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Broad Topics > 2D Geometry, Shape and Space > Radius (radii) & diameters

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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Far Horizon

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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Circle-in

Stage: 4 Challenge Level: Challenge Level:1

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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Sports Equipment

Stage: 2 Challenge Level: Challenge Level:1

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

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Perfect Eclipse

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Use trigonometry to determine whether solar eclipses on earth can be perfect.

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Circle Box

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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Flower

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

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Circles in Circles

Stage: 5 Challenge Level: Challenge Level:1

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

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Arrh!

Stage: 4 Challenge Level: Challenge Level:1

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. . . .

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The Eyeball Theorem

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Two tangents are drawn to the other circle from the centres of a pair of circles. What can you say about the chords cut off by these tangents. Be patient - this problem may be slow to load.

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Incircles Explained

Stage: 5

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.

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Not So Little X

Stage: 3 Challenge Level: Challenge Level:1

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?

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Coins on a Plate

Stage: 3 Challenge Level: Challenge Level:1

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

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Escriptions

Stage: 5 Challenge Level: Challenge Level:1

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

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Sangaku

Stage: 5 Challenge Level: Challenge Level:1

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

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Medallions

Stage: 4 Challenge Level: Challenge Level:1

I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized. . . .

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Incircles

Stage: 5 Challenge Level: Challenge Level:1

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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Just Touching

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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Just Rolling Round

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Some(?) of the Parts

Stage: 4 Challenge Level: Challenge Level:1

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle