# Geometry – Definitions of Terms

Acute Angle: An angle of less than 90 degrees. The figure shows an example, in this case the angle is 45°.

Acute Triangle: A triangle having 3 acute angles.

Altitude: In a triangle, an altitude is a line segment that extends from one vertex to the side opposite the vertex, and which is perpendicular to that side. The figure shows the altitude of a triangle, designated as h (for “height”), extending from the top vertex to the side opposite that vertex, designated as b (for “base”). The formula for the area of the triangle makes use of the altitude, as shown.

Base of a Cylinder: One of the two circles that form the top or the bottom of a cylinder. The figure shows the two bases shaded blue.

Base of a Triangle: The side of a triangle to which an altitude is perpendicular, used to calculate the area of the triangle. If h is an altitude, and b is its corresponding base, then the area A of the triangle is calculated by A = ½bh. The figure shows the base of a triangle, designated as b, to which the line h (for “height”) extends from the top vertex.

Center of a Circle: The point in the middle of the circle that is equidistant from all the points on the circle. The figure shows a circle of radius 5 on the coordinate plane, with its center at the point (-3, 4).

Circle: A set of points equidistant from a given point. The given point is the center of the circle, and the distance from the center to any point on the circle is the radius. The figure shows a circle of radius 5 on the coordinate plane, with its center at the point (-3, 4).

Circumference of a Circle: The distance around (or the perimeter of) the circle.

Given by the formula:

\begin{align} C = \pi d\end{align}

where d is the diameter of the circle, and \begin{align}\pi\end{align} is a constant value approximately equal to 3.14.

Circumscribed Circle: A circle which is drawn around a polygon in such a way that all the vertices of the polygon are on the circle. In this case the polygon is inscribed in the circle. The figure shows a circle circumscribed around an octagon.

Circumscribed Polygon: A polygon drawn around a circle in such a way that each side of the polygon is tangent to the circle. In this case, the circle is inscribed in the polygon.

Complementary Angles: Two angles whose sum is 90 degrees.

Cone: A 3-dimensional geometric figure whose base is a circle, and which moves up from the base until it converges to a single point. Its volume is calculated from the formula:

\begin{align} V = \frac{1}{3} \pi r^2 h\end{align}

where r is the radius of the circle at the base, and h is the height from the base to the point at the top.

Congruent Triangles: Two triangles which are exactly equal in their corresponding sides and angles. The figure below shows two congruent triangles, one on the left, the other on the right, created by dividing an equilateral triangle in half. Note the equality of corresponding parts: Each triangle has a side of length 1 and a side of length 2. The vertical line in the middle is a side common to both triangles. Also, the three angles of both triangles are equal to each other: Each triangle has an angle of size 60° and an angle of 30°. Finally, the angle the vertical line makes with the base of the triangle is a right angle for each triangle.

Coordinate Axes: The straight lines, one horizontal (the x-axis), and the other vertical (the y-axis), that define a plane (the Coordinate Plane) for graphing equations. Every real number is on each of these two axes, and the point – called the Origin – at which the axes intersect is 0 on both axes. Moving to the right on the x-axis, the numbers increase, and moving to the left they decrease. Moving up on the y-axis numbers increase, and moving down they decrease. Every point on the coordinate plane has an x-coordinate and a y-coordinate – two real numbers which define exactly where the point is on the plane. The figure shows the x and y axes with two lines, a and b.

Coordinate Plane: See Coordinate Axes

Coordinates: An ordered pair of real numbers (x, y) which identify a point on the coordinate plane. For example, in the figure the coordinates (-3, 4) identify the point which is 3 units to the left of the y-axis and 4 units above the x-axis. This point is the center of a circle of radius 5.

Cube: A 3-dimensional figure of 6 sides, each of which is a square with the same dimensions. Each line where two faces meet is called an edge, and each edge has the same length. The cube’s volume is given by the formula:

\begin{align} V = s^3\end{align}

where s is the length of each edge of the cube.

Cylinder: A 3-dimensional figure whose base is a circle, and which continues straight up to height h until it ends in a circle of the same size as the base. Its volume is given by the formula:

\begin{align} V = \pi r^2 h\end{align}

where r is the radius of the base circle, and h is the height of the cylinder.

Diameter of a Circle: Any line segment from one point on the circle to a point on the opposite side of the circle, passing through the center of the circle. Its length (also called “the diameter”) is always twice the length of a radius.

Edge of a Solid Figure: A solid figure, like a cube, has multiple faces which meet to form the solid. The lines where they meet are edges. Examples: A cube has 6 faces, and each face is a square. The sides of each square are edges of the cube. The figure shows a cube with 3 edges marked as s. As you can see, a cube has 12 edges altogether.

Endpoints: The two points – the red dots in the figure below – which mark the beginning and end of a line segment.

Equilateral Triangle: A triangle whose 3 sides have equal length.

Face of a Geometric Solid: A side of a 3-dimensional geometric object. For example, a cube has 6 faces. These can be seen in the figure which also shows the edges and the formula for calculating the cube’s volume.

Hexagon: A 6-sided polygon.

Hypotenuse: The side of a right triangle opposite the right angle. The figure shows a right triangle with the side marked ‘C’ as the hypotenuse.

Inscribed Circle: A circle within a polygon that is tangent to all sides of the polygon.

Inscribed Polygon: A polygon inside a circle in which every vertex of the polygon is on the circle. In this case, the circle is circumscribed around the polygon, as shown in the figure.

Intercept: The point at which a line passes through the x- or the y-axis. At the y-intercept, the x coordinate for the point will be 0, and at the x-intercept the y coordinate will be 0. For example, in the equation, as shown in the figure:

\begin{align}y = 3x + 12\end{align}

the line passes through the y-axis at y = 12, and through the x-axis as x = -4. Thus, the y-intercept is 12 and the x-intercept is -4.

Isosceles Triangle: A triangle with two equal sides.

Legs of a Right Triangle: The two sides of the right triangle whose intersection forms the right angle. The figure shows a right triangle in which A and B are its legs.

Line: A set of points on the coordinate plane that is defined by a linear equation with x and y as variables. Every pair of (x, y) points that satisfies the equation is on the line, and every pair that does not satisfy the equation is not on the line. An example of such an equation is

\begin{align}y = 3x + 5\end{align}

This equation defines a straight line which passes through the y axis at y = 5.

Line Segment: A finite portion of a line between two endpoints.

Linear Equation: An equation with x, or x and y, as variables, but with every variable having an exponent of 1. Examples:

\begin{align}4x + 2 = 6\end{align}

\begin{align}5x – 3y = 17\end{align}

They are called “linear” because the graph of such an equation on the coordinate plane is a straight line.

Obtuse Angle: An angle larger than 90 degrees but less than 180 degrees.

Obtuse Triangle: A triangle with one obtuse angle.

Origin: The point (0, 0) on the coordinate plane. In the figure the origin is the point at which the vertical or y-axis and the horizontal or x-axis cross.

Parabola: A geometric figure which results by graphing a quadratic formula on the coordinate plane. The figure below shows the general form of the quadratic expression which defines the parabola. Note that if a is less than 0 the parabola rises to a maximum point, whereas if a is positive the parabola goes down to a minimum point.

Parallel Lines: Two lines in the same plane which never intersect.

Parallelogram: A 4-sided geometric figure in which opposite sides are parallel.

Pentagon: A 5-sided polygon.

Perimeter: The distance around a geometric figure, usually a polygon. (The perimeter of a circle is called its circumference.) Example: The perimeter of a rectangle is the sum of the lengths of its 4 sides. If l is the length of the rectangle, and w is its width, then the perimeter p is calculated by p = 2l + 2w. The figure below shows perimeters of several polygons, as well as the perimeter or circumference of a circle.

Perpendicular Lines: Two lines which intersect at a right angle. In the figure, the lines a and b are perpendicular.

Point of Tangency: The single point on a circle and a tangent line where the circle and the tangent line meet. In the figure the point of tangency is where the tangent line and the radius shown intersect.

Point on the Coordinate Plane: A place on the coordinate plane which is exactly identified by two real numbers, the x-coordinate and the y-coordinate of the point. The figure shows a point on the plane with coordinates (-3, 4). It is the center of a circle.

Other examples:

• (4, 3) is the point 4 units to the right on the x-axis, and 3 units up on the y-axis. It’s in the first quadrant.
• (-4, 3) is the point 4 units to the left on the x-axis, and 3 units up on the y-axis. It’s in the second quadrant.
• (-4, -3) is the point 4 units to the left on the x-axis, and 3 units down on the y-axis. It’s in the third quadrant.
• (4, -3) is the point 4 units to the right on the x-axis, and 3 units down on the y-axis. It’s in the fourth quadrant.
• (0, 0) is the origin. It is not in any quadrant, but is the point where the x and y axes intersect.

Polygon: A closed geometric figure consisting of connected line segments. If all its angles are less than 180°, then it is convex. Examples: Triangles, squares, rectangles, pentagons, hexagons. The figure shows an example of a hexagon.

Pyramid: A 3-dimensional object, with a base which is usually a square or an equilateral triangle, and each vertex of which is connected via a straight line to an apex. The figure shows a pyramid with a rectangular base, as well as the formula for calculating its volume.

Pythagorean Theorem: A statement that shows the relationship between the hypotenuse and the legs of a right triangle as a famous formula:

\begin{align}a^2 + b^2 = c^2\end{align}

where a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse.

Quadrants: The four sections of the coordinate plane – the first quadrant is the upper right, the second is the upper left, the third is the lower left, and the fourth is the lower right. A point in any of the quadrants will have positive or negative coordinates as follows: Quadrant I: x positive, y positive; Quadrant II: x negative, y positive; Quadrant III: x negative, y negative; Quadrant IV: x positive, y negative. Points on the two axes do not lie in any quadrant.

Quadrilateral: A 4-sided polygon. Examples: Rectangle, square, parallelogram. The figure shows a rectangle.

Radius of a Circle: Any line segment from the center of the circle to a point on the circle. Its length (also called “the radius”) is always half the length of a diameter.

Rectangle: A 4-sided geometric figure with each angle a right angle.

Rectangular Solid: A 3-dimensional figure with 6 sides, all of which are rectangles. The figure shows an example, together with the formula for calculating its surface area.

Right Angle: An angle of 90 degrees.

Right Triangle: A triangle with a right angle.

Scalene Triangle: A triangle with 3 unequal sides. The figure shows an acute triangle which is also scalene. An obtuse triangle can also be a scalene triangle.

Secant: A line which passes through a circle, intersecting 2 points on the circle.

Similar Triangles: Two triangles with corresponding angles equal to each other. Their corresponding sides are in the same ratio, but they aren’t necessarily equal.

Slope: The rate at which a line is rising or falling. If a line’s slope is positive, then the line is rising as it moves from left to right. If the slope is negative, then the line is falling as it moves from left to right. A simple example is:

\begin{align}y = 3x\end{align}

This line increases its y value by 3 every time its x value increases by 1. Thus, if x = 0, then y = 0. If x = 1, y = 3, etc. We say in this case that the slope of the line is 3.

For an opposite example, consider:

\begin{align}y = -3x\end{align}

On this line if x = 0, then y = 0, but if x = 1, then y = -3. So every time x increases by 1, y decreases by 3. So the slope of this line is -3. The figure below shows 4 possible values for a line’s slope: positive, negative, zero, or undefined.

Slope-Intercept Form: The form of the equation of a line that shows the value of y in terms of x. It is usually given as

\begin{align} y = mx + b\end{align}

where m is the slope of the line, and b is the value of the y-intercept. Notice that when x is 0, then y = b, so the line crosses the y-axis at b. That is, b is the y-intercept.

Special Triangles: These triangles occur frequently in problems because of their “natural” structure.

30-60-90: This triangle is one-half of an equilateral triangle. A line from the apex of an equilateral triangle to the midpoint of the base results in two congruent right triangles (see diagram). If each side of the equilateral triangle is of length 2, then the three sides of each half of the triangle have lengths as shown in the diagram.

45-45-90: This triangle is an isosceles right triangle. If we assume that the legs of the triangle are each of length 1, then its 3 sides have lengths as shown in the diagram.

Sphere: A 3-dimensional figure consisting of a set of points, all of which are the same distance from a single, central point. The distance from the center point to any point on the sphere is the radius of the sphere. The figure shows a sphere of radius r, together with the formula for the surface area of the sphere.

Square: A 4-sided figure with each side the same length and each angle a right angle.

Supplementary Angles: Two angles whose sum is 180 degrees. In the figure angles 1 and 2 are supplementary, as are angles 1 and 4, 2 and 3, and 3 and 4.

Surface Area: The external area of a 3 dimensional geometric object.

Tangent: A line which intersects or “touches on” a circle at exactly one point.

Transversal: A line which intersects two other lines. If the other two lines are parallel, then the transversal makes the same angle with both of them.

Trapezoid: A 4-sided figure with two opposite sides parallel, but whose other two opposite sides are not parallel. The figure shows a trapezoid together with the formula for its area.

Triangle: A 3-sided polygon. The figure shows a triangle, together with the formula for its area.

Vertex: The point of intersection of two sides of a polygon. In the figure the points A, D, and C are vertices of the larger triangle.

Vertical Angles: Two angles, as angles 1 and 3 in the figure, which are on opposite sides of the intersection of 2 lines.

They are always equal to each other, because they are both supplementary to the same angle. For example, in the figure, angles 1 and 3 are vertical angles and are equal because they’re both supplementary to angle 2. Angles 2 and 4 are also vertical angles and are equal.

Vertices: The plural of vertex.

Volume: The measure of the interior capacity of a 3-dimensional geometric object. Example: The volume of a sphere is 4/3 πr³, where r is the radius of the sphere.

x-axis: The horizontal line used on the coordinate plane to identify the horizontal distance of points from the y-axis.

x-coordinate: The first of two numbers which together identify the location of a point on the coordinate plane. The x-coordinate identifies the horizontal distance and direction of the point from the y-axis. If the x-coordinate is positive, then the point is to the right of the y-axis. If negative, the point is to the left. Examples: (2, 3) has x-coordinate 2, which means that the point is 2 units to the right of the y-axis. (-4, 5) has x-coordinate -4, which means the point is 4 units to the left of the y-axis.

x-intercept: The point at which a straight line crosses the x-axis.

y-axis: The vertical line used on the coordinate plane to identify the vertical distance of points from the x-axis.

y-coordinate: The second of two numbers which together identify the location of a point on the coordinate plane. The y-coordinate identifies the vertical distance and direction of the point from the x-axis. If the y-coordinate is positive, then the point is above the x-axis. If negative, the point is below. Examples: (2, 3) has y-coordinate 3, which means that the point is 3 units above the y-axis. (-4, -5) has y-coordinate -5, which means the point is 5 units below the x-axis.

y-intercept: The point at which a straight line crosses the y-axis.