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There are **53** NRICH Mathematical resources connected to **Area - triangles, quadrilaterals, compound shapes**, you may find related items under Measuring and calculating with units.

Problem
Primary curriculum
Secondary curriculum
### More Isometric Areas

Isometric Areas explored areas of parallelograms in triangular units. Here we explore areas of triangles...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Isometric Areas

We usually use squares to measure area, but what if we use triangles instead?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Farmers' Field Boundary

The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle in a Trapezium

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

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Completing Quadrilaterals

We started drawing some quadrilaterals - can you complete them?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadrilaterals in a Square

What's special about the area of quadrilaterals drawn in a square?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangles in a Square

What are the possible areas of triangles drawn in a square?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Kite in a Square

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Growing Rectangles

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Maths Filler

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gutter

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Trapezium Four

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Areas of Parallelograms

Can you find the area of a parallelogram defined by two vectors?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Isosceles Triangles

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tilted Squares

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shear Magic

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Inscribed in a Circle

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pick's Theorem

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Of All the Areas

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quad in Quad

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Doesn't Add Up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### So Big

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 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Areas and Ratios

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Towering Trapeziums

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bicentric Quadrilaterals

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Maths Filler 2

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Uncanny Triangles

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dotty Triangles

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Biggest Enclosure

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squ-areas

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 triangular areas are enclosed. What is the area of this convex hexagon?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Halving the Triangle

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle Island

You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rati-o

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Disappearing Square

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. Do you have any interesting findings to report?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Overlap

A red square and a blue square overlap. Is the area of the overlap always the same?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Pizza

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rhombus in Rectangle

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Linkage

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Isosceles

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Same Height

A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Arrowhead

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### From All Corners

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

Problem
Primary curriculum
Secondary curriculum
### Kissing Triangles

Determine the total shaded area of the 'kissing triangles'.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dividing the Field

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 trapeziums each of equal area. How could he do this?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Equilateral Areas

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

Problem
Primary curriculum
Secondary curriculum
### Diagonals for Area

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Triangle

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Six Discs

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?

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
### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Age 14 to 16

Challenge Level