If you take a cotton reel and hold the end of the cotton, then you roll the reel across the ground, the cotton will unwind from the reel. Each time the reel turns once, the length of cotton unrolled is equal to the circumference of the reel. The same holds for a bicycle or car […]

## Constructing an Angle of 45 Degrees

To construct an angle of 45°, start by constructing a right angle. To construct a right angle use the circle theorem which says that a triangle inscribed in a circle, one edge of which is a diameter, is always a right angled triangle. Therefore, start by drawing a circle with a compass. The centre of […]

## Nets and Surface Areas

To find the surface area of a solid shape, we should often first draw the net of the solid. This involves ‘unfolding’ the solid onto a flat surface and finding the areas of individual parts of the net, which are often simple shapes like rectangles, squares, triangles and circles. To find the area of the […]

## Constructing an Equilateral Triangle/60 Degree Angle

We can easily construct an equilateral triangle with a pair of compasses. Start by drawing a line AB. Open the compass to the same length as AB and put the compass needle at B. Draw an arc as shown. Now put the compass needle at A and draw another arc that crosses the first arc […]

## Constructing the Locus of Points a Fixed Distance From a Given Line

To construct the set of points a fixed distance x from a line AB, with a set of compasses draw circles of radius x with centres at A and B, then draw tangents from one circle to another, parallel to the line. For example, construct the set of points 2 cm from the line AB […]

## Constructing a Right Angle

To construct a right angle, we can use the circle theorem which says that a triangle inscribed in a circle, one edge of which is a diameter, is always a right angled triangle. Therefore, start by drawing a circle with a compass. The centre of the circle will be at the position of the compass […]

## Scales and Maps

London is a very big place, and a map of it must be small enough to hold in your hand – an area of almost 1000 square miles must be scaled down so that it can fit on much less than 1 square metre of paper. The scaling here is typically 1:200,000 but may not […]

## Lengths, Areas and Volumes

Ratios of Lengths, Areas and Volumes Imagine two cubes, one with sides of length 4cm and one with sides of length 8cm. The ratio of these lengths is 4 : 8 (= 1 : 2). Since these are cubes, each face has base 4 and 8 respectively, and height 4 and 8 respectively. The area […]

## Lapping Racing Drivers

Good – or more dangerous – racing drivers are faster than other racing drivers. Towards the end of a race, the slower drivers may find themselves being lapped by other racing drivers who have completed one more circuit than they have. If the speeds of the racing drivers areandwithon a circuit ofmiles, how long will […]

## Surface Areas of Different Shapes

Name Shape Surface Area Circular Cone (including base) Cylinder (including top and base) Cuboid Square Based Pyramid (including base)