Looking for some Geometry Help? Our materials here review the basic terms and concepts in geometry and provide further lessons to help you develop your understanding of geometry and its applications to solving problems in real life. Geometry is about the shape and size of things. It is the study of points, lines, angles, shapes, their relationships, and their properties.
Examples, solutions, and videos have been included in almost all the following topics to help reinforce your understanding.
Basic Geometry Terms
Points, Lines, Collinear, Line Segments, Midpoints, Rays, Planes, Coplanar, Space
Pairs of Lines
Intersecting Lines, Parallel Lines, Perpendicular Lines, Skew Lines
Angles
How to Name an Angle. Angles Around a Point.
Measuring Angles
How to use a protractor to measure an angle.
Drawing Angles
How to use a protractor to draw different types of angles
Types of Angles
Right Angles, Acute Angles, Obtuse Angles, Straight Angles, Reflex Angles and Full Angles
Pairs of Angles
Complementary, Supplementary, Vertical, Corresponding, Alternate Interior, Alternate Exterior and Adjacent Angles
Complementary & Supplementary Angles
Solve problems using Complementary and Supplementary Angles
Vertical Angles
Solving problems using Vertical Angles, Proof of the Vertical Angle Theorem
Corresponding Angles
Corresponding Angle Theorem, Converse of the Corresponding Angle Postulate
Alternate Interior & Exterior Angles
Alternate Interior Angle and Alternate Exterior Angle Theorems, Proofs and Converse
“Find the angle” problems
Summary of all the different angle properties and how they can be used to find missing angles
Types of Triangles
Right, Acute, Obtuse, Equilateral, Equiangular, Isosceles, Scalene. Oblique Triangles
Triangles
Right, Acute, Obtuse
Triangles
Equilateral, Isosceles, Scalene
Special Right Triangles
3-4-5 Triangles, 5-12-13 Triangles, 45-45-90 Triangles, 30-60-90 Triangles
3-4-5 Triangles
45-45-90 Triangles
30-60-90 Triangles
Pythagorean Theorem
How to use the Pythagorean Theorem, Converse of the Pythagorean Theorem, Worksheets, Pythagorean Theorem Proofs
Pythagorean Triples
Examples of Pythagorean Triples, Families of Pythagorean Triples, Pythagorean Triples and Right Triangles,
Solving Problems using the Pythagorean Triples, How to generate Pythagorean Triples
Pythagorean Theorem Word Problems
How to use the Pythagorean Theorem to solve word problems
Converse of the Pythagorean Theorem
Explain how to use the Converse of the Pythagorean Theorem. Proof of the Converse of the Pythagorean Theorem
Congruent Triangles
SSS rule, SAS rule, AAA rule, AAS rule, HL rule for congruent triangles, CPCTC
SSS rule, SAS rule, ASA rule, AAS rule
Explain the rules, How to use two-column proofs to prove triangles congruent
Hypotenuse Leg (HL)
Why HL is sufficient to prove two right triangles congruent and How to use HL postulate in two-column proofs
Similar Triangles
Properties of similar triangles, AA rule, SAS rule, SSS rule, Solving problems with similar triangles
Triangle Inequality
Triangle Inequality Theorem, Angle-Side Relationship
Triangle Sum Theorem
Proof of the Triangle Sum Theorem. How to use the Theorem to solve geometry problems involving triangles
Exterior Angle Theorem
How to use the Exterior Angle Theorem, How to prove the Exterior Angle Theorem
Interior Angles of a Triangle
Properties of Interior Angles, Solve problems involving interior angles
Exterior Angles of a Triangle
Find unknown exterior angles, Proof the sum of exterior angles
Angles of a Triangle
Summary of the properties of angles in a triangle
Law of Sines or Sine Rule
How to use the Law of Sines, Ambiguous case, Proof for the Law of Sines, Applications using the Law of Sines
Law of Cosines or Cosine Rule
How to use the Law of Cosines, Proof for the Law of Cosines, Applications using the Law of Cosines
Solving a Triangle - SAS - Finding Missing Sides/Angles
How to solve triangles using the Law of Sines
Polygons
Types of Polygons:
simple or complex, convex or concave, equilateral,
equiangular, regular or irregular, Naming Polygons
Angles in Polygons
Sum of Angles
in a Triangle, Dividing Polygons into Triangles, Formula
for the Sum of Interior and Exterior Angles of a Polygon
Quadrilaterals
Parallelogram,
Square, Rhombus, Rectangle, Trapezoid, Kite, Trapezium
Polygons
Describe the characteristics of a polygon
Area of Polygons
Formulas for the area of Square, Rectangle, Parallelogram, Triangle, Rhombus, Kite, Trapezoid, any Regular Polygon
Area of Squares and Rectangles
Formulas and practice for the area of Square and Rectangle
Area of Parallelograms
Formula for the area of a parallelogram, Derive the formula for the area of a parallelogram, Word problems using
parallelograms
Area of Triangles
Use of the different formulas to calculate the area of triangles, given base and height, given three sides,
given side angle side, given equilateral triangle, given triangle drawn on a grid, given three vertices
on coordinate plane, given three vertices in 3D space
Area of Rhombus
Use of the different formulas to calculate the area of rhombus, given base and height, given lengths of diagonals,
given side and angle
Area of Trapezoids
Area of trapezoids, Derive area formula of trapezoids, Solve problems using area of trapezoids
Area of Shaded Region
How to calculate the area of shaded regions involving polygons and circles.
Perimeters of Polygons
Squares, Rectangles, Parallelograms, Triangle, Rhombus, Trapezoids, Word Problems involving perimeters of polygons
Circles
Diameter, chord, radius, arc, semicircle, minor arc, major arc, tangent, secant, circumference, area, sector
Parts of a circle
Diameter, Chord, Radius, Arc, Tangent, Intersecting Circles, Internal and External Tangents
Circumference of circle
Find pi, Formula for circumference of circle, Find circumference, Find radius, diameter and area when
given circumference
Arc of a Circle
Arc of a circle, Central Angle, Arc Measure, Arc Length Formulas for arc
measure given in degrees or in radians.
Area of circle
Formula for area of circle, Find area, Find radius, diameter and circumference when given area
Area of Sector
Area of a sector formula in degrees and radians, area of segment
Area of Shaded Region
How to calculate the area of shaded regions involving polygons and circles.
[Tangent to a Circle
Point of Tangency, Tangent to a Circle Theorem, Secant, Two-Tangent Theorem, Common Internal
and External Tangents
Angles, Tangents & Circles
Find angles involving Tangents and Circles
Degrees and Radians
Measure angles in degrees, minutes and seconds, Convert to decimal notation, Add and subtract
angles, Measure angles in radians, Convert between degrees and radians
Arc Length of Circle in Radians
Formula for arc length when arc measure is in radians, Solving problems using arc length formula
Area of Sector
Area of a sector formula in degrees and radians, area of segment
Chords of a Circle
Perpendicular bisector of a chord passes through the center of a circle, Congruent chords are
equidistant from the center of a circle, If two chords in a circle are congruent, then their
intercepted arcs are congruent, If two chords in a circle are congruent, then they determine
two central angles that are congruent.
Angles and Intercepted Arcs
Formulas relating the angles and the intercepted arcs of circles.
Measure of a central angle.
Measure of an inscribed angle (angle with its vertex on
the circle)
Measure of an angle with vertex inside a circle.
Measure of an angle with vertex outside a circle.
The Inscribed Angle Theorem
Inscribed angles and central angles, The Inscribed Angle Theorem or The Central Angle Theorem or
The Arrow Theorem.
The Bow Theorem
Inscribed angles subtended by the same arc or chord are equal.
Thales' Theorem
Triangle inscribed in semicircle or Angle inscribed in semicircle or ӹ0 degrees in Semicircle
Theorem or Thales' Theorem
Alternate Segment Theorem
An angle between a tangent and a chord through the point of contact is equal to the angle in the
alternate segment.
Quadrilaterals in a Circle
Cyclic Quadrilateral, the opposite angles of a cyclic quadrilateral are supplementary, the
exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
Angles in a Circle
A review and summary of the properties of angles that can be formed in a circle and their theorems.
Volume of Cubes
What is volume, how to find the volume of a cube, how to solve word problems about cubes, nets
of a cube.
Volume of Rectangular Prisms
How to find the volume of a rectangular prism, how to solve word problems about rectangular
prisms
Volume of Prisms
What is a prism, how to find the volume of prisms, how to solve word problems about prisms.
Volume of Cylinders
How to find the volume of cylinders, how to find the volume of hollow cylinders or tubes, how
to solve problems about cylinders.
Volume of Spheres
How to find the volume of a sphere, how to find the volume of a hemisphere, how to prove the
formula for the volume of a sphere.
Volume of Cones
What is a cone, how to calculate the volume of a cone, how to solve word problems about cones,
how to prove the formula of the volume of a cone.
Volume of Pyramids
What is a pyramid, how to find the volume of a pyramid, how to solve word problems about
pyramids, the relationship between the volume of a pyramid and the volume of a prism with the
same base and height.
Volume & Surface Area of Solids
Volume and surface area of cubes, cuboids, prisms, cylinders, spheres, cones, pyramids
Surface Area of a Cube
How to calculate the surface area of a cube, how to find the length of a cube given the
surface area, nets of a cube.
Surface Area of a Cuboid
How to calculate the surface area of a cuboid, how to solve word problems about cuboids,
nets of a cuboid.
Surface Area of a Prism
Calculate the surface area of prisms: rectangular prisms, triangular prisms, trapezoidal
prisms, hexagonal prisms etc., solve problems about prisms. calculate the surface area
of prisms using nets.
Surface Area of a Cylinder
Calculate the surface area of solid cylinders, calculate the surface area of hollow cylinders,
solve word problems about cylinders, calculate the surface area of cylinders using nets.
Surface Area of a Cone
Calculate the surface area of a cone when given the slant height, calculate the surface area
of a cone when not given the slant height, solve word problems about cones, derive the
formula for the surface area of a cone
Surface Area of a Sphere
Calculate the surface area of a sphere, calculate the surface area of a hemisphere, solve
problems about surface area of spheres, prove the formula of the surface area of a sphere.
Surface Area of a Pyramid
Find the surface area of any pyramid, find the surface area of a regular pyramid, find the
surface area of a square pyramid, find the surface area of a pyramid when the slant height
is not given.
Geometric Nets
Nets of solids: cubes, cuboids, triangular prisms, prisms, pyramids, cylinders, cones.
Surface Area of Solids
Using nets to calculate the surface area of solids: cube, rectangular prism or cuboid,
triangular prism, cylinder, and pyramids
Area Formula:
Gives the area formula for square, rectangle, parallelogram. rhombus, triangle, regular polygon,
trapezoid (trapezium), circle and ellipse.
Surface Area Formula:
Gives the surface area formula for cube, cuboid, prism, solid cylinder, hollow cylinder, cone,
pyramid, sphere and hemisphere
Volume Formula:
Gives the volume formula for cube, cuboid, prism, solid cylinder, hollow cylinder, cone, pyramid,
sphere and hemisphere
Formulas Derived:
Area of Cone, Volume of Cone, Volume of Sphere
Summary of shapes and formulas
Describes the common geometrical shapes and the formulas to calculate their area and perimeter.
It also includes the use of the Pythagorean Theorem and Heron’s formula.
Coordinate Geometry
Coordinate plane, Slope Formula, Equation of a Line, Slopes of parallel lines, Slope of
perpendicular lines, Midpoint Formula, Distance Formula
Coordinate Plane
The coordinate plane or Cartesian plane, points on the Cartesian Plane, quadrants
Slope of a Line
Slope of line from the graph (rise over run), using the slope formula, negative slope, y-intercept
Equation of a Line
The slope-intercept form for the equation of a line, how to write equations in slope-intercept
form, how to write equations of horizontal and vertical lines, how to get the equation of a
line given two points on the line.
Forms of Linear Equations
The slope-intercept form, the point-slope form, the general form, the standard form, how to convert
between the different forms of linear equations.
Explore the straight line graph
Activity to investigate how the change of the slope and y-intercept can affect the straight line graph.
Equation of a Line Parallel to the X-axis or Y-axis
Equation of a Line Given its Slope and a Point on the Line
Equation of a Line Given Two Points on the Line
Slopes of Parallel and Perpendicular Lines
How to determine if two lines are parallel or perpendicular when given their slopes, how to find
the equation of a line given a point on the line and a line that is parallel or perpendicular to
it, how to find parallel or perpendicular lines using Standard Form.
Graphing Linear Equations
How to graph linear equations by plotting points, how to graph linear equations by finding
the x-intercept and y-intercept, how to graph linear equations using the slope and y-intercept.
Slope and Intercept of a Linear Equation
How to graph a linear equation when the equation is given in slope-intercept form or when
the equation is given in general form.
Midpoint Formula
The midpoint formula, how to find the midpoint given two endpoints, how to find one endpoint
given the midpoint and another endpoint, how to proof the midpoint formula.
Distance Formula
How to derive the distance formula from the Pythagorean Theorem, how to use the distance formula.
Graphing Linear Inequalities
Graph of linear inequalities, how to graph linear inequalities, how to graph systems of linear
inequalities.
Graphing Inequalities
Linear Programming
Linear programming, how to use linear programming to solve word problems.
Quadratic Functions
The different forms of quadratic functions, general form, factored form, vertex form, convert
from general form to factored form, convert from the general form to the vertex form using
the vertex formula, convert from the general form to the vertex form using completing the square.
Graphing Quadratic Functions
How to graph of quadratic functions by plotting points, how to graph quadratic function of the form y =
ax^{2}, the properties of the graph y = ax^{2}, how to graph a quadratic
function given in general form, how to graph a quadratic function given in factored form, how to
graph a quadratic function given in vertex form.
Graphical Solutions of Quadratic Equations
How the solutions of a quadratic equation is related to the graph of the quadratic function,
how to use the graphical method to solve quadratic equations.
Graphing Cubic Functions
How to graph of cubic functions by plotting points, how to graph cubic functions of the
form y = a(x− h)^{3} + k.
Graphing Exponential Functions
How to graph exponential functions by plotting points, the characteristics of exponential
functions, how to use transformations to graph an exponential function.
Graphing Reciprocal Functions
How to graph reciprocal functions by plotting points, the characteristics of graphs of
reciprocal functions, how to use transformations to graph a reciprocal function, how to
get the equation of a reciprocal function when given its graph.
Sketching the Graphs of some Functions
How to graph functions of the form y = ax^{n} + c, how to sketch some basic or common graphs.
Geometric Construction
Construct and Copy a Line Segment
Construct the Perpendicular Bisector of a Line Segment
Construct a Perpendicular Line through a Point
How to construct a perpendicular to a line through a point on a line. how to construct a
perpendicular to a line through a point not on a line.
Construct Parallel Lines
How to construct parallel lines, how to construct a line parallel to another line and through a
given point.
Construct a 60° Angle by Constructing An Equilateral Triangle
Construct an Angle Bisector
How to construct an angle bisector of a given angle, how to use an angle bisector to construct some
angles for example, 90°, 45°, 60°, 30°, 120°, 135°, 15°.
Construct a 30-Degree Angle
Construct a 45° Angle
Construct A Triangle
Given the Length of its Three Sides (SSS)
Construct A Triangle
Given One Side and Two Angles (ASA)
Construct A Triangle
Given Two Sides and an Angle (SAS)
Construct Parallelogram, Square & Rectangle
How to construct a parallelogram given the lengths of its sides and an angle, how to construct a
parallelogram given the lengths of its diagonals, how to construct a square given the length of
the diagonal, how to construct a square given the length of one side.
Construct Various Angles & Shapes
30° 45° 90° 120° hexagon, triangle
Locus of a Moving Point
The rules of the Locus Theorem, how the rules of the Locus Theorem can be used in real world
examples, how to determine the locus of points that will satisfy more than one condition.
Geometry / Math Transformations
Translation, Reflection, Rotation, Dilation or Enlargement
Translation
Involves sliding the object from one position to another.
Reflection
Involves flipping the object over a line called the line of reflection.
Rotation
Involves turning the object about a point called the center of rotation.
Dilation
Involves a resizing of the object. It could result in an increase in size (enlargement) or a
decrease in size (reduction).
2-Dimensional Proofs
Triangle Medians and Centroids
Area Circumradius Formula Proof
Proof that the diagonals of a rhombus are perpendicular bisectors of each other
Free SAT Practice Questions (with Hints & Solutions) - Geometry
Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.
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