A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
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?
What is the same and what is different about these circle
questions? What connections can you make?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
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.
Can you find the area of a parallelogram defined by two vectors?
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.
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 areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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
Determine the total shaded area of the 'kissing triangles'.
Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A tower of squares is built inside a right angled isosceles
triangle. The largest square stands on the hypotenuse. What
fraction of the area of the triangle is covered by the series of
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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?
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .
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.
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
Can you maximise the area available to a grazing goat?
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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. . . .
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What happens to the area and volume of 2D and 3D shapes when you
A task which depends on members of the group noticing the needs of
others and responding.
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?
Derive a formula for finding the area of any kite.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
What fractions of the largest circle are the two shaded regions?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Can you work out the area of the inner square and give an
explanation of how you did it?
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in
the middle. Cut the carpet into two pieces to make a 10 by 10 foot
Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Draw two circles, each of radius 1 unit, so that each circle goes
through the centre of the other one. What is the area of the
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
How efficiently can you pack together disks?