What is the difference between a sector and a segment of a circle?

A sector is the region enclosed between two radii and an arc, like a slice of pizza. A segment is the region enclosed between a chord and an arc, which is what a single straight cut across a circle leaves behind.

Define a tangent to a circle.

A tangent is a straight line that touches the circumference at one point only . It never passes inside the circle, which is what separates it from a chord.

Define an arc of a circle.

An arc is a portion of the circumference of a circle. An arc is a curved length rather than a region, which is what separates it from the parts of a circle that have an area.

True or False? Every diameter of a circle is a line of symmetry.

True. Any straight line through the centre cuts the circle into two identical halves, so a circle has infinitely many lines of symmetry. Every one of those lines is a diameter.

A prism is a solid with the same cross-section all the way through its length. It is the cross-section that gives a prism its name, so a prism whose cross-section is a triangle is called a triangular prism.

What is the cross-section of a cube, a cuboid and a cylinder?

A cube has a square cross-section, a cuboid a rectangular one, and a cylinder a circular one. Identifying the cross-section is what lets the same volume method be used for all three.

A pyramid has a flat base with sloping sides that all meet at a single point at the top. A pyramid is named after its base, so a square-based pyramid is one whose base is a square.

What is a tetrahedron ?

A tetrahedron is a triangular-based pyramid. All four of its faces are triangles, so any one of them can be treated as the base.

A triangular prism has 2 triangular faces and 3 rectangular ones. When are all three of the rectangles equal?

When the triangular faces are equilateral , because then all three sides of the triangle are the same length. If the triangles are isosceles only two of the rectangles are equal, since only two sides of the triangle match.

What faces does a square-based pyramid have?

One square face, which is the base, and four equal triangular faces sloping up from it. That makes five faces in total.

True or False? A cube and a cuboid both have six faces.

True. A cube has six equal square faces, and a cuboid has six rectangular faces arranged as three matching pairs. The difference between them is in the shape of the faces, not in how many there are.

Define a net of a solid.

A net is a two-dimensional drawing that can be cut out and folded up to make a particular three-dimensional shape. It shows every face of the solid, arranged so that the faces meet correctly once it is folded.

What is the connection between the area of a net and the solid it folds into?

The area of the net is exactly the surface area of the solid. Nothing is added or lost in the folding, so adding up the areas of the faces on the flat drawing gives the surface area directly.

True or False? Any arrangement of six squares joined edge to edge will fold up into a cube.

False. Only 11 different arrangements of six squares fold up into a cube, and the rest overlap or leave a gap. The easiest of the eleven to remember is the cross shape.

What is the net of a square-based pyramid made from?

One square, which becomes the base, and four congruent triangles that fold up to meet at the apex. The perpendicular height of each of those triangles is the slant height of the pyramid, not its vertical height.

How many rectangles does the net of a cuboid have, and how do they pair up?

Six rectangles, forming three pairs of identical faces, one pair for each of the three different pairs of dimensions. A cuboid measuring 6 cm by 3 cm by 2 cm therefore has two 6 by 3 rectangles, two 6 by 2 and two 3 by 2.

True or false? To convert between two metric units of measure, you must multiply or divide by powers of 10 .

True. To convert between two metric units of measure, you must multiply or divide by powers of 10 . E.g. to convert from centimetres to metres, divide by 100.

What type of quantity is measured using cubic units?

Cubic units are units used to measure volume .

True or False? To convert between cubic units, you must multiply or divide by the normal unit conversion rate .

False. To convert between cubic units, you must multiply or divide by the 'cubed' conversion rate . E.g. to convert from cm 3 to m 3 you must divide by 100 3 .

What type of units are used to measure area ?

Square units are used to measure area .

True or False? To convert between square units, you must multiply or divide by the 'squared' conversion rate .

True. To convert between square units, you must multiply or divide by the 'squared' conversion rate . E.g. to convert from km 2 to m 2 you must multiply by 1000 2 .

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Geometry Toolkit (Cambridge (CIE) O Level Maths): Flashcards

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FrontRotational Symmetry
What is rotational symmetry?

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Cards in this collection (53)

What is rotational symmetry?
A shape is said to have rotational symmetry if, during a 360° rotation about its centre, it looks the same as it did in its original position.
What is meant by the order of rotational symmetry?
The order of rotational symmetry refers to the number of times a shape looks the same as it is rotated 360° about its centre.
True or False?
A shape can have order 0 rotational symmetry.
False.
A shape can never have order 0 rotational symmetry.
A shape is said to have no rotational symmetry when the order of rotational symmetry is 1, i.e. it only looks like its original position when it has been rotated through the full 360°.
How can tracing paper be used to help work out the order of rotational symmetry of a shape?
You can use tracing paper to help you work out the order of symmetry for a particular shape by doing the following:
1. Sketch the shape onto the tracing paper.
2. Draw an arrow pointing upwards on the tracing paper.
3. Place your pencil in the centre of the shape and rotate the tracing paper until the arrow is pointing upwards again (360º).
4. Count how many times the shape on the tracing paper maps over the shape on the paper beneath exactly; this is the order of rotational symmetry.
What is line symmetry?
A shape is said to have line symmetry when one half of the shape is a mirror image of the other half.
What is a line of symmetry?
A line of symmetry is a line across which a shape can be folded or reflected so that the two halves match exactly.
True or False?
A line of symmetry is also known as a line of reflection.
True.
A line of symmetry is also known as a line of reflection.
It can also be called a mirror line.
How can tracing paper be used to help sketch the reflection of a shape?
You can use tracing paper to help you sketch the reflection of a shape:
1. Trace the shape and the line of symmetry on the tracing paper.
2. Flip the paper over the line of symmetry.
3. The traced image will now be in its reflected position.
What is a plane of symmetry in a 3D shape?
A plane of symmetry in a 3D shape is a plane (flat 2D shape) that splits a 3D shape into two congruent (identical) halves that are mirror images of each other.
How many planes of symmetry does a cube have?
A cube has 9 planes of symmetry.
How many planes of symmetry does a cuboid have?
A cuboid has 3 planes of symmetry.
How many planes of symmetry does a square-based pyramid have?
A square-based pyramid has 4 planes of symmetry.
True or False?
A cylinder has infinite rotational symmetry.
True.
A cylinder has infinite rotational symmetry about the axis that runs through the centre point of both circular faces.
Complete the names of these polygons by their number of sides:
A polygon with 5 sides is a pentagon, one with 7 sides is a $______$ , one with 9 sides is a nonagon, and one with 10 sides is a $______$ instead.
The completed list is:
A polygon with 5 sides is a pentagon, one with 7 sides is a heptagon, one with 9 sides is a nonagon, and one with 10 sides is a decagon instead.
The other two names to know are the hexagon with 6 sides and the octagon with 8.
How are equilateral, isosceles and scalene triangles told apart by their sides?
An equilateral triangle has three equal sides, an isosceles triangle has two, and a scalene triangle has three sides all of different lengths.
The fourth type, right-angled, is named by its 90° angle rather than by its sides.
What makes a square a special kind of rectangle?
A square is a rectangle in which all four sides have the same length.
Everything that is true of a rectangle is therefore true of a square as well, but not the other way round.
What makes a rhombus a special kind of parallelogram?
A rhombus is a parallelogram in which all four sides have the same length.
Both shapes have two pairs of parallel sides, and the rhombus adds the equal side lengths on top of that.
Define a kite.
A kite is a quadrilateral with two pairs of equal adjacent sides.
Unlike every other named quadrilateral it has no parallel sides at all.
In which quadrilaterals are the two diagonals perpendicular to each other?
The diagonals are perpendicular in a square, a rhombus and a kite.
In a rectangle or a parallelogram that is not one of those, the diagonals cross at an angle that is not 90°.
Complete the sentence about the diagonals of quadrilaterals:
Both diagonals bisect each other in a square, a rectangle, a parallelogram and a $______$ , whereas in a kite only $______$ of them bisects the other.
The completed sentence is:
Both diagonals bisect each other in a square, a rectangle, a parallelogram and a rhombus, whereas in a kite only one of them bisects the other.
In a trapezium neither diagonal bisects the other.
What does an isosceles trapezium have that an ordinary trapezium does not?
Its two non-parallel sides are the same length, and its two diagonals are equal in length as well.
Both kinds have exactly one pair of parallel sides, and in each case those parallel sides are of different lengths.
True or False?
A parallelogram has two pairs of parallel sides, and so does a trapezium.
False.
A parallelogram has two pairs of parallel sides, but a trapezium has exactly one pair.
That single pair of parallel sides is the defining feature of a trapezium.
Define the circumference of a circle.
The circumference is the perimeter of a circle, the distance all the way round its edge.
It is the circle's own name for what any other shape would call its perimeter.
Complete the relationship between the two lengths:
$diameter =______\times radius$
The completed relationship is:
$diameter = 2 \times radius$
A radius runs from the centre out to the circumference, while a diameter runs the whole way across, passing through the centre.
Define a chord of a circle.
A chord is a straight line joining two points on the circumference of a circle.
The longest chord that any circle has is its diameter.
What is the difference between a sector and a segment of a circle?
A sector is the region enclosed between two radii and an arc, like a slice of pizza.
A segment is the region enclosed between a chord and an arc, which is what a single straight cut across a circle leaves behind.
Define a tangent to a circle.
A tangent is a straight line that touches the circumference at one point only.
It never passes inside the circle, which is what separates it from a chord.
What do you get if you divide any circle's circumference by its diameter?
You always get $π$ , which is about 3.14159, no matter how big or small the circle is.
That constant ratio is the reason $π$ turns up in every circle formula there is.
Define an arc of a circle.
An arc is a portion of the circumference of a circle.
An arc is a curved length rather than a region, which is what separates it from the parts of a circle that have an area.
True or False?
Every diameter of a circle is a line of symmetry.
True.
Any straight line through the centre cuts the circle into two identical halves, so a circle has infinitely many lines of symmetry.
Every one of those lines is a diameter.
Define a prism.
A prism is a solid with the same cross-section all the way through its length.
It is the cross-section that gives a prism its name, so a prism whose cross-section is a triangle is called a triangular prism.
What is the cross-section of a cube, a cuboid and a cylinder?
A cube has a square cross-section, a cuboid a rectangular one, and a cylinder a circular one.
Identifying the cross-section is what lets the same volume method be used for all three.
Complete the three terms used to describe a solid:
A $______$ is a single flat surface of the shape, a vertex is a corner of it, and an $______$ is the line joining one vertex to another.
The completed terms are:
A face is a single flat surface of the shape, a vertex is a corner of it, and an edge is the line joining one vertex to another.
The plural of vertex is vertices.
Define a pyramid.
A pyramid has a flat base with sloping sides that all meet at a single point at the top.
A pyramid is named after its base, so a square-based pyramid is one whose base is a square.
What is a tetrahedron?
A tetrahedron is a triangular-based pyramid.
All four of its faces are triangles, so any one of them can be treated as the base.
A triangular prism has 2 triangular faces and 3 rectangular ones. When are all three of the rectangles equal?
When the triangular faces are equilateral, because then all three sides of the triangle are the same length.
If the triangles are isosceles only two of the rectangles are equal, since only two sides of the triangle match.
What faces does a square-based pyramid have?
One square face, which is the base, and four equal triangular faces sloping up from it.
That makes five faces in total.
True or False?
A cube and a cuboid both have six faces.
True.
A cube has six equal square faces, and a cuboid has six rectangular faces arranged as three matching pairs.
The difference between them is in the shape of the faces, not in how many there are.
Define a net of a solid.
A net is a two-dimensional drawing that can be cut out and folded up to make a particular three-dimensional shape.
It shows every face of the solid, arranged so that the faces meet correctly once it is folded.
What is the connection between the area of a net and the solid it folds into?
The area of the net is exactly the surface area of the solid.
Nothing is added or lost in the folding, so adding up the areas of the faces on the flat drawing gives the surface area directly.
True or False?
Any arrangement of six squares joined edge to edge will fold up into a cube.
False.
Only 11 different arrangements of six squares fold up into a cube, and the rest overlap or leave a gap.
The easiest of the eleven to remember is the cross shape.
Complete the description of the net of a cylinder:
It is made from two circles and one $______$ , whose length is equal to the $______$ of the circles and whose width is the height of the cylinder.
The completed description is:
It is made from two circles and one rectangle, whose length is equal to the circumference of the circles and whose width is the height of the cylinder.
The rectangle rolls round to become the curved surface.
Why is the rectangle in a cylinder's net as long as the circle's circumference?
The rectangle is the curved surface unrolled flat, and its long edge is the one that wraps exactly once around the circular face.
That edge therefore has to measure $2 \times π \times radius$ for the net to close up without a gap or an overlap.
What is the net of a square-based pyramid made from?
One square, which becomes the base, and four congruent triangles that fold up to meet at the apex.
The perpendicular height of each of those triangles is the slant height of the pyramid, not its vertical height.
How many rectangles does the net of a cuboid have, and how do they pair up?
Six rectangles, forming three pairs of identical faces, one pair for each of the three different pairs of dimensions.
A cuboid measuring 6 cm by 3 cm by 2 cm therefore has two 6 by 3 rectangles, two 6 by 2 and two 3 by 2.
True or false?
To convert between two metric units of measure, you must multiply or divide by powers of 10.
True.
To convert between two metric units of measure, you must multiply or divide by powers of 10.
E.g. to convert from centimetres to metres, divide by 100.
What is the equation used to convert kilometres to metres?
The equation used to convert kilometres to metres is:
$Metres = Kilometres \times 1000$ .
What is the equation used to convert millilitres to litres?
The equation used to convert millilitres to litres is:
$Litres = \frac{Millilitres}{1000}$ .
What is the equation used to convert kilograms to milligrams?
The equation used to convert kilograms to milligrams is:
$Milligrams = Kilograms \times 1, 000, 000$ .
What type of quantity is measured using cubic units?
Cubic units are units used to measure volume.
True or False?
To convert between cubic units, you must multiply or divide by the normal unit conversion rate.
False.
To convert between cubic units, you must multiply or divide by the 'cubed' conversion rate.
E.g. to convert from cm³ to m³ you must divide by 100³.
What type of units are used to measure area?
Square units are used to measure area.
True or False?
To convert between square units, you must multiply or divide by the 'squared' conversion rate.
True.
To convert between square units, you must multiply or divide by the 'squared' conversion rate.
E.g. to convert from km² to m² you must multiply by 1000².

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