Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
Prove Pythagoras' Theorem using enlargements and scale factors.
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
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?
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
This is the second article on right-angled triangles whose edge lengths are whole numbers.
A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.
What is the same and what is different about these circle questions? What connections can you make?
Can you make sense of these three proofs of Pythagoras' Theorem?
Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
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
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
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 ...?
Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?
A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.
Can you make sense of the three methods to work out the area of the kite in the square?
Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.
Can you work out the dimensions of the three cubes?
Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.
Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.
Describe how to construct three circles which have areas in the ratio 1:2:3.
What remainders do you get when square numbers are divided by 4?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?
A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...
Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
If a ball is rolled into the corner of a room how far is its centre from the corner?