Playing with Constructions · Lesson 5 of 8
Planning Patterns with Squares and Circles
“Turn a finished design into a construction plan using equal lengths, repeated parts, and symmetry.”
• Use rough sketches and equality ticks to plan compound constructions. • Construct rectangles made from two or three identical squares. • Position a square centrally inside a rectangle and align falling squares. • Reproduce square subdivisions and shaded patterns. • Locate centres for matching circular holes and four uniform curved boundaries.
Plan the relationships before choosing numbers
Some construction tasks specify a finished pattern but give no side lengths. That freedom does not mean every arrangement will work. The parts still need to fit together. Draw a rough sketch, identify lengths that must match, and then decide which one length you can choose freely. Other lengths follow from those relationships.
If a rectangle must contain three identical squares in one row, simplify the task by first considering two squares. The easier case reveals how each square side connects to the rectangle dimensions. A rough sketch does not have to be accurately measured; its job is to show the shared sides and the planned sequence.
Two squares, then three
Name the corners of the two-square arrangement A, C, D, F around its outer rectangle. B lies halfway along the top and E halfway along the bottom. The left square is ABEF and the right square is BCDE. Their common side is BE. Because it belongs to both squares, it connects the equalities in the two parts.
In the left square AF = AB = BE = FE. In the right square BE = BC = CD = ED. Therefore every short segment in the diagram has the same length. A small matching tick on those segments records their equality. Use the same tick pattern only for lengths intended to match; a tick is a statement about length, not a decoration.
Problem
Construct a rectangle that divides into two identical squares with AF = 4 cm.
- 1.Draw AF = 4 cm as one upright side. Draw a perpendicular ray through A long enough for the whole top edge.
- 2.Set the compass opening to AF. From A mark B on the top ray, then from B mark C with the same opening. Thus AB = BC = 4 cm and AC = 8 cm.
- 3.Complete the outer rectangle with height 4 cm, using perpendiculars and equal side transfer. Draw the divider BE perpendicular to AC.
- 4.The two pieces each have four 4 cm sides and four right angles. They are identical squares.
You can also begin with an unmeasured segment AF. Copy it twice along the perpendicular top ray and complete the rectangle. The result still has the required relationships even if you never read a centimetre value. A compass preserves a distance; the ruler helps you maintain straight lines.
Problem
Construct a rectangle divided into three identical squares, each of side 3 cm.
- 1.Choose the square side as 3 cm. Three squares in one row need three 3 cm spans, so the long side must be 3 + 3 + 3 = 9 cm.
- 2.Construct a 9 cm by 3 cm rectangle. On the long side, mark divisions 3 cm and 6 cm from the start.
- 3.Draw perpendicular dividers through the two marked points to the opposite side. Each region is 3 cm by 3 cm with right angles, so each is a square.
- 4.Check that all three regions are the same size and cover the rectangle without gaps or overlap.
For this row arrangement, the short side is the square side s. If there are n identical squares in the row, the long side L contains n copies of s. Saying the relationship in words first makes the symbols easier to read.
Problem
Could a 4 cm by 2.5 cm rectangle divide into two identical squares in one row? Could a 7 cm by 2 cm rectangle divide into three?
- 1.For two squares with side 2.5 cm, the required long side is 2 × 2.5 = 5 cm. The given long side is 4 cm, so the first rectangle does not fit.
- 2.For three squares with side 2 cm, the required long side is 3 × 2 = 6 cm. The given long side is 7 cm, so the second does not fit.
- 3.Equal areas alone would not establish square pieces. The arrangement must also have the correct side lengths and right angles.
Putting a square in the middle
A square inside an 8 cm by 4 cm rectangle must be placed carefully if both figures are to have the same centre. In the displayed design the square fills the rectangle height, so its side is 4 cm. The remaining horizontal length is shared equally on the left and right. This is a placement condition as well as a size condition.
Problem
Construct the centred square that fills the height of an 8 cm by 4 cm rectangle.
- 1.The square side is 4 cm because it matches the rectangle height.
- 2.The unused width is 8 − 4 = 4 cm. Divide it equally between the two ends: each gap is 4 ÷ 2 = 2 cm.
- 3.On the top and bottom edges mark points 2 cm from each end. Join matching points vertically. The enclosed central region is 4 cm by 4 cm.
- 4.Check the equal left and right gaps. They place the square centre at the same position as the rectangle centre.
A correct 4 cm square placed against one end of the 8 cm rectangle has the right size but the wrong centre. Check corresponding gaps or a common centre, rather than only checking that the square fits.
Falling squares and repeated shading
In a falling-square pattern, adjacent squares meet at one corner. Their sides keep the same horizontal and vertical directions. Draw the largest or lowest square first, then use its top-right corner as the next square lower-left corner. That shared point does not turn the two squares into one square; each retains its own boundary and side length.
For three equal 4 cm squares, transfer the same opening for every new side. For the unequal design, use 7 cm for the first, 5 cm for the next, and 3 cm for the last. Keep the corner-contact sequence and directions correct. The displayed unequal diagram can be built from the lowest square upwards, with the small square lower-left corner at the preceding square top-right corner.
A shading pattern adds another planning task: the larger boundary is a square, and its subdivisions also include squares. First reproduce the subdivision lines. Then draw diagonals in the selected small squares in the same direction and shade the specified triangular halves. Shading too soon can hide lines you still need to construct.
Problem
You choose a large square of side 8 cm for a pattern based on four equal divisions across each edge. What is a small division length?
- 1.Four equal lengths must together make 8 cm. Each is 8 ÷ 4 = 2 cm.
- 2.Mark 2 cm intervals on the relevant edges and draw the selected perpendicular subdivision lines. Keep any larger combined square regions intact.
- 3.Draw the chosen small-square diagonals and shade the same indicated halves. Changing the overall size preserves the design when the division relationships are copied.
A square with one hole or four
A circular hole centred in a square needs the square centre, not a point guessed near the middle. Draw the two diagonals lightly. Their intersection locates the centre. Set a radius small enough for the circle to stay inside the square, put the compass point at that intersection, and draw the circle.
For four matching holes, first divide the larger square into four equal smaller squares using its middle horizontal and vertical lines. Find each smaller square centre by its diagonals. Keep the same compass opening for all four circles. The circles match in size, and their centre positions match within their respective squares.
Problem
A square has side 6 cm. Can a centred circle of radius 2 cm fit inside it?
- 1.The square centre is 3 cm from each side, half of the 6 cm side length.
- 2.A radius of 2 cm is smaller than that 3 cm available distance, so the circle remains strictly inside the square.
- 3.Find the centre using the diagonals and draw the 2 cm circle. A radius larger than 3 cm would extend beyond a side; radius 3 cm would touch the side midpoints.
Four matching arcs inside a square
The square-with-curves design has four equal inward-bending arcs joining neighbouring corners of an 8 cm square. The arcs have to match, so choose compass centres in matching positions outside the four sides. Each centre lies on the line through its side midpoint perpendicular to that side. This makes its distances to the two side endpoints equal.
One workable choice puts each centre 4 cm outside its side midpoint. For the top side, mark the midpoint, draw a perpendicular outward, and place the centre 4 cm along it. Set the opening from that centre to a top corner and draw the short lower arc between the two top corners. Repeat the construction at the other sides with the same opening and the same outside offset.
The centre is outside, but the short arc bends into the square. Choosing the wrong compass centre or drawing the opposite portion of the circle can produce an outward bulge. Keep only the intended short arcs as final lines. The four equal offsets, equal side spans, and equal openings explain the uniform appearance; you do not need an advanced formula for the radius.
Quiz
A rectangle made from three identical squares in one row has square side 4 cm. What is its long side?
What does a matching tick on two segments mean?
A 4 cm square fills the height of an 8 cm by 4 cm rectangle and has the same centre. What is each end gap?
Which rectangle can be divided into two identical squares in one row?
How do you locate the centre for a hole in a square?
What helps make four inward arcs uniform?
Practice Problems
- Construct a rectangle divided into two identical squares with square side 4 cm. Mark all equal short segments with matching ticks.
- Construct a rectangle divided into three identical squares with square side 2.5 cm. Calculate its dimensions first.
- Begin with an unmeasured segment and construct a two-square rectangle by copying that segment with a compass.
- Explain why 4 cm by 2.5 cm cannot give two identical squares in one row, and why 7 cm by 2 cm cannot give three.
- Construct the centred 4 cm square inside an 8 cm by 4 cm rectangle. Verify the equal gaps.
- Draw the falling-square patterns with sides 4, 4, 4 cm and then 7, 5, 3 cm. Keep shared corners and side directions correct.
- Reproduce the shading design using a large square size of your choice. Record its small division length and shade only after checking.
- Construct a square with one centred circular hole, then one with four matching holes. Explain how you located the centres.
- Construct an 8 cm square with four matching inward arcs. Leave the supporting midpoint lines and outside centres visible for checking.
Three 2.5 cm squares need a rectangle 7.5 cm long and 2.5 cm wide. A two-square rectangle with side 4 cm is 8 cm by 4 cm.
Key Takeaways
• A rough sketch reveals the lengths and positions a design requires. • Matching ticks record equal segments; a compass copies those lengths. • A row of n identical squares has long side n times the square side. • Centred shapes require equal corresponding gaps or a shared centre. • Pattern construction copies directions and positions as well as lengths. • Matching holes and arcs use matching centre placements and compass openings.