Geometry: Circles and Polygons 171

angles

The space formed between two lines or surfaces that meet at a common point. Angles are used to define part features, hole layouts, and tool paths.

arcs

A portion of the circumference of a circle. Arcs appear as curved edges, paths, or features on parts and tools.

area

The amount of space contained within a closed shape. Area is used in manufacturing to determine material usage, surface coverage, and part size.

bearings

A machine element that supports and guides a moving part while reducing friction. A bearing is commonly used to support shafts, wheels, or rotating assemblies in machinery.

belt lengths

The total distance that a belt travels around a pulley system or belt drive. Knowing belt length ensures proper tension, alignment, and motion transmission.

bolt circles

A set of holes arranged in a circular pattern around a central point. Bolt circles, also called bolt hole patterns, are used to space fasteners or mounting holes for assemblies.

bolt hole pattern

A set of holes arranged in a circular pattern around a central point. Bolt hole patterns, also known as bolt circles, are used to space fasteners or mounting holes for assemblies.

CAD

computer-aided design. Software used to create precise digital models of parts and assemblies. CAD allows engineers and designers to visualize geometry, plan tool paths, and simulate machining operations.

center

The point that is equidistant from all points along the edge of a circle. The center is used as a reference for measurements, radii, and circular layouts in manufacturing.

central angle

An angle formed at the center of a circle by two radii extending to the circumference. Central angles are often used to locate features, define spacing, or control motion in circular parts.

circle

A shape consisting of a set of points that are all equidistant from a central point. A circle measures 360 degrees around its center and is commonly used to define round features such as holes and shafts.

circumference

The total distance around the outside edge of a circle. Circumference is used in manufacturing to calculate rotations, belt lengths, and material coverage.

closed shape

A shape whose sides connect to form a continuous boundary with no openings. Closed shapes define the interior space used for measurements, material calculations, and part layouts.

coating

A layer of material applied to a surface to protect or enhance it. Coatings are used to prevent corrosion, reduce wear, and improve part performance.

computer-aided design

CAD. Software used to create precise digital models of parts and assemblies. Computer-aided design allows engineers and designers to visualize geometry, plan tool paths, and simulate machining operations.

concave polygons

A polygon that has at least one interior angle greater than 180°, creating an indentation. Concave polygons may appear in complex part shapes or cutouts.

convex polygon

A polygon in which all interior angles are less than 180° and no sides cave inward. Convex polygons provide structural stability and are easier to machine and measure accurately.

corresponding

Located in the same relative position in two or more related figures or objects. Corresponding angles or points are often used to compare shapes or establish alignment in drawings and measurements.

cutouts

An area removed from a part to reduce weight, allow clearance, or create a specific shape. Cutouts are often used in brackets, plates, and housings for functional or design purposes.

diameter

A straight line that passes through the center of a circle and connects two points on its edge. The diameter is twice the length of the radius and is used to specify hole and shaft sizes.

dimensions

A measurable extent of length, width, height, or angle. Dimensions specify the size and position of features on a part to ensure proper fit and function.

equilateral triangles

A triangle in which all three sides and angles are equal. Equilateral triangles are commonly used in brackets, supports, and structural layouts for stability.

exterior angles

The angle formed outside a polygon by extending one of its sides. Exterior angles are used to help calculate interior angles and analyze polygon geometry.

features

A specific geometric element or detail of a part that has a defined shape and size. A feature represents functional aspects such as holes, slots, or surfaces that are critical to part design and manufacturing.

formula

A mathematical expression that shows the relationship between variables or quantities. Formulas are used to calculate dimensions, angles, areas, and other key values in manufacturing.

frames

A rigid structure that supports or encloses a component or assembly. Frames provide strength, alignment, and stability in manufactured products.

geometry

A branch of mathematics that studies the size, shape, and spatial relationships of objects. Geometry is used in manufacturing to design, create, and measure parts.

hexagon

A polygon with six sides and six angles. Hexagons are often seen in fastener heads.

holes

An opening or cavity through a part or material. A hole is often used to allow fasteners, shafts, or fluids to pass through a component.

indexing

The process of positioning a part or tool at precise intervals around a central axis. Indexing is used in machining and assembly to create evenly spaced features, such as holes, slots, or bolt patterns.

infinite decimal number

A number with a decimal expansion that never ends or repeats. Infinite decimal numbers can be rounded down to make them easier to use in calculations.

interior angles

The angle formed inside a polygon where two sides meet at a vertex. Interior angles are used to calculate polygon geometry and ensure accurate layout and fit of manufactured parts.

intersect

To meet or cross at a single point. Intersecting points define vertices and angles in geometric layouts and part features.

line

A straight, one-dimensional path that extends infinitely in both directions. Lines are used as references for measurements, edges, and toolpaths.

line segment

A straight portion of a line with two endpoints. Line segments represent finite edges or features on a part.

linear pair

Two adjacent angles whose non-common sides form a straight line. The angles in a linear pair always add up to 180 degrees and are used to find unknown angle values.

material estimation

The process of calculating the amount of raw material needed to produce a part. Accurate material estimation reduces waste and helps control production costs.

models

A digital or physical representation of a part or assembly. Models allow you to visualize features, verify dimensions, and plan manufacturing operations.

n-gons

A polygon with more than twelve sides, where “n” represents the number of sides. N-gons are rare in manufacturing but may appear in specialized designs or large patterns.

parallel

Describes lines or surfaces that are always the same distance apart and never intersect. Parallel geometry helps maintain uniform spacing and consistent part dimensions.

pentagon

A polygon with five sides and five angles. Pentagons can appear in design layouts or decorative components but are less common than triangles and quadrilaterals.

perpendicular

Describes two lines or surfaces that meet to form a 90-degree angle. Perpendicular features are common in prints that specify exact corner or hole placement.

pi

π. A mathematical constant representing the ratio of a circle’s circumference to its diameter. Pi is often rounded down to 3.14 for calculating the measurements of circles and arcs.

point of tangency

The single point where a tangent touches a curve or circle. The point of tangency is critical for calculating angles, radii, and tool paths in circular features.

points

An exact location in space represented by a single position with no size, length, or width. A point is used to define positions such as the center of a circle or the endpoints of lines in geometric constructions.

polygon

A closed shape with three or more straight sides. A polygon has the same number of sides and interior angles, which helps define shapes used in flat surfaces and structural features.

prints

A technical drawing that represents a part’s geometry, dimensions, and features. Prints guide manufacturing, inspection, and assembly processes.

quadrilateral

A polygon with four sides and four angles. Quadrilaterals include squares, rectangles, and parallelograms.

radius

The distance from the center of a circle to any point on its edge. Radii are used to define circular features and calculate measurements such as diameter, circumference, and area.

ray

A straight line that has one endpoint and extends infinitely in one direction. Rays are used in geometry to define directions, angles, and projections.

regular polygons

A polygon in which all sides and angles are equal. Regular polygons are often used in manufacturing for symmetrical parts and evenly spaced features.

right triangles

A triangle that contains one 90-degree angle. Right triangles are often used in calculations involving perpendicular lines, measurements, and trigonometric relationships.

semicircle

A shape that represents exactly half of a full circle. A semicircle measures 180 degrees and is commonly used when calculating angles within half-round features.

shafts

A cylindrical component that transmits torque or rotation. Shafts are critical in mechanical systems for motion and power transfer.

specifications

A documented requirement that defines the properties, dimensions, or performance of a part. Specifications guide design, manufacturing, and quality control.

square

The result of multiplying a number by itself. Squares are used in calculations for areas, radii, and other geometric measurements.

square units

A unit used to measure the area of a shape, representing the space inside it. Square units are literally squares with sides of that unit length, such as square inches or square centimeters, used to measure total area.

squares

A four-sided polygon in which all sides are of equal length and all angles are 90 degrees. Squares are used in plates, frames, and part layouts that require right angles and symmetry.

structural components

A part designed to carry loads and provide strength or stability to an assembly or structure. Structural components are critical in maintaining shape, alignment, and integrity under forces such as tension, compression, or bending.

tangent

A line, line segment, or ray that touches a circle or curve at exactly one point. Tangents are used in manufacturing to define contact points and angles between circular and straight features.

toolpaths

The route that a cutting or forming tool follows to shape a part. Toolpaths are programmed in CNC machines to produce accurate and consistent features.

triangle

A polygon with three sides and three angles. Triangles are commonly used in brackets, supports, and structural components in manufacturing.

vertex

The point where two lines or edges meet to form an angle. Vertices are key reference points for measuring angles and positioning features on a part.

vertical angles

Two angles that are opposite each other when two lines intersect. Vertical angles always have equal measures and help calculate unknown angle values.