2D Shapes

Triangles are categorized by both their side lengths and their internal angles. An equilateral triangle has three equal sides and three equal angles of $60^\circ$ each.

An isosceles triangle features at least two equal sides and two equal base angles, while a scalene triangle has no equal sides and no equal angles.

When classified by angles, a right-angled triangle contains one $90^\circ$ angle. The side opposite this right angle is always the longest and is known as the hypotenuse.

Regardless of the type, the sum of the interior angles in any triangle always equals exactly $180^\circ$.

A circle is a set of all points in a plane that are at a fixed distance, the radius ($r$), from a central point. The diameter ($d$) is a straight line passing through the center, equal to $2r$.

The boundary of the circle is the circumference. The ratio of the circumference to the diameter is the mathematical constant $\pi \approx 3.14159$.

Internal regions include the sector (a 'pie slice' bounded by two radii and an arc) and the segment (the area between a chord and an arc).

A tangent is a straight line that touches the circumference at exactly one point and is always perpendicular to the radius at that point of contact.

Check for Regularity: When a question mentions a 'regular' shape, immediately apply the rule that all sides and angles are equal. This is often the key to solving missing angle problems.

Radius vs. Diameter: Always verify which measurement is provided in a circle problem. A common error is using the diameter in a formula that requires the radius, or vice versa.

Quadrilateral Properties: Memorize the diagonal properties. For example, knowing that diagonals of a rhombus and a kite meet at $90^\circ$ can help identify the shape from a coordinate grid.

Sanity Check: Ensure your identified shape matches the visual cues. For instance, an obtuse angle in a triangle means it cannot be equilateral or right-angled.