Geometry: Useful Things to Remember

Age 16 to 18 Challenge Level:
Check that you are familiar with the facts and formulas below, and that you know how and when to apply them. Some of these formulas appear on the formula sheet, so make sure you know which are given and which you have to recall or derive!

Area and Volume
Surface area of a sphere: $4\pi r^2$
Volume of a sphere: $\frac{4}{3} \pi r^3$
Area of curved surface of cone: $\pi r \times$ slant height
Volume of pyramid: $\frac{1}{3} \times$ base area $\times$ height

Angles
Lots of useful trig relations can be applied in geometry questions.
In particular, the cosine rule: $a^2=b^2+c^2-2bc \cos A$
Area of a triangle $=\frac{1}{2}ab \sin C$

Circle theorems
Angle in a semicircle = $90^{o}$
Angles in the same segment are equal
Opposite angles in a cyclic quadrilateral add up to $180^{o}$
Angle at the centre is twice the angle at the circumference

Coordinate geometry
Equation of a line:
1) $y=mx+c$, where $m$ is the gradient and $c$ is the y-intercept
2) $y-y_1=m(x-x_1)$ where $m$ is the gradient and $x_1, y_1)$ is a point on the line
3) $ax+by+c=0$ for constants a, b and c.
It is useful to be able to represent lines in these different ways, and switch between the different representations.

Equation of a circle passing through $(a,b)$ with radius $r$: $(x-a)^2+(y-b)^2=r^2$

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is an equation of an ellipse.

The perpendicular distance from $(h,k)$ to $ax + by + c = 0$ is $\frac{ah + bk + c}{\sqrt{a^2+b^2}}$

The acute angle between lines with gradients $m_1$ and $m_2$ is $\tan ^{-1}|\frac{m_1-m_2}{1+m_1m_2}|$