Other tags that relate to Lunar Angles
Circle properties and circle theorems. Visualising. Investigations. Creating and manipulating expressions and formulae. 2D shapes and their properties. Similarity and congruence. Pythagoras' theorem. Angles - points, lines and parallel lines. Regular polygons and circles. Sine, cosine, tangent.There are 81 results
Broad Topics > Pythagoras and Trigonometry > Pythagoras' theorem
Baby Circle
Age 16 to 18 Challenge Level:
A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
Incircles Explained
Age 16 to 18
This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.
Two Circles
Age 14 to 16 Challenge Level:
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?
Under the Ribbon
Age 14 to 16 Challenge Level:
A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?
Where Is the Dot?
Age 14 to 16 Challenge Level:
A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?
Crescents and Triangles
Age 14 to 16 Challenge Level:
Can you find a relationship between the area of the crescents and the area of the triangle?
Pythagorean Triples I
Age 11 to 16
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!
Pythagorean Triples II
Age 11 to 16
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Some(?) of the Parts
Age 14 to 16 Challenge Level:
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
Belt
Age 16 to 18 Challenge Level:
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.
Compare Areas
Age 14 to 16 Challenge Level:
Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?
Fitting In
Age 14 to 16 Challenge Level:
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. . . .
Napkin
Age 14 to 16 Challenge Level:
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
Strange Rectangle
Age 16 to 18 Challenge Level:
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.
Retracircles
Age 16 to 18 Challenge Level:
Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.
Circle Scaling
Age 14 to 16 Challenge Level:
Describe how to construct three circles which have areas in the ratio 1:2:3.
Circle Box
Age 14 to 16 Challenge Level:
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?
Where to Land
Age 14 to 16 Challenge Level:
Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
Pythagoras Proofs
Age 14 to 16 Challenge Level:
Can you make sense of these three proofs of Pythagoras' Theorem?
Zig Zag
Age 14 to 16 Challenge Level:
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?
The Dodecahedron
Age 16 to 18 Challenge Level:
What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?
The Pillar of Chios
Age 14 to 16 Challenge Level:
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.
Ball Packing
Age 14 to 16 Challenge Level:
If a ball is rolled into the corner of a room how far is its centre from the corner?
Circumnavigation
Age 14 to 16 Challenge Level:
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
Medallions
Age 14 to 16 Challenge Level:
Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?
Orthogonal Circle
Age 16 to 18 Challenge Level:
Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
Square World
Age 16 to 18 Challenge Level:
P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?
Logosquares
Age 16 to 18 Challenge Level:
Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.
Round and Round
Age 14 to 16 Challenge Level:
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Matter of Scale
Age 14 to 16 Challenge Level:
Prove Pythagoras' Theorem using enlargements and scale factors.
The Medieval Octagon
Age 14 to 16 Challenge Level:
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Circle Packing
Age 14 to 16 Challenge Level:
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? ...
Rhombus in Rectangle
Age 14 to 16 Challenge Level:
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.
Grid Lockout
Age 14 to 16 Challenge Level:
What remainders do you get when square numbers are divided by 4?
Get Cross
Age 14 to 16 Challenge Level:
A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
Holly
Age 14 to 16 Challenge Level:
The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.
At a Glance
Age 14 to 16 Challenge Level:
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
Chord
Age 16 to 18 Challenge Level:
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.
Ladder and Cube
Age 14 to 16 Challenge Level:
A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?
Inscribed in a Circle
Age 14 to 16 Challenge Level:
The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
Floored
Age 14 to 16 Challenge Level:
A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?
Incircles
Age 16 to 18 Challenge Level:
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 ...?
Squ-areas
Age 14 to 16 Challenge Level:
Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .
Are You Kidding
Age 14 to 16 Challenge Level:
If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?
Semi-square
Age 14 to 16 Challenge Level:
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?
Slippage
Age 14 to 16 Challenge Level:
A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .
Three Four Five
Age 14 to 16 Challenge Level:
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.
Rectangular Pyramids
Age 14 to 18 Challenge Level:
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?
Nicely Similar
Age 14 to 16 Challenge Level:
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?
Partly Circles
Age 14 to 16 Challenge Level:
What is the same and what is different about these circle questions? What connections can you make?
NRICH is part of the family of activities in the Millennium Mathematics Project.