Create designs using circle and triangle relationships

Ontario Grade 9 Mathematics

Geometry and Measurement

Use circle centres, radii, and triangle side lengths to build a repeating geometric motif.

Imagine making a pattern for a tile or a greeting card. Two equal circles can overlap to make a lens-shaped space. If you mark their centres and one point where they cross, those three points can form a triangle. By choosing the circles carefully, you control the triangle’s side lengths. In this lesson, you will use that relationship to create a simple design.

What you will learn

Identify the centre and radius of a circle.
Use equal radii and a measured distance between circle centres to create triangles.
Plan and describe a design that uses circle and triangle relationships.

Start with the parts of a circle

A circle is the set of points that are all the same distance from one fixed point. That fixed point is the centre. A radius is a line segment from the centre to any point on the circle.

All radii in the same circle have the same length. If two circles have the same radius, they have the same size. When circles overlap, a point where their outlines meet is an intersection point.

A triangle is a shape with three straight sides. An isosceles triangle has at least two equal sides. An equilateral triangle has three equal sides. These names help describe the triangles that appear in a design.

The centre is equally far from every point on its circle.
Radii in one circle have equal lengths.
A triangle’s side lengths can be compared to describe its shape.

How overlapping circles make triangles

Draw two circles with the same radius. Label their centres

A

and

B

, and label one point where the circles cross as

P

. Connect

A

to

B

,

A

to

P

, and

B

to

P

. These three straight segments form triangle

ABP

.

Because

P

lies on both circles,

AP

and

BP

are radii. They have equal lengths. So triangle

ABP

is isosceles, even if the distance between the centres is different from the radius.

A special design happens when the distance between the centres matches the radius. Then the segment

AB

has the same length as

AP

and

BP

. All three sides are equal, so triangle

ABP

is equilateral.

This is a useful design rule: equal circle sizes give two equal sides in the triangle made by the centres and an intersection point. Matching the centre distance to the radius makes all three sides equal.

AP=BP=r

The two centre-to-intersection segments are radii.
Equal radii give an isosceles triangle.
If the centre distance also equals the radius, the triangle is equilateral.

Plan a clear design

Before drawing, decide what relationship you want viewers to notice. For this lesson’s motif, use equal circles whose centres are one radius apart. Their overlap gives two intersection points. Connecting the centres to either intersection point makes an equilateral triangle.

Use a ruler to mark the centre distance. Use a compass to draw the circles, if one is available. A compass is a drawing tool that keeps the point-to-pencil distance fixed, which helps make a circle with a chosen radius. If you do not have one, trace a round object and mark its centre and radius carefully.

For a larger design, repeat the same motif in a row or around a central point. Keep the circle sizes and centre spacing consistent. Add colour to the circles or triangles, but keep the construction lines clear enough that the relationships can still be seen.

A labelled sketch is part of the design process. Mark the centres, radii, and intersection points. Then add a short note explaining which lengths are equal. This makes the geometric idea visible rather than relying only on decoration.

Choose the relationship first, then draw to match it.
Keep repeated circles and distances consistent.
Labels and a brief explanation help show the design’s structure.

Check the design against its goal

A drawing may look balanced but still miss the intended relationship. Check lengths instead of relying only on appearance. Confirm that the two radii to an intersection point are equal. If you want an equilateral triangle, also check that the distance between the centres is the same.

The design does not need to be complicated. A small number of carefully placed circles can show a strong relationship. The important question is whether the circle parts and triangle sides follow the rule you chose.

Measure or construct equal lengths; do not assume they look equal.
Connect the points that define the triangle.
Make sure the finished design shows the relationship you intended.

What the circle placement creates

Circle relationship	Triangle relationship
Two equal circles; centres joined to a shared intersection	Two equal sides, so the triangle is isosceles
The centre distance also equals the radius	Three equal sides, so the triangle is equilateral

Worked example

Create an equilateral-triangle motif

Make a motif from two circles. Each circle has radius 4 cm. Place their centres 4 cm apart. Explain why connecting the two centres to one circle intersection makes an equilateral triangle.

Set the circle sizes
Draw two circles with radius $4$ cm and label their centres $A$ and $B$ . Choose one point where the circles cross and call it $P$ .
$AP=4\text{ cm},\quad BP=4\text{ cm}$
Use the centre spacing
The centres are $4$ cm apart, so the segment joining them has the same length as each radius.
$AB=4\text{ cm}$
Identify the triangle
The three sides of triangle $ABP$ are all $4$ cm long. A triangle with three equal sides is equilateral. Draw the three segments to show the triangle in the motif.
$AB=AP=BP$

Answer: The motif contains an equilateral triangle because all three sides are 4 cm long.

Check: The two centre-to-intersection sides are radii. The centre-to-centre side was set to the same length, so all three sides match.

Common mistakes and how to avoid them

Calling the triangle equilateral just because the two circles are the same size.

Correction: Equal circles guarantee two equal radii to a shared intersection. The centre distance must also equal the radius for all three triangle sides to be equal.

Measuring from a circle’s edge instead of its centre when setting the centre spacing.

Correction: Mark each centre first. Measure the distance from one centre to the other, not from one circle outline to another.

Drawing circles that only look the same size.

Correction: Use a compass set to the same radius for both circles, or measure carefully before drawing.

Leaving out the segments that show the triangle.

Correction: Connect the two centres to the chosen circle intersection. The circle outlines alone do not make the triangle visible.

Lesson summary

A radius joins a circle’s centre to a point on its outline.
Two equal circles give equal radii to a shared intersection, creating two equal sides in a triangle.
If the centre distance equals the radius, the triangle has three equal sides.
Use careful construction, labels, and repeated spacing to make the relationship clear in a design.

Check your understanding

Question 1

Two equal circles cross at point

P

. Their centres are

A

and

B

. What is always true about triangle

ABP

?

$AP=BP$
$AB=AP$
All three sides are different
correctIndex": 0,
"explanation": "Segments $AP$ and $BP$ are radii of equal circles, so they have equal lengths. The centre distance $AB$ is not always the same as a radius."

Show answer and explanation

AP=BP

Segments

AP

and

BP

are radii of the circles, so they have equal lengths. The centre distance

AB

is not automatically equal to a radius.

Question 2

Each circle has radius 3 cm, and the centres are 3 cm apart. What kind of triangle is formed by the two centres and one shared intersection?

Equilateral
Isosceles, but not equilateral
A shape with four sides
correctIndex": 0,
"explanation": "The two radii are each 3 cm, and the centre distance is also 3 cm. All three sides are equal, so the triangle is equilateral."

Show answer and explanation

Equilateral

The two radii are each 3 cm, and the centre distance is also 3 cm. All three sides are equal, so the triangle is equilateral.

Key terms

Centre: The fixed point in the middle of a circle, equally far from every point on its outline.
Radius: A line segment from a circle’s centre to its outline.
Intersection point: A point where two lines or shapes meet.
Isosceles triangle: A triangle with at least two equal sides.
Equilateral triangle: A triangle with three equal sides.
Motif: A small shape or group of shapes that can be repeated in a design.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation E1.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E1.2 · Create designs using circle and triangle relationships