Other tags that relate to Diagonals for Area
Complex numbers. Circumference and arc length. Trigonometric functions and graphs. 2D shapes and their properties. Trigonometric identities. Area - triangles, quadrilaterals, compound shapes. Regular polygons and circles. Pythagoras' theorem. Non Euclidean Geometry. Sine, cosine, tangent.There are 58 results
Broad Topics > Pythagoras and Trigonometry > Sine, cosine, tangent
Diagonals for Area
Age 16 to 18 Challenge Level:
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
Geometric Trig
Age 16 to 18 Short Challenge Level:
Trigonometry, circles and triangles combine in this short challenge.
The History of Trigonometry- Part 1
Age 11 to 18
The first of three articles on the History of Trigonometry. This takes us from the Egyptians to early work on trigonometry in China.
History of Trigonometry - Part 2
Age 11 to 18
The second of three articles on the History of Trigonometry.
History of Trigonometry - Part 3
Age 11 to 18
The third of three articles on the History of Trigonometry.
Over the Pole
Age 16 to 18 Challenge Level:
Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.
So Big
Age 16 to 18 Challenge Level:
One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2
Gold Again
Age 16 to 18 Challenge Level:
Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.
Pythagoras on a Sphere
Age 16 to 18 Challenge Level:
Prove Pythagoras' Theorem for right-angled spherical triangles.
Trig Reps
Age 16 to 18 Challenge Level:
Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?
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. . . .
Sine and Cosine
Age 14 to 16 Challenge Level:
The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?
Six Discs
Age 14 to 16 Challenge Level:
Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
Figure of Eight
Age 14 to 16 Challenge Level:
On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?
Round and Round a Circle
Age 14 to 16 Challenge Level:
Can you explain what is happening and account for the values being displayed?
Octa-flower
Age 16 to 18 Challenge Level:
Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?
Small Steps
Age 16 to 18 Challenge Level:
Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.
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?
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.
Sine and Cosine for Connected Angles
Age 14 to 16 Challenge Level:
The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.
From All Corners
Age 14 to 16 Challenge Level:
Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.
Strange Rectangle 2
Age 16 to 18 Challenge Level:
Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.
Raising the Roof
Age 14 to 16 Challenge Level:
How far should the roof overhang to shade windows from the mid-day sun?
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?
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?
A Scale for the Solar System
Age 14 to 16 Challenge Level:
The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?
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?
Spokes
Age 16 to 18 Challenge Level:
Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
Degree Ceremony
Age 16 to 18 Challenge Level:
What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?
30-60-90 Polypuzzle
Age 16 to 18 Challenge Level:
Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.
After Thought
Age 16 to 18 Challenge Level:
Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?
Muggles, Logo and Gradients
Age 11 to 18
Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.
Three by One
Age 16 to 18 Challenge Level:
There are many different methods to solve this geometrical problem - how many can you find?
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?
Circle Scaling
Age 14 to 16 Challenge Level:
Describe how to construct three circles which have areas in the ratio 1:2:3.
Flight Path
Age 16 to 18 Challenge Level:
Use simple trigonometry to calculate the distance along the flight path from London to Sydney.
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.
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.
8 Methods for Three by One
Age 14 to 18 Challenge Level:
This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .
Dodecawhat
Age 14 to 16 Challenge Level:
Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.
Farhan's Poor Square
Age 14 to 16 Challenge Level:
From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.
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.
Why Stop at Three by One
Age 16 to 18
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
Doesn't Add Up
Age 14 to 16 Challenge Level:
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
Far Horizon
Age 14 to 16 Challenge Level:
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
Trigonometric Protractor
Age 14 to 16 Challenge Level:
An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.
Shape and Territory
Age 16 to 18 Challenge Level:
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Ball Bearings
Age 16 to 18 Challenge Level:
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
Moving Squares
Age 14 to 16 Challenge Level:
How can you represent the curvature of a cylinder on a flat piece of paper?
Cosines Rule
Age 14 to 16 Challenge Level:
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
NRICH is part of the family of activities in the Millennium Mathematics Project.