Symmetry and Grids: 11 Applications on Mirror Lines, Folds, Punched Holes and Tilings

The Lines of Symmetry of a Regular Hexagon and of a Rectangle

propertyTry Each Fold: It Is a Line of Symmetry Only If One Half Lands Exactly on the Other

(a) How many lines of symmetry does a regular hexagon have? (b) How many lines of symmetry does a rectangle that is not a square have?

A fold along a line joining two opposite corners lays one half on the other. There are 3 such lines.

A regular hexagon has 6 equal sides and 6 equal angles. A fold along the line joining two opposite corners lays one half exactly on the other, and there are 3 lines like this.

step 1 of 5

A line of symmetry is a fold line that lays one half of a shape exactly on the other half. Try each fold that could work, and count only the folds where the two halves match.

All steps

A regular hexagon has 6 equal sides and 6 equal angles. A fold along the line joining two opposite corners lays one half exactly on the other, and there are 3 lines like this.
A fold along the line joining the middles of two opposite sides also lays one half exactly on the other, and there are 3 lines like this too.
(a) The regular hexagon has 3 + 3 = 6 lines of symmetry.
A rectangle folds exactly in half along the line joining the middles of its two long sides, and along the line joining the middles of its two short sides.
(b) A fold along a diagonal does not work: the folded half sticks out past the edges of the other half. So a rectangle that is not a square has 2 lines of symmetry.

answer(a) 6; (b) 2

techniqueSymmetry

examsPSLE · O-Level · GCSE Higher

Common pitfalls

Counting the two diagonals of the rectangle and answering 4. A diagonal cuts the rectangle into two equal triangles, but when you fold along it, the folded triangle sticks out past the edges of the other one. Only a square folds exactly onto itself along its diagonals.
Answering 3 for the hexagon by counting only the lines through opposite corners. The lines through the middles of opposite sides also fold the hexagon exactly in half, so there are 3 + 3 = 6.

The Order of Rotational Symmetry of a Regular Pentagon and of a Rectangle

propertyTurn the Shape Once All the Way Round and Count the Times It Fits Its Outline

(a) What is the order of rotational symmetry of a regular pentagon? (b) What is the order of rotational symmetry of a rectangle that is not a square?

Turn the pentagon about its centre by 360° ÷ 5 = 72°: each corner moves to where the next corner was.

Turn: 36°

A regular pentagon has 5 equal sides and 5 equal angles. Turn it about its centre by 360^° ÷ 5 = 72^°: each corner moves to where the next corner was, so the pentagon fits its outline.

step 1 of 5

Turn the shape about its centre, once all the way round. The order of rotational symmetry is the number of times the shape fits exactly on its own outline during that full turn, and the fit at the end of the turn counts.

All steps

A regular pentagon has 5 equal sides and 5 equal angles. Turn it about its centre by 360^° ÷ 5 = 72^°: each corner moves to where the next corner was, so the pentagon fits its outline.
It fits again after 2 × 72^°, 3 × 72^°, 4 × 72^° and 5 × 72^° = 360^°, the full turn. That is 5 times in one full turn.
(a) The order of rotational symmetry of a regular pentagon is 5.
Turn the rectangle by a quarter turn, 90^°. Its long sides now stand where its short sides were, so it does not fit its outline.
(b) After a half turn, 180^°, the rectangle fits its outline, and after the full turn it fits again. That is 2 times, so the order of rotational symmetry is 2.

answer(a) 5; (b) 2

techniqueRotational Symmetry

examsGCSE Higher

Common pitfalls

Giving the rectangle order 4, as for a square. After a quarter turn the rectangle's long sides stand where its short sides were, so it does not fit. It fits only after a half turn and after the full turn, so its order is 2.
Leaving out the fit at the end of the turn and giving the rectangle order 1. The rectangle fits its outline after a half turn and again after the full turn, when it is back where it started. Both fits count, so the order is 2.

A Square Sheet Folded Once, with One Hole Punched

propertyThe Fold Line Is a Line of Symmetry: Each Hole Has a Twin as Far from the Fold on the Other Side

A square sheet of paper is folded in half, so that its left edge lands on its right edge. A hole is punched through both layers, 2 cm from the fold and 3 cm from the top edge. (a) When the sheet is opened out, how many holes are there? (b) How far apart are the holes?

Folded left onto right, the sheet is 2 layers thick.

The fold lays the left half of the sheet exactly on the right half, so the folded paper is 2 layers thick.

step 1 of 4

The punch goes through every layer of the folded paper, so it makes one hole in each layer. When the sheet is opened, the fold line is a line of symmetry: each hole has a twin the same distance from the fold, on the other side of it.

All steps

The fold lays the left half of the sheet exactly on the right half, so the folded paper is 2 layers thick.
(a) One punch through 2 layers makes 2 holes.
Open the sheet. The fold line is a line of symmetry, so the hole in the right half has a twin in the left half. The twin is also 2 cm from the fold, at the same height, 3 cm from the top.
(b) The holes are 2 + 2 = 4 cm apart. Check: each hole is 2 cm from the fold, one on each side of it, so the fold is exactly halfway between them.

answer(a) 2 holes; (b) 4 cm apart

techniqueSymmetry · Reflection

examsPSLE · O-Level · GCSE Higher

Common pitfalls

Answering 2 cm apart. That is the distance from one hole to the fold, which is only half of the gap. The other hole is 2 cm from the fold on the other side, so the holes are 2 + 2 = 4 cm apart.
Answering 1 hole. The folded paper is 2 layers thick and the punch goes through both, so when the sheet is opened there is a hole in each half.

A Square Sheet Folded Twice, with One Hole Punched

propertyUnfold One Fold at a Time: Each Fold Line Gives Every Hole a Twin on Its Other Side

A square sheet of paper is folded in half from left to right, and then in half again from bottom to top, making a square a quarter of the size. A hole is punched through all the layers, 1 cm from each of the two folded edges. (a) When the sheet is opened out, how many holes are there? (b) The holes are the corners of a shape. What shape is it, and how long is each of its sides?

Folded twice, the sheet is 4 layers thick. The two folded edges meet at the centre of the sheet.

The first fold makes 2 layers, and the second fold doubles them to 4 layers. The two folded edges meet at the corner that is the centre of the sheet.

step 1 of 5

Each fold doubles the number of layers, and the punch makes a hole in every layer. Open the folds one at a time: each fold line is a line of symmetry, so every hole gets a twin as far from the fold line on its other side.

All steps

The first fold makes 2 layers, and the second fold doubles them to 4 layers. The two folded edges meet at the corner that is the centre of the sheet.
(a) One punch through 4 layers makes 4 holes.
Open the second fold: the hole has a twin 1 cm on the other side of that fold line. Then open the first fold: both holes have twins 1 cm on the other side of the first fold line.
So each of the 4 holes is 1 cm from both fold lines. Two holes on opposite sides of a fold line are 1 + 1 = 2 cm apart.
(b) The 4 holes are the corners of a square, around the centre of the sheet, with sides of 2 cm.

answer(a) 4 holes; (b) a square with sides of 2 cm

techniqueSymmetry · Reflection

examsPSLE · O-Level · GCSE Higher

Common pitfalls

Answering 2 holes by opening only one fold. The sheet was folded twice, so it is 4 layers thick, and the punch goes through all 4 layers.
Giving the side of the square as 1 cm. 1 cm is the distance from a hole to a fold line. The hole's twin is 1 cm on the other side of the same line, so the two holes are 1 + 1 = 2 cm apart.

A Square Folded Along a Diagonal, with a Corner Cut Off

propertyCorners That Lie on Top of Each Other After the Fold Are Cut Together

A square sheet of paper ABCD is folded along the diagonal AC, so that corner B lands on corner D. A small triangle is cut off the folded corner, where B and D lie together. (a) When the sheet is opened out, how many corners of the square have been cut off? (b) How many sides does the shape have now?

Folded along AC, corner B lies exactly on corner D.

Folding along the diagonal AC lays triangle ABC exactly on triangle ADC, so corner B lands on corner D.

step 1 of 5

After the fold, corners B and D lie one on top of the other, so a single cut through both layers removes both of them. Each corner that is cut off is replaced by a short new side.

All steps

Folding along the diagonal AC lays triangle ABC exactly on triangle ADC, so corner B lands on corner D.
The cut goes through both layers at the corner where B and D lie together, so it cuts off both corners at once.
(a) When the sheet is opened, corners B and D have been cut off. Corners A and C are at the two ends of the fold line, away from the cut, so they are still there. That is 2 corners cut off.
Cutting off a corner leaves the two sides at that corner in place, only shorter, and makes one new short side between them. So each cut adds 1 side.
(b) The square had 4 sides, and the two cuts add 2 more, so the shape has 4 + 2 = 6 sides.

answer(a) 2 corners; (b) 6 sides

techniqueSymmetry · Completing a Symmetric Figure

examsPSLE · O-Level · GCSE Higher

Common pitfalls

Answering 1 corner. At the folded corner the paper is 2 layers thick, with B on top of D, so the one cut goes through both layers and removes both corners.
Answering 4 sides, as if cutting off a corner took a side away. The two sides at the corner only get shorter, and the cut makes one new side between them, so each cut corner adds a side: 4 + 2 = 6.

A Strip of Paper Folded Off Centre

propertyThe Fold Is Halfway Between Where the End Starts and Where It Lands

A strip of paper is 12 cm long. Its left end is folded over to the right, so that the end lands 2 cm from the right end of the strip. (a) How far from the left end is the fold? (b) A hole is punched through both layers, 1 cm from the fold. When the strip is opened out, how far from the left end is each hole?

The left end lands 2 cm short of the right end: 12 − 2 = 10 cm from where it started.

The left end lands 2 cm from the right end, so it lands 12 − 2 = 10 cm from where it started.

step 1 of 5

The fold line is a line of symmetry: the left end is as far from the fold before folding as it is after. So the fold is halfway between the place where the end starts and the place where it lands.

All steps

The left end lands 2 cm from the right end, so it lands 12 − 2 = 10 cm from where it started.
The end is the same distance from the fold before and after folding, so the fold is halfway along those 10 cm.
(a) Half of 10 cm is 10 ÷ 2 = 5 cm, so the fold is 5 cm from the left end.
The hole is 1 cm from the fold. When the strip is opened, one hole is 1 cm to the right of the fold and its twin is 1 cm to the left of it.
(b) The holes are 5 + 1 = 6 cm and 5 − 1 = 4 cm from the left end. Check: the part from 0 cm to 5 cm folds onto the part from 5 cm to 10 cm, so the point 4 cm from the left end lands exactly on the point 6 cm from it.

answer(a) 5 cm; (b) 4 cm and 6 cm

techniqueReflection · Finding a Missing Side

examsGCSE Higher

Common pitfalls

Answering 6 cm, the middle of the strip. The fold would be in the middle only if the left end landed on the right end. Here the end lands 2 cm short of it, so the fold is halfway along the 10 cm the end moves: 5 cm.
Answering 10 cm, the place where the end lands. The end is 5 cm before the fold when it starts and 5 cm past the fold when it lands, so the fold is only half of the 10 cm from the left end.

The Perimeter and Area of a Staircase Shape

propertyThe Step Edges Add Up to the Rectangle's Edges; the Area Is the Rectangle Less the Pieces Above the Steps

A staircase shape fits inside a rectangle 12 cm long and 8 cm high. Its three steps rise from right to left. Going up from the bottom right corner, the edges of the steps are 3 cm up and 4 cm across, then 3 cm up and 4 cm across, then 2 cm up and 4 cm across. (a) Find the perimeter of the shape. (b) Find its area.

The bottom edge is 12 cm and the left edge is 8 cm. The rest of the walk is up and across the steps.

Walk round the shape. The bottom edge is 12 cm and the left edge is 8 cm. The rest of the walk is up and across the steps.

step 1 of 5

The flat edges of the steps add up to the length of the rectangle, and the upright edges add up to its height, so the walk round the staircase is as long as the walk round the rectangle. The area is not the same: take away the rectangles above the steps.

All steps

Walk round the shape. The bottom edge is 12 cm and the left edge is 8 cm. The rest of the walk is up and across the steps.
The flat edges of the steps are 4 + 4 + 4 = 12 cm altogether, as long as the top of the rectangle. The upright edges are 3 + 3 + 2 = 8 cm altogether, as tall as its right side.
(a) So the perimeter is the same as the rectangle's: 12 + 8 + 12 + 8 = 40 cm.
The rectangle is 12 × 8 = 96 cm². Two rectangles above the steps are not part of the shape: above the lowest step, one 4 cm wide and 8 − 3 = 5 cm tall, and above the middle step, one 4 cm wide and 8 − 6 = 2 cm tall.
(b) Take them away: 96 − 4 × 5 − 4 × 2 = 96 − 20 − 8 = 68 cm². Check: cut the shape into three strips instead, 12 × 3 + 8 × 3 + 4 × 2 = 36 + 24 + 8 = 68 cm².

answer(a) 40 cm; (b) 68 cm²

techniqueArea and Perimeter · Finding a Missing Side

examsPSLE · SAT · GCSE Higher

Common pitfalls

Giving the perimeter as 12 + 8 = 20 cm, the bottom edge and the left edge only. The perimeter is the whole walk round the shape, and the edges of the steps are part of it: they add another 12 + 8 = 20 cm.
Giving the area as 12 × 8 = 96 cm², because the perimeter is the same as the rectangle's. The step edges have the same total length as the rectangle's edges, but the spaces above the steps are not part of the shape, so the area is less: 68 cm².

The Area of a Pentagon on a Dot Grid, and the Dots Inside It

propertySplit the Shape into a Rectangle and Triangles Along the Grid; Count the Dots Inside Row by Row

On a grid of dots 1 cm apart, each dot is named by how far it is across and then up from the bottom left dot, so (2, 5) is 2 across and 5 up. A pentagon has its corners at (0, 0), (5, 0), (5, 3), (2, 5) and (0, 3). (a) Find the area of the pentagon. (b) How many dots lie inside the pentagon, not counting the dots on its edges?

A line across at 3 cm up leaves a rectangle 5 by 3 below it: 15 cm².

Draw a line across from (0, 3) to (5, 3). Below it is a rectangle 5 cm long and 3 cm high, with an area of 5 × 3 = 15 cm².

step 1 of 6

A line across the pentagon at the height of 3 cm splits it into a rectangle and a triangle, the roof. A line from the top corner straight down splits the roof into two right-angled triangles. Then count the dots inside one row at a time, leaving out every dot on an edge.

All steps

Draw a line across from (0, 3) to (5, 3). Below it is a rectangle 5 cm long and 3 cm high, with an area of 5 × 3 = 15 cm².
Above the line is the roof. Draw a line from its top corner, (2, 5), straight down to (2, 3). It cuts the roof into two right-angled triangles, each 5 − 3 = 2 cm tall.
The left triangle is 2 cm wide, so its area is 12 × 2 × 2 = 2 cm². The right triangle is 3 cm wide, so its area is 12 × 3 × 2 = 3 cm².
(a) The area of the pentagon is 15 + 2 + 3 = 20 cm². Check: the roof is one triangle with a base of 5 cm and a height of 2 cm, and 12 × 5 × 2 = 5 = 2 + 3.
Now count the dots inside. The bottom row is on an edge, and the only dot in the row 5 up is the top corner. In the rows 1 up and 2 up, the dots 1, 2, 3 and 4 across are inside: 4 in each row. The line drawn across at 3 up is not an edge of the pentagon, so in that row only the corners, at 0 and 5 across, are on an edge, and the 4 dots between them are inside.
In the row 4 up, the dot 1 across is on the left edge of the roof, and the right edge of the roof crosses that row between 3 and 4 across. So only the dots 2 and 3 across are inside. (b) There are 4 + 4 + 4 + 2 = 14 dots inside.

answer(a) 20 cm²; (b) 14 dots

techniqueArea of a Composite Figure · Area of a Triangle

examsPSLE · SAT · GCSE Higher

Common pitfalls

Counting dots on the edges as inside, such as (1, 4) on the left edge of the roof, or the corners (0, 3) and (5, 3). A dot on an edge is on the outline of the pentagon, not inside it, so it is left out.
Taking the height of the roof as 5 cm, the height of its top corner above the bottom row. The roof stands on the line 3 cm up, so its height is 5 − 3 = 2 cm, and its area is 12 × 5 × 2 = 5 cm².

A Ring Made of Identical Trapeziums

propertyEach Piece Takes the Angle Between Its Slanted Sides at the Centre; the Pieces Share 360°

Identical trapeziums are fitted together to make a closed ring round a centre point, with the slanted sides of neighbouring trapeziums touching. Every slanted side, extended inward, passes through the centre. The longer parallel side of each trapezium is on the outside of the ring, and the two angles at that side are each 75^°. (a) How many trapeziums make the ring? (b) What is the size of each angle at the shorter parallel side?

Extend the slanted sides of one trapezium to the centre: with the longer side they make a triangle.

Extend the two slanted sides of one trapezium inward to the centre. With the longer parallel side they make a triangle, and its two angles at the longer side are the trapezium's two 75^° angles.

step 1 of 5

Extend the two slanted sides of one trapezium to the centre. With the longer parallel side they make a triangle, so the angle at the centre is what is left of 180^°. The angles at the centre of all the pieces make one full turn, 360^°.

All steps

Extend the two slanted sides of one trapezium inward to the centre. With the longer parallel side they make a triangle, and its two angles at the longer side are the trapezium's two 75^° angles.
The angles of a triangle add up to 180^°, so the angle at the centre is 180^° − 75^° − 75^° = 30^°.
(a) Each trapezium takes 30^° of the angle at the centre, and the angles at a point add up to 360^°. So there are 360^° ÷ 30^° = 12 trapeziums in the ring.
In a trapezium the shorter and the longer sides are parallel. So along each slanted side, the angle at the shorter side and the angle at the longer side add up to 180^°.
(b) Each angle at the shorter parallel side is 180^° − 75^° = 105^°. Check: the four angles of the trapezium add up to 75^° + 75^° + 105^° + 105^° = 360^°, as the angles of every four-sided shape do.

answer(a) 12 trapeziums; (b) 105^°

techniqueAngles at a Point · Area of a Trapezium

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

Dividing 360^° by 75^°. That is not even a whole number, and 75^° is not the angle a trapezium takes at the centre. Each trapezium takes the angle between its two slanted sides at the centre, which is 30^°.
Answering 75^° for the angles at the shorter side, as if all four angles were equal. The two parallel sides make the angles along each slanted side add up to 180^°, so each angle at the shorter side is 180^° − 75^° = 105^°.

Covering a Rectangle with L-Shaped Tiles

propertyTiles Needed = Area to Cover ÷ Area of One Tile; a Tile's Perimeter Is Only Its Outside Edges

An L-shaped tile is made of 3 unit squares joined edge to edge. A rectangle 6 units long and 4 units wide is covered with these tiles, with no gaps and no overlaps. (a) How many tiles are used? (b) What is the perimeter of one tile?

The rectangle is 6 × 4 = 24 unit squares.

The rectangle is made of 6 × 4 = 24 unit squares.

step 1 of 5

With no gaps and no overlaps, the areas of the tiles add up to the area of the rectangle. The perimeter of one tile is the walk round its outside, so the edges where two of its squares are joined are not counted.

All steps

The rectangle is made of 6 × 4 = 24 unit squares.
Each tile covers 3 unit squares, and the tiles cover the rectangle with no gaps and no overlaps, so the number of tiles is 24 ÷ 3 = 8.
(a) 8 tiles are used. They really do fit: two tiles fit together to make a rectangle 3 units long and 2 units wide, and four of those fill the 6 by 4 rectangle.
Now walk round the outside of one tile and count the unit edges: 2 along the bottom, 1 up, 1 back in, 1 more up, 1 across the top and 2 down the far side.
(b) The perimeter of one tile is 2 + 1 + 1 + 1 + 1 + 2 = 8 units. Check: 3 separate squares have 3 × 4 = 12 edges, and each of the 2 joins hides 2 edges, one from each square, so 12 − 4 = 8.

answer(a) 8 tiles; (b) 8 units

techniqueArea and Perimeter · Making Shapes from Smaller Shapes

examsPSLE · SAT · GCSE Higher

Common pitfalls

Giving the perimeter of a tile as 12 units, the edges of 3 separate squares. Where two squares are joined, the shared edge is inside the tile, not on its outline, so each of the 2 joins takes away 2 edges: 12 − 4 = 8.
Dividing the rectangle's perimeter by the tile's perimeter: 20 ÷ 8 is not even a whole number. The tiles cover the space inside the rectangle, so it is the areas that are divided: 24 ÷ 3 = 8 tiles.

A Sailboat and Its Reflection in the Water, Completed on Squared Paper

propertyEach square's twin is in the same column and the same number of rows from the line, on the other side

Mia draws a sailboat on squared paper, above a straight line across the page that stands for the water. The boat is made of 12 shaded squares: 6 in the row just above the line for the hull, then 3, 2 and 1 in the three rows above that for the sail. She completes the picture by drawing the boat's reflection in the water, with the line as a line of symmetry. (a) The top square of the sail is in the 4th row above the line. In which row below the line is its twin? (b) How tall is the finished picture, in squares, and how many squares are shaded in it?

The water line is the line of symmetry. Each square's twin is in the same column, the same number of rows away, on the other side.

The line is the line of symmetry. Each shaded square has a twin in the same column, the same number of rows from the line, on the other side of it.

step 1 of 5

A reflection keeps every square in its own column, and the same number of rows from the line of symmetry, but on the other side. Count the rows from the square to the line, cross the line, and count the same number of rows again.

All steps

The line is the line of symmetry. Each shaded square has a twin in the same column, the same number of rows from the line, on the other side of it.
The hull is in the 1st row above the line, so its twin is in the 1st row below the line. The reflected hull touches the real hull at the water line.
The sail is in the 2nd, 3rd and 4th rows above the line, so its twin is in the 2nd, 3rd and 4th rows below it. The reflected sail points down.
(a) The top square of the sail is in the 4th row above the line, so its twin is in the 4th row below the line, in the same column. It is the lowest square in the picture.
(b) The picture reaches 4 rows above the line and 4 rows below it, so it is 4 + 4 = 8 squares tall. Each of the 6 + 3 + 2 + 1 = 12 squares has one twin, so 12 + 12 = 24 squares are shaded.

answer(a) The 4th row below the line, in the same column; (b) 8 squares tall, with 24 squares shaded

techniqueCompleting a Symmetric Figure · Symmetry

examsPSLE · O-Level · GCSE Higher

Common pitfalls

Copying the boat below the line the same way up, with its hull at the bottom and the top of its sail touching the line. A reflection turns the boat over: the hull touches the line from above, so its twin touches the line from below.
Counting 12 + 12 − 6 = 18 shaded squares, as if the hull and its twin were the same squares. The hull is in the row above the line and its twin is in the row below it. A square is its own twin only when the line runs through it, and no square here has the line through it.