A frustum is that past of a pyramid or cone that lines between two parallel planes cutting it. If one of the planes is the base and the other is horizontal then the result is the shape shown below right (for a cone). The smaller cone top right and the original cone left are similar […]

## How Many Times Does a Wheel Turn When Travelling One Kilometre

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 […]

## Ratios and Similar Shapes

If a shape is enlarged, so that all the sides are multiplied by the same factor, it will remain the same basic shape, the only difference being that all the sides are larger by the same factor. Sometimes though, we may deconstruct a diagram into similar shapes so that an enlargement exists from one shape […]

## Proof of Formula for Curved Surface Area of Cone

For the cone of slant heightand base radiusthe curved surface areais give by the equation To prove this equation, draw the net of the cone. The net of the cone is a sector of a circle, with radiusand arc length equal to The area of the sector is(1) whereis in radians. The circumference of the […]

## Simultaneous Equations With One a Quadratic

When solving ordinary – linear – simultaneous equations we multiply the equations by constant factors to make the coefficient of some variable the same in magnitude, then add or subtract the equations to eliminate that variable. For example, solve (1) (2) (1)*2-(2)*3 eliminatesto give Substitution of this value ofinto (1) to find a gives If […]

## When Hour and Minute Clock Hands Coincide

The hour and minute hands of a clock coincide at midday and midnight. At one am and one pm, they do not coincide because the minute hand has made one complete turn but has to make just over one complete turn to coincide with the hour hand again. One compete turn is 360 degrees, so […]

## Solving Sine Equations for More Than One Solution

To solve the equationinvert both sides, obtaining This is not the only the solution. There are an infinite number of solutions. Typically though, we want solutions in the range 0° – 360°. We can use the sine curve below to find the solutions. Draw the horizontal line y=0.2 (because the equation to solve is sin x […]

## Proof of Formula for the Area of a Triangle

A triangle doesn’t need to be right – angled for the area to be easily found. For the triangle – not rightangled, equliateral or isosceles – labelled as below, The area is calculated from To prove this formula, draw a line from B to meet the line AC (the side labelled b) at right angles. […]

## 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 […]