Sine rule and cosine rule

There are 17 NRICH Mathematical resources connected to Sine rule and cosine rule
Pythagoras for a Tetrahedron
problem
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Pythagoras for a tetrahedron

Age
16 to 18
Challenge level
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In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

Hexy-Metry
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Hexy-metry

Age
14 to 16
Challenge level
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A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
Three by One
problem
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Three by one

Age
16 to 18
Challenge level
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There are many different methods to solve this geometrical problem - how many can you find?
Cubestick
problem
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Cubestick

Age
16 to 18
Challenge level
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Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.
Bendy Quad
problem
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Bendy quad

Age
14 to 16
Challenge level
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Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.
Quadarc
problem
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Quadarc

Age
14 to 16
Challenge level
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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.
Calculating with cosines
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Calculating with cosines

Age
14 to 18
Challenge level
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If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
Square World
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Square world

Age
16 to 18
Challenge level
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P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?
30-60-90 Polypuzzle
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30-60-90 polypuzzle

Age
16 to 18
Challenge level
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Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.
Darts and Kites
problem

Darts and kites

Age
14 to 16
Challenge level
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Explore the geometry of these dart and kite shapes!