E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

Creating and Analysing Designs with Circles and Triangles

E1.2 E1.5 A1 A2About 75 minutes

Learning goals

I can construct a regular hexagon, an equilateral triangle, a square and a rosette in a circle with a compass and straightedge or with geometry software.
I can analyse a design by finding central angles, isosceles triangles and inscribed angles.
I can calculate lengths and areas in a design, using the Pythagorean theorem where I need a height.
I can describe how designers, builders and artists use circle and triangle properties.

Words to know

construction
A drawing made with only a compass and a straightedge (or the same tools in software), so it is exact.

regular polygon
A polygon with all sides equal and all angles equal.

central angle of a regular polygon
The angle at the centre between two neighbouring vertices: 360° ÷ n.

inscribed polygon
A polygon with every vertex on a circle.

circumscribed polygon
A polygon whose sides all touch a circle inside it.

rosette
A flower-like design made from equal circular arcs.

dynamic geometry software
Apps like GeoGebra where you construct shapes and drag points to see which properties stay true.

A hook: where are the circles?

Look at a bicycle wheel, a stained-glass window, a hubcap, a Ferris wheel, a sports logo or a snowflake. Designers use circles for balance and triangles for strength. Every one of those designs hides the properties you learned in the last three lessons.

Today you become a designer and a detective: you will create designs and then analyse them with math.

Classic constructions

Regular hexagon: draw a circle. Keep the compass at the same opening (the radius) and step it around the circle. It fits exactly 6 times. Join the marks.
Equilateral triangle: join every second mark of the hexagon.
Square: draw two diameters that are perpendicular (use the perpendicular bisector of the first diameter). Join the 4 ends.
Any regular polygon with n sides: divide 360° by n to get the central angle, measure it with a protractor around the centre, and join the points.
Rosette: at each hexagon point, draw an arc with the same radius through the centre.

Constructing a regular hexagon and an equilateral triangle in a circle using only a compass and a straightedge.

Analysing a design

To analyse a circle design, look for:

Central angles: n equal parts give 360° ÷ n.
Isosceles triangles: two radii make an isosceles triangle, so the base angles are (180° − central angle) ÷ 2.
Inscribed angles: half the central angle on the same arc.
Right triangles: a height or a half-chord gives a right triangle, so you can use the Pythagorean theorem to find missing lengths and areas.

Left: a rosette made of six arcs, each with the same radius as the circle. Right: a 5-spoke wheel. Each central angle is 360° ÷ 5 = 72°, and each triangle made by two spokes is isosceles with base angles of 54°.

Worked example

Example 1: The 5-spoke wheel

A wheel has 5 equally spaced spokes. Find the central angle between spokes, the base angles of the triangle made by two spokes and the chord joining their ends, and the interior angle of the pentagon made by joining all the spoke ends.

Central angle = 360° ÷ 5 = 72°.
The two spokes are equal radii, so the triangle is isosceles: base angles = (180° − 72°) ÷ 2 = 54°.
At each pentagon vertex, two of these base angles meet: 54° + 54° = 108°.

72°, 54°, 108°

Worked example

Example 2: How much glass?

A stained-glass window is a circle of radius 10 cm with a regular hexagon of clear glass inside it. The 6 pieces outside the hexagon are coloured. Find the coloured area.

The hexagon is 6 equilateral triangles with sides of 10 cm.
Height of one triangle: h² + 5² = 10², so h = √75 ≈ 8.66 cm.
Area of one triangle = ½ × 10 × 8.66 ≈ 43.30 cm². Hexagon ≈ 6 × 43.30 = 259.8 cm².
Circle = π × 10² ≈ 314.16 cm².
Coloured glass ≈ 314.16 − 259.8 ≈ 54.4 cm².

About 54.4 cm² (about 17% of the window).

An equilateral triangle with its circumscribed circle and its inscribed circle. The outer radius is always exactly twice the inner radius.

Worked example

Example 3: A logo ring

A logo is an equilateral triangle inside a circle of radius 6 cm, with the triangle's inscribed circle shaded. Find the area of the ring between the two circles.

In an equilateral triangle, the centre is ⅔ of the way down each median, so the inner radius is half the outer radius: r = 3 cm.
Ring area = π × 6² − π × 3² = 36π − 9π = 27π.
27π ≈ 84.8 cm².

27π ≈ 84.8 cm²

Designs from many communities

Circle and triangle designs appear in art and architecture all over the world: Islamic geometric tile patterns, Gothic rose windows, quilts, and much more.

For many First Nations on the Plains, star blankets (star quilts) are an important art form, often made to honour a person at a special event. The large star is built from many small diamond-shaped pieces arranged around a centre. Analysing it with geometry shows why the pieces fit perfectly.

A simplified eight-pointed star made of 8 diamonds (rhombuses) around a centre. Each diamond has a 45° angle at the centre, so 8 of them fill 360°.

Worked example

Example 4: The eight-pointed star

Eight identical diamonds (rhombuses) meet at the centre of a star. Find all the angles of each diamond.

8 equal angles fill 360°, so each angle at the centre is 360° ÷ 8 = 45°.
Opposite angles of a rhombus are equal, so the far tip is also 45°.
Neighbouring angles of a rhombus add to 180°, so the other two angles are 180° − 45° = 135° each.

45°, 135°, 45°, 135°

Technology: test your design

In GeoGebra or Desmos Geometry, build the design with the circle, intersect, polygon and perpendicular bisector tools, then drag the points. If your hexagon stays a regular hexagon when you drag the centre or change the radius, your construction is correct. Use the angle and area tools to check your calculations.

Careers

Graphic designers build logos from circles and triangles. Architects and engineers use triangles in roof trusses and bridges because a triangle cannot change shape without changing a side length. Machinists and jewellers space holes and stones using central angles. Landscape designers lay out round gardens and paths.

Hands-on activity

Design your own rosette window

You need: paper, a pencil, a ruler, a compass (the point is sharp: a grown-up should supervise) or a string-and-pencil compass (tie a pencil to a string, hold the other end down with your finger), and coloured pencils.

Draw a circle with a radius of 6 cm.
Step the radius around the circle to mark 6 points. Join them to make a hexagon.
Draw a 6 cm arc centred at each point, inside the circle, to make the rosette.
Colour your design so it has rotational symmetry (it looks the same after turning 60°).
Analyse it: label one central angle, one isosceles triangle and its base angles, and find the area of the hexagon (use Example 2 with r = 6).

Model answer for step 5: central angle 60°; the triangles are equilateral (60°, 60°, 60°); triangle height = √(36 − 9) ≈ 5.20 cm; hexagon area ≈ 6 × ½ × 6 × 5.20 ≈ 93.5 cm².

Just for fun

Bees were right all along: in 1999 a mathematician proved that hexagons divide a flat surface into equal cells using the least total wall length of any shape. Honeycomb is the most efficient design.

Your turn

A wheel has 8 equally spaced spokes. What is the angle between neighbouring spokes?

A regular 12-sided polygon is inscribed in a circle. Find its central angle, the base angles of each isosceles triangle, and its interior angle.

A regular hexagon is inscribed in a circle of radius 7 cm. Find its perimeter.

An equilateral triangle is inscribed in a circle of radius 10 cm. What is the radius of its inscribed circle?

A square is inscribed in a circle of radius 8 cm. Find its side length and area.

Why does the radius step around a circle exactly 6 times?

In a logo, a triangle is drawn in a circle with one side along a diameter. What is the angle opposite the diameter?

More practice: warm-up

What is the central angle of a regular pentagon?

What is the central angle of a regular hexagon?

A circle has radius 3 cm. What is its diameter?

How many equilateral triangles make up a regular hexagon (from the centre)?

What tool do you use to draw an exact circle?

More practice: on your own

Find the area of a regular hexagon inscribed in a circle of radius 4 cm.

A five-pointed star is drawn by joining every second vertex of a regular pentagon in a circle. Find the angle at each point of the star.

Find the central angle, base angles and interior angle of a regular octagon.

An equilateral triangle is inscribed in a circle of radius 10 cm. Find the area of the ring between that circle and the triangle's inscribed circle.

A window is a rectangle 80 cm wide and 120 cm tall with a semicircle on top (diameter 80 cm). Find its total area.

Challenge

In a rosette made in a circle of radius 6 cm, each petal is made of two pieces, each a 60° sector minus an equilateral triangle with side 6 cm. Find the area of one petal and of all six.

A hubcap has 5 spokes of length 20 cm. A designer joins the spoke ends to make a pentagon. Explain why the pentagon is regular and find its interior angle.

A logo is a right triangle with legs 6 cm and 8 cm and its inscribed circle. Find the radius of the circle. (Hint: the triangle splits into 3 triangles with the incentre, each with height r.)

Check your understanding

Quiz

A design has 9 equally spaced spokes. What is the central angle between neighbours?

A regular hexagon is inscribed in a circle of radius 5 cm. What is its perimeter?

Two radii and a chord make a triangle with a central angle of 50°. What are the base angles?

An equilateral triangle is inscribed in a circle of radius 8 cm. What is the radius of its inscribed circle?

Eight identical diamonds fill the centre of a star. What is the angle of each diamond at the centre?

Why do engineers use triangles in bridges and roof trusses?

For parents and teachers

Big idea: E1.2 asks students to create and analyse designs with geometric relationships and circle and triangle properties, using various tools. This lesson brings together the previous three lessons in design work, with connections to careers and to art from different communities (A2).

Common difficulties:

Imprecise constructions. Keep the compass opening fixed; a loose compass gives a hexagon that doesn't close.
Forgetting the isosceles triangle. Remind your teen that two radii always make an isosceles triangle.
Areas needing a height. The Pythagorean theorem gives the height of each triangle.

Safety: compass points are sharp; supervise, or use the string-and-pencil method.

How to help: Go on a "design hunt" at home or in the neighbourhood: wheels, logos, windows, quilts. Ask your teen to estimate a central angle or find the regular polygon hidden in each design.