Other tags that relate to All about Ratios
Similarity and congruence. Angle properties of polygons. Fractal. Pythagoras' theorem. Golden ratio. Quadratic equations. Enlargements and scale factors. Ratio. 2D shapes and their properties. Area - triangles, quadrilaterals, compound shapes.There are 30 results
Broad Topics > Measuring and calculating with units > Area - triangles, quadrilaterals, compound shapes
Golden Triangle
Age 16 to 18 Challenge Level:
Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.
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.
Same Height
Age 14 to 16 Challenge Level:
A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?
Another Triangle in a Triangle
Age 16 to 18 Challenge Level:
Can you work out the fraction of the original triangle that is covered by the green triangle?
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.
Dividing the Field
Age 14 to 16 Challenge Level:
A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .
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?
Trapezium Four
Age 14 to 16 Challenge Level:
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
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?
Halving the Triangle
Age 16 to 18 Challenge Level:
Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.
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. . . .
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?
Kite in a Square
Age 14 to 16 Challenge Level:
Can you make sense of the three methods to work out the area of the kite in the square?
Bicentric Quadrilaterals
Age 14 to 16 Challenge Level:
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
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
Equilateral Areas
Age 14 to 16 Challenge Level:
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
Towering Trapeziums
Age 14 to 16 Challenge Level:
Can you find the areas of the trapezia in this sequence?
Of All the Areas
Age 14 to 16 Challenge Level:
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Biggest Bendy
Age 16 to 18 Challenge Level:
Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?
Diagonals for Area
Age 16 to 18 Challenge Level:
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
Areas and Ratios
Age 16 to 18 Challenge Level:
Do you have enough information to work out the area of the shaded quadrilateral?
Maths Filler 2
Age 14 to 16 Challenge Level:
Can you draw the height-time chart as this complicated vessel fills with water?
Gutter
Age 14 to 16 Challenge Level:
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Areas of Parallelograms
Age 14 to 16 Challenge Level:
Can you find the area of a parallelogram defined by two vectors?
Pick's Theorem
Age 14 to 16 Challenge Level:
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Quad in Quad
Age 14 to 16 Challenge Level:
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
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?
Square Pizza
Age 14 to 16 Challenge Level:
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
Arrowhead
Age 14 to 16 Challenge Level:
The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?
Biggest Enclosure
Age 14 to 16 Challenge Level:
Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?
NRICH is part of the family of activities in the Millennium Mathematics Project.