Math formulas

What is the volume of a cube?

V = a^3 (where a is the side length)

What is the surface area of a cube?

SA = 6a^2

What is the volume of a sphere?

V = \frac{4}{3}\pi r^3

What is the surface area of a sphere?

SA = 4\pi r^2

What is the volume of a hemisphere?

V = \frac{2}{3}\pi r^3

What is the surface area of a hemisphere?

SA = 3\pi r^2 (includes flat circular base)

What is the volume of a pyramid?

V = \tfrac{1}{3} \times \text{Base Area} \times \text{Height}

What is the surface area of a pyramid?

SA = \text{Base Area} + \tfrac{1}{2} \times \text{Perimeter of Base} \times \text{Slant Height}

What is the volume of a cone?

V = \tfrac{1}{3}\pi r^2 h

What is the surface area of a cone?

SA = \pi r^2 + \pi r l (where l = \text{slant height})

What is the area of a circle?

A = \pi r^2

What is the circumference of a circle?

C = 2\pi r

What is the area of a semicircle?

A = \tfrac{1}{2}\pi r^2

What is the perimeter of a semicircle?

P = \pi r + 2r

What is the area of a trapezium?

A = \tfrac{1}{2}(a+b)h (where a and b are parallel sides)

What is the perimeter of a trapezium?

P = a + b + c + d

What is the area of a parallelogram?

A = b \times h (base × height)

What is the perimeter of a parallelogram?

P = 2(a+b) (where a,b are adjacent sides)

What is the area of a rhombus?

A = \tfrac{1}{2} d1 d2 (product of diagonals)

What is the perimeter of a rhombus?

P = 4a (where a is side length)

How do you find an angle in a right triangle using sine?

\sin \theta = \tfrac{\text{opposite}}{\text{hypotenuse}}

How do you find an angle in a right triangle using cosine?

\cos \theta = \tfrac{\text{adjacent}}{\text{hypotenuse}}

How do you find an angle in a right triangle using tangent?

\tan \theta = \tfrac{\text{opposite}}{\text{adjacent}}

How do you find a side of a right triangle using sine?

Opposite = Hypotenuse × sin(\theta)

How do you find a side of a right triangle using cosine?

Adjacent = Hypotenuse × cos(\theta)

How do you find a side of a right triangle using tangent?

\text{Opposite} = \text{Adjacent} \times \tan(\theta) or \text{Adjacent} = \text{Opposite} \div \tan(\theta)

What is the formula for direct proportion?

y \propto x \to y = kx (where k is the constant of proportionality)

What is the formula for inverse proportion?

y \propto \tfrac{1}{x} \to y = \tfrac{k}{x}

How do you find an angle using sine inverse?

\theta = \sin^{-1}\Big(\tfrac{\text{opposite}}{\text{hypotenuse}}\Big)

How do you find an angle using cosine inverse?

\theta = \cos^{-1}\Big(\tfrac{\text{adjacent}}{\text{hypotenuse}}\Big)

How do you find an angle using tangent inverse?

\theta = \tan^{-1}\Big(\tfrac{\text{opposite}}{\text{adjacent}}\Big)

When do you use sine, cosine, or tangent (normal trig functions)?

When you want to find the length of a side in a right triangle and you already know an angle.

When do you use inverse sine, cosine, or tangent?

When you want to find the angle in a right triangle and you already know two sides.

Example — If you know opposite and hypotenuse, how do you find the angle?

\theta = \sin^{-1}(\tfrac{\text{opposite}}{\text{hypotenuse}})

Example — If you know adjacent and hypotenuse, how do you find the angle?

\theta = \cos^{-1}(\tfrac{\text{adjacent}}{\text{hypotenuse}})

Example — If you know opposite and adjacent, how do you find the angle?

\theta = \tan^{-1}(\tfrac{\text{opposite}}{\text{adjacent}})