Geometry Terms

Points, lines and planes 5

• LineA line is a perfectly straight path of points that goes on forever in both directions. It has no thickness and no endpoints, so it has no length you can measure. Through any two points there is exactly one line, which is named by those two points, as in line AB, or by one lowercase letter.Grade 4, Grade 8, Geometry
• Line SegmentA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB bar. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Grade 4, Grade 5, Grade 6
• PlaneA plane is a perfectly flat surface that has no thickness and extends forever in every direction, like a tabletop that never ends. It is one of the undefined terms of geometry, along with point and line. Any three points that are not on one line lie in exactly one plane.Grade 7, Grade 8, Geometry
• PointA point is an exact location in space. It has no length, no width and no thickness, so it has no size at all. You draw a point as a dot and name it with a capital letter, such as point A. On a coordinate plane, an ordered pair such as (3, 2) names a point.Grade 4, Grade 5, Grade 6
• RayA ray is part of a line that starts at one endpoint and goes on forever in one direction, like a beam of light from a flashlight. It is named by its endpoint first and then any other point on it, so ray AB starts at A and passes through B. Two rays sharing an endpoint form an angle.Grade 4, Geometry

Angles and lines 14

• Acute AngleAn acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.Grade 4, Grade 5, Grade 7
• Alternate Interior AnglesAlternate interior angles are two angles that lie between two lines and on opposite sides of a transversal that crosses them. When the two lines are parallel, alternate interior angles are equal. They often trace a Z shape: in the diagram, angles 3 and 6 form one pair and angles 4 and 5 the other.Grade 8, Geometry
• AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Grade 4, Grade 5, Grade 6
• Complementary AnglesComplementary angles are two angles whose measures add up to 90 degrees. Each angle is the complement of the other, so the complement of an angle x is 90° − x. They can share a side and form a right angle, or sit apart. For example, 35° and 55° are complementary.Grade 4, Grade 7, Geometry
• Corresponding AnglesCorresponding angles are two angles in the same relative position at each place where a transversal crosses two lines, such as the top-left corner at both crossings. When the two lines are parallel, corresponding angles are equal. A transversal across two lines makes four pairs of corresponding angles.Grade 8, Geometry
• DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1/360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Grade 4, Grade 5, Grade 7
• Obtuse AngleAn obtuse angle is an angle that measures more than 90 degrees and less than 180 degrees. It opens wider than a square corner but does not flatten into a straight line. Angles of 100, 135 and 179 degrees are obtuse. A triangle can have at most one obtuse angle.Grade 4, Grade 5, Grade 7
• Parallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Grade 4, Grade 5, Grade 6
• Perpendicular LinesPerpendicular lines are lines that intersect to form right angles: their intersection makes a 90-degree angle. The symbol ⊥ means is perpendicular to. On a coordinate plane, two perpendicular lines that are not vertical have slopes that multiply to −1, such as 2 and −1/2.Grade 4, Grade 5, Grade 6
• Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Grade 2, Grade 3, Grade 4
• Supplementary AnglesSupplementary angles are two angles whose measures add up to 180 degrees. Each one is the supplement of the other, so the supplement of an angle x is 180° − x. When they share a side, they form a straight line, called a linear pair. For example, 118° and 62° are supplementary.Grade 7, Grade 8, Geometry
• TransversalA transversal is a line that crosses two or more other lines at different points. Crossing two lines makes eight angles, four at each intersection. When the two lines are parallel, those angles come in equal groups: corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to 180 degrees.Grade 8, Geometry
• VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Grade 4, Grade 5, Grade 6
• Vertical AnglesVertical angles are the pair of opposite angles formed when two lines intersect. They share a vertex but no sides, and they are always equal in measure. Two crossing lines make two pairs of vertical angles: if one angle is 120°, the angle across from it is also 120°.Grade 7, Grade 8, Geometry

Triangles 6

• Equilateral TriangleAn equilateral triangle is a triangle whose three sides all have the same length. Its three angles are also equal, so each one measures 60 degrees. With side length a, its perimeter is 3a and its area is (square root of 3)/4a^2. Every equilateral triangle is also isosceles and acute.Grade 4, Grade 5, Grade 7
• Isosceles TriangleAn isosceles triangle is a triangle with at least two sides of equal length. The equal sides are called legs, and the third side is the base. The two angles touching the base, the base angles, are always equal. If the angle between the legs is 40 degrees, each base angle is 70 degrees.Grade 4, Grade 5, Grade 7
• Median of a TriangleA median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side. Every triangle has three medians, and they always meet at one point, the centroid. The centroid divides each median in a 2 : 1 ratio, with the longer part next to the vertex.Geometry
• Scalene TriangleA scalene triangle is a triangle whose three sides all have different lengths. Because no two sides match, no two of its angles are equal either. A scalene triangle can be acute, right or obtuse: the 3-4-5 right triangle is scalene, and so is a triangle with sides 4, 6 and 7.Grade 4, Grade 5, Grade 7
• TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Grade 3, Grade 4, Grade 5
• Triangle Inequality TheoremThe triangle inequality theorem says that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side. So 3, 4 and 5 can form a triangle, but 2, 3 and 6 cannot, because 2 + 3 < 6. Shortcut: the two shorter sides must add to more than the longest.Grade 7, Geometry

Right triangles 6

• 30-60-90 TriangleA 30-60-90 triangle is a right triangle whose angles measure 30°, 60° and 90°. Its sides are always in the ratio 1 : square root of 3 : 2: if the short leg (opposite 30°) is x, the long leg is xsquare root of 3 and the hypotenuse is 2x. So knowing one side gives the other two.Geometry, Precalculus
• 45-45-90 TriangleA 45-45-90 triangle is a right triangle with two 45° angles and two equal legs. Its sides are always in the ratio 1 : 1 : square root of 2: the hypotenuse is a leg times square root of 2. Legs of 5 give a hypotenuse of 5square root of 2. It is half of a square cut along a diagonal.Grade 8, Geometry, Algebra 2
• HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a^2 + b^2 = c^2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Grade 8, Algebra 1
• Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a^2 + b^2 = c^2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Grade 8, Algebra 1
• Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the pythagorean theorem: a^2 + b^2 = c^2.Grade 4, Grade 6, Grade 8
• Special Right TrianglesSpecial right triangles are the two right triangles whose angles are 45°-45°-90° or 30°-60°-90°. Their sides always come in fixed ratios: x : x : xsquare root of 2 for the 45-45-90 triangle and x : xsquare root of 3 : 2x for the 30-60-90 triangle. From one side you can find the other two without the pythagorean theorem.Grade 8, Geometry, Algebra 2

Circles 9

• Arc LengthArc length is the distance along a curved part of a circle, measured in units of length. An arc with central angle θ in a circle of radius r is θ/360 of the circumference: s = θ/360 × 2π r when θ is in degrees, or simply s = rθ when θ is in radians.Geometry, Precalculus
• Central AngleA central angle is an angle whose vertex is the center of a circle and whose sides are two radii. It cuts off an arc, and the arc's degree measure equals the central angle. A 90° central angle marks off a quarter of the circle, and the central angles around the center add to 360 degrees.Grade 7, Geometry
• ChordA chord is a line segment whose two endpoints both lie on a circle. The diameter is the longest chord, because it passes through the center. A chord at distance d from the center of a circle with radius r has length 2square root of (r^2 − d^2), so chords closer to the center are longer.Grade 7, Geometry
• CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r^2.Kindergarten, Grade 1, Grade 7
• CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle's perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Grade 6, Grade 7, Grade 8
• DiameterThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Grade 6, Grade 7, Grade 8
• Equation of a CircleThe equation of a circle with center (h, k) and radius r is (x − h)^2 + (y − k)^2 = r^2. It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x^2 + y^2 = r^2.Geometry, Algebra 2, Precalculus
• PiPi (π) is the number you get when you divide any circle's circumference by its diameter. It is the same for every circle, about 3.14159. Pi is an irrational number: its decimal never ends or repeats, so we round it to 3.14 or use 22/7.Grade 7, Grade 8, Geometry
• RadiusThe radius is a line segment from the center of a circle to any point on the circle, or the length of that segment. The radius is half the diameter: if the diameter is 10, the radius is 5. All radii in the same circle have equal length, and r is the radius in C = 2π r.Grade 6, Grade 7, Grade 8

Area, perimeter and volume 5

• AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle's area is length times width, so a 5 by 3 rectangle has area 15 square units.Grade 3, Grade 4, Grade 5
• Area of a CircleThe area of a circle is the amount of flat space inside the circle, measured in square units. It equals pi times the radius squared: A = π r^2. A circle with radius 5 cm has area 25π, about 78.5 square centimeters. If you know the diameter, halve it first to get the radius.Grade 7, Grade 8, Geometry
• PerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Grade 3, Grade 4, Grade 5
• Surface AreaSurface area is the total area of all outside faces of a three-dimensional figure, measured in square units. To find it, unfold the solid into a flat net, find the area of each face and add them. A box with length l, width w and height h has surface area 2lw + 2lh + 2wh.Grade 6, Grade 7, Grade 8
• VolumeVolume measures the amount of space inside a three-dimensional figure, in cubic units such as cubic centimeters or cubic feet. You can find it by counting the unit cubes that fill the solid. For a rectangular prism, V = l × w × h; for any prism, it equals the area of the base multiplied by the height.Grade 5, Grade 6, Grade 7

Coordinate geometry 9

• Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Grade 5, Grade 6, Grade 7
• Distance FormulaThe distance formula finds the straight-line length between two points on the coordinate plane: d = square root of ((x2 − x1)^2 + (y2 − y1)^2). It is the pythagorean theorem in disguise: the horizontal and vertical differences are the legs of a right triangle, and the distance is the hypotenuse.Grade 8, Algebra 1, Geometry
• MidpointA midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2).Grade 8, Algebra 1, Geometry
• Midpoint FormulaThe midpoint formula gives the point halfway between (x1, y1) and (x2, y2): M = ((x1 + x2)/2, (y1 + y2)/2). You average the two x-coordinates and, separately, the two y-coordinates. For (2, 1) and (10, 7), the midpoint is (6, 4).Grade 8, Algebra 1, Geometry
• Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Grade 5, Grade 6, Grade 7
• OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Grade 5, Grade 6, Grade 8
• QuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point's coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Grade 5, Grade 6, Grade 7
• X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Grade 5, Grade 6, Grade 7
• Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x axis at the origin, (0, 0). A point's y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Grade 5, Grade 6, Grade 7

Transformations 8

• Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Grade 7, Grade 8
• DilationA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Grade 8, Geometry
• Line of SymmetryA line of symmetry is a line that divides a figure into two halves that are mirror images of each other. If you fold the figure along that line, the two halves match exactly. A square has 4 lines of symmetry, a rectangle has 2, and a parallelogram that is not a rectangle has none.Grade 4, Grade 8, Geometry
• ReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Grade 8, Geometry
• RotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Grade 8, Geometry
• SymmetrySymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Grade 4, Grade 8, Geometry
• TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Grade 7, Grade 8, Algebra 1
• TranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Grade 8, Geometry

Congruence and similarity 2

• CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB bar ≅ CD bar reads "segment AB is congruent to segment CD."Grade 8, Geometry
• Similar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Grade 7, Grade 8, Algebra 1

Reasoning and proof 4

• CounterexampleA counterexample is a single example that shows a general statement is false. It fits all the conditions of the statement but does not have the claimed result. To disprove 'every prime number is odd,' one counterexample is enough: 2 is prime and even. Examples that agree with a claim never prove it, but one counterexample disproves it.Grade 8, Algebra 1, Geometry
• ProofA proof is a step-by-step logical argument that shows a math statement is true in every case it covers. It starts from given facts and uses definitions, postulates and statements already proved, with a reason for each step. A statement that has been proved is called a theorem.Geometry, Algebra 1
• TheoremA theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.Grade 8, Geometry
• Two-Column ProofA two-column proof is a way of writing a proof as a table: numbered statements in the left column and the reason for each one in the right column. It begins with the given facts and ends with the statement to be proved. Every reason must be a given, a definition, a postulate, a property or a proved theorem.Geometry

Polygons and quadrilaterals 10

• Interior AngleAn interior angle is an angle inside a polygon, formed at a vertex by two sides that meet there. A polygon with n sides has n interior angles, and they add up to (n − 2) × 180 degrees. A triangle's interior angles total 180 degrees, and a rectangle's four right angles total 360 degrees.Grade 7, Grade 8, Geometry
• ParallelogramA parallelogram is a four-sided shape whose opposite sides are parallel. Its opposite sides are also equal in length, its opposite angles are equal, and its diagonals cut each other in half. Rectangles, rhombuses and squares are special parallelograms. Its area is base times height: A = bh.Grade 3, Grade 5, Grade 6
• PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Grade 2, Grade 3, Grade 5
• Polygon Angle SumThe polygon angle sum is the total of the interior angles of a polygon. For a polygon with n sides it equals (n − 2) × 180°, because diagonals from one vertex cut the polygon into n − 2 triangles. A pentagon's angles add to 540°, and a hexagon's add to 720°.Grade 8, Geometry
• QuadrilateralA quadrilateral is a polygon with four straight sides, four vertices and four angles. Its four interior angles always add to 360 degrees. Squares, rectangles, rhombuses, parallelograms, trapezoids and kites are special quadrilaterals, each named for extra properties of its sides or angles.Grade 2, Grade 3, Grade 5
• Regular PolygonA regular polygon is a polygon whose sides are all the same length and whose angles are all the same size. The equilateral triangle, the square and the stop-sign octagon are regular. Each interior angle of a regular n-gon is ((n − 2) × 180°)/n, and each exterior angle is (360°)/n.Grade 5, Grade 8, Geometry
• RhombusA rhombus is a quadrilateral with four sides of equal length. It is a special parallelogram, so its opposite sides are parallel and its opposite angles are equal. Its diagonals cross at right angles and cut each other in half, and its area is half the product of the diagonals.Grade 3, Grade 4, Grade 5
• SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Kindergarten, Grade 1, Grade 2
• SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s^2, and each diagonal is ssquare root of 2.Kindergarten, Grade 2, Grade 3
• TrapezoidA trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases, and the other two sides are the legs. Its area is the average of the bases times the height: A = (b1 + b2)/2 × h. In British English it is called a trapezium.Grade 3, Grade 5, Grade 6

Solids and three-dimensional figures 5

• ConeA cone is a three-dimensional solid with one flat circular base and a curved surface that narrows to a single point, the apex, like a party hat. Its volume is one third of a cylinder with the same base and height: V = 1/3π r^2 h, where r is the base radius and h the height.Kindergarten, Grade 1, Grade 8
• CylinderA cylinder is a three-dimensional solid with two parallel, equal circular bases joined by one curved surface, like a soup can. Its size is set by the radius r of a base and the height h between the bases. Its volume is π r^2 h, the base area times the height.Grade 1, Grade 8, Geometry
• PyramidA pyramid is a solid figure with one polygon as its base and triangular faces that rise from the base edges and meet at a single top point, the apex. A pyramid is named by its base, such as a square pyramid. Its volume is one third of the base area times the height: V = 1/3Bh.Grade 6, Grade 7, Grade 8
• Rectangular PrismA rectangular prism is a solid shape with six rectangular faces, where opposite faces match and every corner is a right angle. A shoebox and a brick are rectangular prisms. It has 12 edges and 8 vertices, and its volume is length times width times height: V = lwh.Grade 5, Grade 6, Grade 7
• SphereA sphere is a perfectly round solid shape, like a ball, in which every point on the surface is the same distance from the center. That distance is the radius. A sphere has no faces, edges or vertices. Its volume is V = 4/3π r^3 and its surface area is 4π r^2.Kindergarten, Grade 8, Geometry

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Updated November 2026·Reviewed by the Effortless Math team