Archive | Geometry

Surface Areas of Frustums

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 […]

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 […]

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. […]

The Ambiguous Case

The ambiguous case arises when using the Sine Rule to find an angle in a triangle. It occurs because the Sin function is symmetric about 90°, so that When we solve forthere is an acute solution,and an obtuse solution, Example: Find the angle A in the triangle below. The Sine Rule states Then This is […]

Tessellations

Tessellating a plane is to cover it with shapes leaving no gaps. The shapes may all be the same, or there may be several shapes. Each shape may be rotated, reflected or translated to fit somewhere else leaving no gaps. The shapes tessellating the planes below are all identical, but have been rotated, translated or […]

Drawing Three Dimensional Solids With a Constant Cross Section

Three dimensional solids with a constant cross section can be easily drawn by drawing a copy of the cross section in the background and joining up corresponding vertices. To draw a three dimensional solid with the cross section below: Draw a copy of the cross section in the background: Now join up the vertices. The […]

Vectors I

A vector is the difference between two points. If two points areandthen we can write the difference(when written in a column between brackets) asThe arrow above means that we are going from the pointto the pointIf we swapandthen we are going fromtoand writeThis is shown below. We can also write points as pairs of coordinates and […]

Parts of a Circle

Labels for the different parts of a circle are given below Radius – the radius of a circle is the distance from the centre labelled O to the circumference. The radius is labelled The circumference of a circle is the perimeter, and is labelled Diameter – diameter of a circle is the length of a […]