What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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. . . .

Can you find the area of the central part of this shape? Can you do it in more than one way?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

What is the same and what is different about these circle questions? What connections can you make?

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Can you find pairs of differently sized windows that cost the same?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

Use the applet to explore the area of a parallelogram and how it relates to vectors.

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?

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

Can you draw a square in which the perimeter is numerically equal to the area?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?

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

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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

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

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Weekly Problem 17 - 2015

A square contains two overlapping squares. What is the total of the shaded regions?

Can you make sense of these three proofs of Pythagoras' Theorem?

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

This selection of problems is a great starting point for learning about Perimeter and Area.

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

This article for teachers gives some food for thought when teaching ideas about area.

The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?

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

Working on these Stage 3 problems will help you develop a better understanding of perimeter, area and volume.

Weekly Problem 23 - 2016

If the area of a face of a cuboid is one quarter of the area of each of the other two visible faces, what is the area of these faces?

Working on these Stage 3 problems will help you develop a better understanding of perimeter, area and volume.

Working on these Stage 4 problems will help you develop a better understanding of perimeter, area and volume.

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

Working on these problems will offer you insights into how areas and perimeters are related.

Working on these Stage 4 problems will help you develop a better understanding of perimeter, area and volume.

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

How much of the square is coloured blue? How will the pattern continue?

Draw a triangle and construct a rectangle on its longest side. Use GeoGebra to explore how the area of the rectangle can change...

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

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

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle 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.