Sign in to your Khan Academy account and then click on the links below to start working on the different geometry categories thus it will ensure you get your points at
Khan Academy Geometry:
Unit 1
Angles Basis and Measurements
- Angle basics
- Measuring angles in degrees
- Measuring angles using a protractor
- More angle measurements using a protractor
- Measuring angles
- Acute right and obtuse angles
- Angle types
- Vertical, adjacent and linearly paired angles
- Angle measurement and circle arcs
- Angles and circumference
- Understanding angles
Angles between intersecting and parallel lines
- Introduction to vertical angles
- Vertical angles
- Find measure of vertical angles
- Equation practice with vertical angles
- Proof: Vertical angles are equal
- Angles formed by parallel lines and transversals
- Identifying parallel and perpendicular lines
- Figuring out angles between transversal and parallel lines
- Congruent angles
- Parallel lines 1
- Using algebra to find measures of angles formed from transversal
- Equation practice with congruent angles
- CA Geometry: Deducing angle measures
- Line and angle proofs
Angles with triangles and polygons
- Proof: Sum of measures of angles in a triangle are 180
- Triangle angle example 1
- Triangle angle example 2
- Triangle angle example 3
- Challenging triangle angle problem
- Proof: Corresponding angle equivalence implies parallel lines
- Finding more angles
- Finding angle measures 1
- Finding angle measures 2
- Sum of interior angles of a polygon
- Angles of a polygon
- Sum of the exterior angles of convex polygon
Sal's old angle videos
These are some of the classic, original angle video that Sal had done way back when (like 2007). Other tutorials are more polished than this one, but this one has charm. Also not bad if you're looking for more examples of angles between intersected lines, transversals and parallel lines.
- Introduction to angles (old)
- Angles (part 2)
- Angles (part 3)
- Angles formed between transversals and parallel lines
- Angles of parallel lines 2
- The angle game
- Angle game (part 2)
- Acute right and obtuse angles
Complementary and supplementary angles
In this tutorial we'll look at the most famous types of angle-pairs--complementary and supplementary angles. This aren't particularly deep concepts, but you'll find they do come in handy!
- Complementary and supplementary angles
- Complementary and supplementary angles
- Find measure of complementary angles
- Find measure of supplementary angles
- Equation practice with angle addition
Unit 2
Congruence:
Transformations and Congruence
- Testing congruence by transformations example
- Another congruence by transformation example
- Exploring rigid transformations and congruence
- Example of rigid transformation and congruence
- Another example of rigid transformations for congruence
- Defining congruence through rigid transformations
We begin to seriously channel Euclid in this tutorial to really, really (no, really) prove things--in particular, that triangles are congruents. You'll appreciate (and love) what rigorous proofs are. It will sharpen your mind and make you a better friend, relative and citizen (and make you more popular in general). Don't have too much fun.
- Congruent triangles and SSS
- SSS to show a radius is perpendicular to a chord that it bisects
- Other triangle congruence postulates
- Two column proof showing segments are perpendicular
- Finding congruent triangles
- Congruency postulates
- More on why SSA is not a postulate
- Perpendicular radius bisects chord
- Congruent triangle proof example
- Congruent triangle example 2
- Figuring out all the angles for congruent triangles example
- Problem involving angle derived from square and circle
- Congruent triangles 1
- Congruent triangles 2
This tutorial uses our understanding of congruence postulates to prove some neat properties of isosceles and equilateral triangles.
- Congruent legs and base angles of isosceles triangles
- Equilateral triangle sides and angles congruent
- Equilateral and isosceles example problems
- Recognizing triangle types
- Triangle angles 1
- Another isosceles example problem
- Example involving an isosceles triangle and parallel lines
Unit 3
Similarity:
- Testing similarity through transformations
- Exploring angle-preserving transformations and similarity
- Quadrilateral similarity by showing congruent angles
- Defining similarity through angle-preserving transformations
This tutorial explains a similar (but not congruent) idea to congruency (if that last sentence made sense, you might not need this tutorial). Seriously, we'll take a rigorous look at similarity and think of some reasonable postulates for it. We'll then use these to prove some results and solve some problems. The fun must not stop!
- Similar triangle basics
- Similarity postulates
- Similar triangles 1
- Similar triangle example problems
- Similar triangles 2
- Similarity example problems
- Solving similar triangles 1
- Similarity example where same side plays different roles
- Solving similar triangles 2
We spend a lot of time in geometry proving that triangles are congruent or similar. We now apply this ability to some really interesting problems (seriously, these are fun)!
Finding area using similarity and congruence
- Golden ratio and Rembrandt's self portrait
- Triangle similarity in pool
- Golden ratio to find radius of moon
- Challenging similarity problem
- Solving problems with similar and congruent triangles
Old school similarity
- These videos may look similar (pun-intended) to videos in another playlist but they are older, rougher and arguably more charming. These are some of the original videos that Sal made on similarity. They are less formal than those in the "other" similarity tutorial, but, who knows, you might like them more.
- Similar triangles
- Similar triangles (part 2)
Unit 4
Named after the Greek philosopher who lived nearly 2600 years ago, the Pythagorean theorem is as good as math theorems get (Pythagoras also started a religious movement). It's simple. It's beautiful. It's powerful. In this tutorial, we will cover what it is and how it can be used. We have another tutorial that gives you as many proofs of it as you might need.
- Pythagorean theorem
- The Pythagorean theorem intro
- Pythagorean theorem 1
- Pythagorean theorem 2
- Pythagorean theorem 3
- Pythagorean theorem
- Pythagorean theorem word problems
- Introduction to the Pythagorean theorem
- Pythagorean theorem II
The Pythagorean theorem is one of the most famous ideas in all of mathematics. This tutorial proves it. Then proves it again... and again... and again. More than just satisfying any skepticism of whether the Pythagorean theorem is really true (only one proof would be sufficient for that), it will hopefully open your mind to new and beautiful ways to prove something very powerful.
- Garfield's proof of the Pythagorean theorem
- Bhaskara's proof of the Pythagorean theorem
- Pythagorean theorem proof using similarity
- Another Pythagorean theorem proof
- Pythagorean Theorem proofs
- Pythagorean Theorem proofs
We hate to pick favorites, but there really are certain right triangles that are more special than others. In this tutorial, we pick them out, show why they're special, and prove it! These include 30-60-90 and 45-45-90 triangles (the numbers refer to the measure of the angles in the triangle).
- 30-60-90 triangle side ratios proof
- 45-45-90 triangle side ratios
- 30-60-90 triangle example problem
- Special right triangles
- Area of a regular hexagon
- 45-45-90 triangles
- Intro to 30-60-90 triangles
----------------
Let's see how to find the volumes of cylinders, spheres and other three dimensional shapes. Common Core Standard: 8.G.C.9
- Cylinder volume and surface area
- Find the volume of a triangular prism and cube
- Solid geometry
- Volume of a sphere
- Volume of a cone
- Volume word problems with cones, cylinders, and spheres
----------------------
In this tutorial, you will learn all the trigonometry that you are likely to remember in ten years (assuming you are a lazy non-curious, non-lifelong learner). But even in that non-ideal world where you forgot everything else, you'll be able to do more than you might expect with the concentrated knowledge you are about to get.
- Basic trigonometry
- Example: Using soh cah toa
- Trigonometry 0.5
- Basic trigonometry II
- Trigonometry 1
- Trigonometry 1.5
- Example: Trig to solve the sides and angles of a right triangle
- Trigonometry 2
- Angle to aim to get alien
- How much of a pyramid is submerged
- Applying right triangles
In this tutorial, we will build on our understanding of similarity to get a deeper appreciation for the motivation behind trigonometric ratios and relationships.
- Similarity to define sine, cosine, and tangent
- Sine and cosine of complements example
- Showing relationship between cosine and sine of complements
- Example with trig functions and ratios
- Example relating trig function to side ratios
- Trigonometric functions and side ratios in right triangles
Unit 5
You first learned about perimeter and area when you were in grade school. In this tutorial, we will revisit those ideas with a more mathy lens. We will use our prior knowledge of congruence to really start to prove some neat (and useful) results (including some that will be useful in our study of similarity).
- Perimeter and area: the basics
- Triangle area proofs
- Area of triangles
- Area of triangles 2
- Interesting perimeter and area problems
- Area of diagonal generated triangles of rectangle are equal
- Area of an equilateral triangle
- Area of shaded region made from equilateral triangles
- Shaded areas
- Challenging perimeter problem
The triangle inequality theorem is, on some level, kind of simple. But, as you'll see as you go into high level mathematics, it is often used in fancy proofs. This tutorial introduces you to what it is and gives you some practice understanding the constraints on the dimensions of a triangle.
- Triangle inequality theorem
- Triangle inequality theorem
Named after Helge von Koch, the Koch snowflake is one of the first fractals to be discovered. It is created by adding smaller and smaller equilateral bumps to an existing equilateral triangle. Quite amazingly, it produces a figure of infinite perimeter and finite area!
- Koch snowflake fractal
- Area of an equilateral triangle
- Area of Koch snowflake (part 1) - advanced
- Area of Koch snowflake (part 2) - advanced
Named after Heron of Alexandria, Heron's formula is a power (but often overlooked) method for finding the area of ANY triangle. In this tutorial we will explain how to use it and then prove it!
Heron's formula
- Part 1 of proof of Heron's formula
- Part 2 of proof of Heron's formula
Circles are everywhere. How can we measure how big they are? Well, we could think about the distance around the circle (circumference). Another option would be to think about how much space it takes up on our paper (area). Have fun!
- Circles: radius, diameter, circumference and Pi
- Labeling parts of a circle
- Radius, diameter, and circumference
- Area of a circle
- Area of a circle
Not everything in the world is a rectangle, circle or triangle. In this tutorial, we give you practice at finding the perimeters and areas of these less-than-standard shapes!
- Quadrilateral overview
- Quadrilateral properties
- Area of a parallelogram
- Area of parallelograms
- Area of a trapezoid
- Area of a kite
- Area of trapezoids, rhombi, and kites
- Perimeter of a parallelogram
- Perimeter and area of a non-standard polygon
- Area of quadrilaterals and polygons
Tired of perimeter and area and now want to measure 3-D space-take-upness. Well you've found the right tutorial. Enjoy!
- Nets of polyhedra
- Nets of 3D figures
- Finding surface area: nets of polyhedra
- Surface area using nets
- Surface area
- Find the volume of a triangular prism and cube
- Cylinder volume and surface area
- Solid geometry
- How many people can a blimp carry
- Surface and volume density word problems
- Volume of a cone
- Volume of a sphere
- Volume word problems with cones, cylinders, and spheres
- Slice a rectangular pyramid
- Slicing 3D figures
- Cross sections of 3D objects
- Rotating 2D shapes in 3D
- Rotate 2D shapes to make 3D objects
Unit 6
This tutorial will review some of the basic of circles and then think about lengths of arcs and areas of sectors.
- Language and notation of the circle
- Circles: radius, diameter, circumference and Pi
- Length of an arc that subtends a central angle
- Finding central angle measure given arc length
- Circles and arcs
- Area of a sector given a central angle
- Areas of circles and sectors
We'll now dig a bit deeper in our understanding of circles by looking at central, inscribed and circumscribed angles. This is fun and beautiful as is, but you'll also see that it shows up on a lot of math standardized tests. Why do people like to put geometry like this on standardized tests? Because it shows deductive reasoning skills which are super important in every walk of life!
- Inscribed and central angles
- Inscribed angles 1
- Measure of circumscribed angle
- Example with tangent and radius
- Hypotenuse of right triangle inscribed in circle
- Central, inscribed, and circumscribed angles
You know that a circle can be viewed as the set of all points that whose distance from the center is equal to the radius. In this tutorial, we use this information and the Pythagorean Theorem to derive the equation of a circle.
- Equation for a circle using the Pythagorean Theorem
- Pythagorean theorem and radii of circles
- Pythagorean theorem and the equation of a circle
- Radius and center for a circle equation in standard form
- Equation of a circle in factored form
- Completing the square to write equation in standard form of a circle
- Equation of a circle in non-factored form
This more advanced (and very optional) tutorial is fun to look at for enrichment. It builds to figuring out the formula for the area of a triangle inscribed in a circle!
- SSS to show a radius is perpendicular to a chord that it bisects
- Perpendicular radius bisects chord
- Area of inscribed equilateral triangle (some basic trig used)
Geometric models
- Let's use what we know about geometry to answer some really, really interesting questions.
- Geometric descriptions of real-world objects
- How far can you see from a plane window
- 2D geometric models
Unit 7
In this tutorial, we study lines that are perpendicular to the sides of a triangle and divide them in two (perpendicular bisectors). As we'll prove, they intersect at a unique point called the cicumcenter (which, quite amazingly, is equidistant to the vertices). We can then create a circle (circumcircle) centered at this point that goes through all the vertices. This tutorial is the extension of the core narrative of the Geometry "course". After this, you might want to look at the tutorial on angle bisectors.
- Circumcenter of a triangle
- Circumcenter of a right triangle
- Three points defining a circle
- Area circumradius formula proof
- 2003 AIME II problem 7
This tutorial experiments with lines that divide the angles of a triangle in two (angle bisectors). As we'll prove, all three angle bisectors actually intersect at one point called the incenter (amazing!). We'll also prove that this incenter is equidistant from the sides of the triangle (even more amazing!). This allows us to create a circle centered at the incenter that is tangent to the sides of the triangle (not surprisingly called the "incircle").
- Point-line distance and angle bisectors
- Incenter and incircles of a triangle
- Inradius, perimeter, and area
- Angle bisector theorem proof
- Angle bisector theorem examples
- Angle bisector theorem
You've explored perpendicular bisectors and angle bisectors, but you're craving to study lines that intersect the vertices of a a triangle AND bisect the opposite sides. Well, you're luck because that (medians) is what we are going to study in this tutorial. We'll prove here that the medians intersect at a unique point (amazing!) called the centroid and divide the triangle into six mini triangles of equal area (even more amazing!). The centroid also always happens to divide all the medians in segments with lengths at a 1:2 ration (stupendous!).
- Triangle medians and centroids
- Triangle medians and centroids (2D proof)
- Medians divide into smaller triangles of equal area
- Exploring medial triangles
- Proving that the centroid is 2-3rds along the median
- Median centroid right triangle example
Ok. You knew triangles where cool, but you never imagined they were this cool! Well, this tutorial will take things even further. After perpendicular bisectors, angle bisector and medians, the only other thing (that I can think of) is a line that intersects a vertex and the opposite side (called an altitude). As we'll see, these are just as cool as the rest and, as you may have guessed, intersect at a unique point called the orthocenter (unbelievable!).
- Proof: Triangle altitudes are concurrent (orthocenter)
- Common orthocenter and centroid
This tutorial brings together all of the major ideas in this topic. First, it starts off with a light-weight review of the various ideas in the topic. It then goes into a heavy-weight proof of a truly, truly, truly amazing idea. It was amazing enough that orthocenters, circumcenters, and centroids exist , but we'll see in the videos on Euler lines that they sit on the same line themselves (incenters must be feeling lonely)!!!!!!!
- Review of triangle properties
- Euler line
- Euler's line proof
Unit 8
Not all things with four sides have to be squares or rectangles! We will now broaden our understanding of quadrilaterals!
- Quadrilateral overview
- Quadrilateral properties
- Quadrilaterals: kites as a geometric shape
- Quadrilaterals: find the type exercise
- Quadrilateral types
- Proof: Opposite sides of parallelogram congruent
- Proof: Diagonals of a parallelogram bisect each other
- Proof: Opposite angles of parallelogram congruent
- Quadrilateral angles
- Proof: Rhombus diagonals are perpendicular bisectors
- Proof: Rhombus area half product of diagonal length
- Area of a parallelogram
- Whether a special quadrilateral can exist
- Rhombus diagonals
Unit 9
Let's get an intuitive understanding for symmetry of two dimensional shapes.
- Axis of symmetry
- Rotating polygons 180 degrees about their center
- Constructing quadrilateral based on symmetry
- Constructing a shape by reflecting over 2 lines
- Symmetry of two-dimensional shapes
Let's use some pretty cool software widgets to see how we can transform a shape through translations, rotations, dilations and reflections.
- Rotating segment about origin example
- Reflecting line across another line example
- Properties of rigid transformations
- Translations of polygons
- Determining a translation for a shape
- Translations of polygons
- Rotation of polygons example
- Rotation of polygons
We understand the idea of scaling/dilation from everyday life (hey, let's make it bigger or smaller keeping the same proportions!). In this tutorial, you'll understand this type of transformation in a much, much deeper way.
- Dilating one line onto another
- Comparing side lengths after dilation
- Thinking about dilations
- Dilating from an arbitrary point example
- Dilations
Let's continue our deep voyage through the world of transformations by thinking about how points map to each other after a transformation.
- Performing a rotation to match figures
- Reflection and mapping points example
- Scaling down a triangle by half
- Performing transformations on the coordinate plane
- Possible transformations example
- Qualitatively defining rigid transformations
- Determining the line of reflection
- Determining a translation between points
- Quantitatively defining rigid transformations
- Defining transformations to match polygons
- Transforming polygons
Unit 10
You are familiar with the ideas of slope and distance on the coordinate plane. You also feel comfortable with congruence an similarity and many of the other core ideas in Euclidean geometry. In this, tutorial, Descartes and Euclid are forced to work together as we tackle geometry problems on the coordinate plane!
- Recognizing points on a circle
- Classifying a quadrilateral on the coordinate plane
- Identifying similar triangles in the coordinate plane
- Geometry problems on the coordinate plane
- Which minions can the wizard reach
- Coordinate plane word problems with polygons
We are now going to leverage our understanding of the coordinate plane to think about distances between points and ratios of lengths of segments between points.
- Distance formula
- Distance formula
- Midpoint formula
- Midpoint formula
- Ratios of distances between colinear points
- Finding a point part way between two points
- Dividing line segments
Unit 11
With just a compass and a straightedge (or virtual versions of them), you'll be amazed by how many geometric shapes you can construct perfectly. This tutorial gets you started with the building block of how to bisect angle and lines (and how to construct perpendicular bisectors of lines).
- Constructing a perpendicular bisector using a compass and straightedge
- Constructing a perpendicular line using a compass and straightedge
- Constructing an angle bisector using a compass and straightedge
- Compass constructions 1
Have you ever wondered how people would draw a square, equilateral triangle or even hexagon before there were computers? Well, now you're going to do just that (ironically, with a computer). Using our virtual compass and straightedge, you'll construct several regular shapes (by inscribing them inside circles).
- Constructing square inscribed in circle
- Constructing equilateral triangle inscribed in circle
- Constructing regular hexagon inscribed in circle
- Compass constructions 2
In our study of triangles, we spent a decent amount of time think about incenters (the intersections of the angle bisectors) and circumcenters (the intersections of the perpendicular bisectors). We'll now leverage this knowledge to actually construct circle inscribed and circumscribed about a triangle using only a compass and straightedge (actually virtual versions of them).
- Constructing circle inscribing triangle
- Constructing circumscribing circle
- Inscribing and circumscribing circles on a triangle
Constructing a line tangent to a circle
- Constructing a line tangent to a circle
Unit 12
Sal does the 80 problems from the released questions from the California Standards Test for Geometry. Basic understanding of Algebra I necessary.
- Interesting perimeter and area problems
- Challenging perimeter problem
- CA Geometry: Deductive reasoning
- CA Geometry: Proof by contradiction
- CA Geometry: More proofs
- CA Geometry: Similar triangles 1
- CA Geometry: More on congruent and similar triangles
- CA Geometry: Triangles and parallelograms
- CA Geometry: Area, pythagorean theorem
- CA Geometry: Area, circumference, volume
- CA Geometry: Pythagorean theorem, area
- CA Geometry: Exterior angles
- CA Geometry: Deducing angle measures
- CA Geometry: Pythagorean theorem, compass constructions
- CA Geometry: Compass construction
- Compass constructions 1
- CA Geometry: Basic trigonometry
- CA Geometry: More trig
- CA Geometry: Circle area chords tangent
