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There are **16** NRICH Mathematical resources connected to **Sine rule and cosine rule**, you may find related items under Pythagoras and trigonometry.

Problem
Primary curriculum
Secondary curriculum
### Cubestick

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hexy-metry

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras for a Tetrahedron

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadarc

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bendy Quad

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Calculating with Cosines

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Raising the Roof

How far should the roof overhang to shade windows from the mid-day sun?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cyclic Triangles

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Xtra

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.

Age 14 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### The Dodecahedron Explained

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### 30-60-90 Polypuzzle

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square World

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Get Cross

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Just Touching

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?

Age 16 to 18

Challenge Level