Playing with Constructions · Lesson 1 of 8
Circles, Arcs, and Compass Artwork
“Discover how a fixed compass opening turns equal distances into circles, waves, and matching curves.”
• Describe a circle using its centre and radius. • Set a compass opening and draw circles and arcs accurately. • Separate a design into straight segments and circular parts. • Choose centres and radii for semicircular waves and shallower arcs. • Use supporting lines and symmetry to make matching eyes and original artwork.
Drawing with a purpose
Try drawing a round face without lifting your pencil. Some parts may bulge farther from the middle than others. A compass helps because it holds one distance steady while the pencil moves. The aim of a construction is more than making a neat picture: each point, line, or curve should satisfy the conditions you intended.
Begin by looking for parts you already recognise. A pair of adjoining circles, a small circle inside a large one, and three overlapping circles are different arrangements of the same basic shape. A face combines circles of different sizes with straight mouth and nose details. A scissors design combines crossing straight segments and circular handles. Before reaching for an instrument, draw a rough version and decide which parts need the ruler and which need the compass.
A line traced by a moving pencil. In this broad sense it may be straight, smoothly bent, circular, or irregular.
A ruler guides a straight stroke between positions. A compass has a pointed leg and a pencil leg. Opening or closing its legs changes the distance between the tips. Place the point securely at the chosen centre, keep the opening unchanged, and turn the pencil leg around it. The pencil must reach the paper without forcing the opening wider.
A circle is a distance rule
Place a point P on paper. We want to find every point exactly 4 cm away from P. A point to the right can qualify, but so can a point above, below, or in a slanting direction. The rule specifies distance, not direction. Set the compass to 4 cm and compare the positions reached by its pencil as you turn it.
The curve consisting of all points in a plane at one fixed distance from a fixed point.
The fixed point is the centre. The distance from the centre to any point on the circle is its radius; a segment joining them is also called a radius.
Problem
How can you construct all points at distance 4 cm from P?
- 1.Place a ruler flat and open the compass until its two tips align with 0 cm and 4 cm. The difference between the readings is the required distance.
- 2.Put the pointed tip at P. Hold the compass by its top and turn the pencil leg through a complete turn without changing the opening.
- 3.The pencil traces a circle. Pick several points on it and measure from P: each should be 4 cm away, allowing for small drawing errors.
A point 3 cm from P is inside this circle, and a point 5 cm from P is outside it. Neither is on the circle. The distance rule also explains why different points on the boundary can be far from each other: the equal distances are measured from P, not from one boundary point to another.
The number on the ruler beneath one leg is not enough. If the tips align with 1 cm and 5 cm, the opening is 4 cm. Keep the ruler still while setting the opening, then avoid pressing the legs together as you turn.
Using only part of a circle
A compass does not have to make a complete turn. Stop after drawing the portion you need. Such a portion is an arc. A semicircle is half of a circle; its straight boundary passes through the centre and joins two opposite points of the circle. This straight boundary is a diameter, which contains two radii placed end to end.
A portion of the boundary of a circle.
A straight segment through the centre whose endpoints are on the circle. Its length is twice the radius.
The point halfway along a segment, dividing it into two equal lengths.
In the Person design, draw the circular head and its short connecting segment first, then sketch the straight body sides. The neckline is a shallow arc rather than a semicircle. Its compass centre can lie above the two ends of the neckline. Both ends must be the same distance from that centre. Experiment with the centre position while keeping the two endpoints fixed.
Problem
You have chosen two neckline endpoints 4 cm apart. How could you draw a matching shallow neckline?
- 1.Use a ruler to put the endpoints on one horizontal supporting line. Choose a compass centre above the midpoint so that it is equally distant from the endpoints.
- 2.Open the compass from that centre to one endpoint. Before drawing, check that the same opening reaches the other endpoint.
- 3.Draw only the lower arc between the endpoints. Add the body sides and base with a ruler. If the curve is too deep, choose a more distant centre and reset the opening to an endpoint.
Making a wavy wave
A wave with two equal semicircular parts has a simple plan. Divide its straight baseline into two equal spans. Each span acts as the diameter of one semicircle. The first arc lies above the baseline and the second below it. The compass centres are the midpoints of the two spans, not the point where the waves meet.
Problem
An 8 cm segment AB carries two equal semicircular waves. Find AX and the compass radius.
- 1.The two equal spans together cover AB, so each is 8 ÷ 2 = 4 cm. Therefore AX = XB = 4 cm.
- 2.Each 4 cm span is a diameter, made from two equal radii. The radius is 4 ÷ 2 = 2 cm.
- 3.Mark the two span midpoints, set the compass to 2 cm, and draw one semicircle above and one below the baseline. Keeping the opening unchanged makes the two parts equal.
Problem
Use a 12 cm baseline for two equal semicircular waves. What distances do you need?
- 1.Divide the baseline into two spans: 12 ÷ 2 = 6 cm each.
- 2.Halve a span to find its midpoint and radius: 6 ÷ 2 = 3 cm.
- 3.Set the compass to 3 cm. The centres are 3 cm and 9 cm from the first baseline endpoint. Draw the arcs on opposite sides of the baseline.
For two shallower waves, keep the spans equal but move each centre off the baseline. Put the centre for the upper arc below its span, and the centre for the lower arc above its span. Give the centres equal offsets from their span midpoints and use the same radius. This makes equal curves on opposite sides. A wave segment with a 4 cm chord need not have radius 2 cm: that radius applies specifically when the arc is a semicircle.
On two copies of the same baseline, draw semicircular waves and shallower waves. For the shallower version, keep both spans and both centre offsets equal. Record what changed and what stayed fixed. Use light supporting lines so the final arcs remain easy to see.
Drawing matching eyes
An eye outline has two curved boundaries that share the same endpoints. Place the centre for the upper curve below the eye and the centre for the lower curve above it. Arrange these centres symmetrically about the horizontal line through the endpoints. Use the same opening for both arcs. The pupil is a much smaller circle centred within the eye.
To repeat the eye beside it, copy the endpoint spacing, centre offsets, and compass opening. Guessing each eye independently often produces unequal widths or bends. Supporting lines are part of your construction method even when you erase them from the final picture. Keep them faint until you have checked the result.
The centre of a shallow arc is often outside the small curved part you see. Do not place the compass point on the arc itself. First check that the chosen centre has equal distances to the required endpoints.
Quiz
What describes the points on a circle of radius 4 cm centred at P?
The compass tips align with 2 cm and 7 cm on a ruler. What is the opening?
A point is 3 cm from the centre of a circle of radius 4 cm. Where is it?
What radius makes two equal semicircular waves on a 10 cm baseline?
Which change preserves the two matching eye-outline arcs?
Why are faint supporting lines useful in artwork?
Practice Problems
- Draw a circle centred at P with radius 4 cm. Label three boundary points and explain how to check them.
- Set the compass using ruler readings 3 cm and 6.5 cm. State its opening and draw a circle with that radius.
- Construct two equal semicircular waves on an 8 cm baseline. Label the span midpoints and all required distances.
- Repeat the wave on a 14 cm baseline. Calculate the new spans and radius before drawing.
- Make two equal shallow waves and compare them with semicircles on the same baseline. Describe your centre positions.
- Draw a Person using a circular head, straight body edges, and a shallow curved neckline. Keep the neckline centre visible until checking.
- Construct two matching eyes. Explain which distances and positions you copied.
- Recreate adjoining, nested, and three overlapping circles, then design a face or scissors using circles and straight segments.
Every chosen boundary point should be 4 cm from P. Directions and distances between boundary points can differ.
Key Takeaways
• A circle contains all points at one fixed distance, its radius, from its centre. • A fixed compass opening draws a circle; a partial turn draws an arc. • For a semicircle the span is a diameter, twice its radius. • Matching curves need matching distances and correctly placed centres. • A rough sketch and light supporting lines help turn an idea into a construction.
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Recognising Squares and Rectangles