Resources tagged with: Area - triangles, quadrilaterals, compound shapes

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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.

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.

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?

Triangle in a Trapezium

Age 11 to 16 Challenge Level:

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

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?

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?

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. . . .

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.

Areas and Ratios

Age 16 to 18 Challenge Level:

Do you have enough information to work out the area of the shaded quadrilateral?

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?

Towering Trapeziums

Age 14 to 16 Challenge Level:

Can you find the areas of the trapezia in this sequence?

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?

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?

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.

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.

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?

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.

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?

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.

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?

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?

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. . . .

Diagonals for Area

Age 16 to 18 Challenge Level:

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

Biggest Enclosure

Age 14 to 16 Challenge Level:

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

Maths Filler 2

Age 14 to 16 Challenge Level:

Can you draw the height-time chart as this complicated vessel fills with water?

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

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?