Geometry & trigonometry cheat sheet

Area and volume

Area uses square units; volume uses cubic units. Select a formula that matches the shape and its dimensions.

Rectangle: A = lw; triangle: A = ½bh; prism: V = base area × height.

Pythagoras

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

a² + b² = c²; legs 3 and 4 give hypotenuse 5.

Right-triangle trigonometry

For a chosen acute angle: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.

Identify the angle and the side roles before substituting.

Circles and radians

A full turn is 2π radians or 360°. Circumference and area depend on the radius.

C = 2πr; A = πr²; 180° = π radians.

Similarity and scale

Similar figures have equal corresponding angles and proportional corresponding lengths. If lengths scale by k, areas scale by k² and volumes by k³.

Doubling all lengths multiplies area by four and volume by eight.

Sine rule

In a triangle with opposite side–angle pairs, a/sin A = b/sin B = c/sin C. In an SSA configuration, check whether the information permits zero, one or two triangles.

An inverse-sine output alone may miss a second angle with the same sine.

Cosine rule

The cosine rule relates two sides and their included angle to the opposite side: a² = b² + c² − 2bc cos A. Label opposite pairs before substituting.

When A = 90°, cos A = 0 and the rule reduces to Pythagoras.

Arc length and sector area

For a circle of radius r and central angle θ in radians, arc length is rθ and sector area is ½r²θ. Degree values must be converted before using these expressions.

A quarter-turn has θ = π/2, so its arc length is πr/2.

Worked example: an angle of elevation

A vertical object is 12 m high and its base is 5 m horizontally from an observer at base level. tan θ = 12/5, so θ ≈ 67.4°.

State the horizontal distance and observer level; using the sloping distance in the tangent ratio would be incorrect.

Common mistake: calculator angle mode

A trigonometric calculation depends on whether angles are in degrees or radians. Check the stated unit and calculator mode before evaluating or taking an inverse function.

sin 30° = 1/2, while sin(30 radians) is a different value.

Check your understanding

A similar solid has three times every length. What is its volume scale factor?

3³ = 27. Area would scale by 3² = 9.

Curriculum connections

Sources