Plane geometry describes figures whose relevant points lie in the same two-dimensional plane. It begins with ideal objects such as points and lines, then builds segments, angles, polygons, and circles. Measurements add a scale to those relationships. Length measures a path, perimeter follows a boundary, and area measures the region enclosed by that boundary.
From a point to a closed figure
A point marks a position but has no length or area. A line extends without end in both directions. A ray has one endpoint, while a segment has two. Two rays with a common endpoint form an angle. Parallel lines never meet in the Euclidean plane; perpendicular lines meet at a right angle. These definitions matter because a construction or proof depends on what may extend and what is bounded.
A polygon is a closed chain of straight segments. Triangles have three sides, quadrilaterals have four, and regular polygons have equal sides and equal interior angles. A circle is not a polygon. It is the set of points at a fixed distance from a center, while the disk includes the interior region. The circumference is the length of that circular boundary.
The measurement decides which formula is relevant
Perimeter uses linear units because it adds boundary lengths. Area uses square units because it counts two-dimensional coverage. For a rectangle, multiplying base by perpendicular height gives area, while adding all four sides gives perimeter. A triangle with the same base and height occupies half the rectangular area. A circle uses its radius for both circumference and area, but the exponent and units distinguish the two quantities.
Do not substitute a sloping side for a perpendicular height. In a parallelogram or triangle, the height meets the chosen base at a right angle and may fall outside the drawn figure. In a trapezoid, the parallel sides are the bases and the height is the perpendicular distance between their supporting lines.
Congruent, similar, and equal-area figures are not the same
Congruent figures match after a rigid motion, so all corresponding lengths and angles agree. Similar figures have equal corresponding angles and proportional lengths. If the length scale is , the area scale is . Two figures can also have equal area without being similar or congruent. A 6 by 4 rectangle and an 8 by 3 rectangle both have area 24 square units, but their perimeters and shapes differ.
Coordinates provide another description. Horizontal and vertical differences give a right triangle, so the distance formula follows from the Pythagorean theorem. Slope describes a line's direction, and polygon area can be calculated from coordinates when no convenient base-height decomposition is visible.
Worked example: remove a corner from a rectangle
Start with an 8 m by 6 m rectangle and remove a 3 m by 2 m rectangle from one corner. Subtract the missing region to obtain the area. For the perimeter, trace the new boundary in order. The two removed outer segments are replaced by inner segments of the same total lengths, so this particular corner cut leaves the perimeter unchanged.
This equality of perimeters is specific to a rectangular cut taken from a corner. A hole cut wholly inside the figure would add a new inner boundary. A notch with slanted or curved edges could change the total in another way. Sketching the boundary prevents a memorized rule from being applied outside its conditions.
Decompose complicated regions before calculating
Many floor plans, plots and mechanical profiles are composite figures. Split the region into non-overlapping rectangles, triangles or circular sectors, or enclose it in a larger shape and subtract the missing pieces. Mark every derived length before calculating and keep full precision until the end. If the problem is genuinely three-dimensional, switch to solid geometry rather than attaching cubic units to a plane-area formula.
