Sine rule and cosine rule

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Quadarc

Age

14 to 16

Challenge level

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.
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Bendy Quad

Age

14 to 16

Challenge level

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.
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Hexy-Metry

Age

14 to 16

Challenge level

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
problem
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Cubestick

Age

16 to 18

Challenge level

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.
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Pythagoras for a Tetrahedron

Age

16 to 18

Challenge level

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.
problem

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 ?
problem

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.
problem

Darts and Kites

Age

14 to 16

Challenge level

Explore the geometry of these dart and kite shapes!
problem

Just Touching

Age

16 to 18

Challenge level

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?
problem

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?