# Geometry - Khan Academy

Videos on geometry. Basic understanding of Algebra I necessary. After this, you'll be ready for Trigonometry.

Geometry on Khan Academy: We are surrounded by space. And that space contains lots of things. And these things have shapes. In geometry we are concerned with the nature of these shapes, how we define them, and what they teach us about the world at large--from math to architecture to biology to astronomy (and everything in between). Learning geometry is about more than just taking your medicine ("It's good for you!"), it's at the core of everything that exists--including you. Having said all that, some of the specific topics we'll cover include angles, intersecting lines, right triangles, perimeter, area, volume, circles, triangles, quadrilaterals, analytic geometry, and geometric constructions. Wow. That's a lot. To summarize: it's difficult to imagine any area of math that is more widely used than geometry.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

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## Lessons hide description |
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## 1) Euclid as the father of geometry | Introduction to Euclidean geometry | Geometry## We don't normally delve too far into history when talking about math, but sometimes it's important to have perspective about how some of our math concepts came about and how influential they have beco ... |

## 2) Basic geometry: language and labels | Introduction to Euclidean geometry | Geometry## Before learning any new concept (mathematical or otherwise), it's important we learn and use a common language and label concepts consistently. |

## 3) Intro to lines, line segments, and rays | Geometry## Let's get familiar with the difference between lines, line segments, and rays. Hint: a ray is somewhere between a line and a line segment! Practice this lesson yourself on KhanAcademy.org right now: ... |

## 4) Language and notation of the circle | Introduction to Euclidean geometry | Geometry## Formal definition of a circle. Tangent and secant lines. Diameters and radii. major and minor arcs |

## 5) Angle basics | Angles and intersecting lines | Geometry## The naming of an angle is related to the line segments that create it. We'll also learn about the angle's vertex. Come on. Let's do this. |

## 6) Measuring angles in degrees | Angles and intersecting lines | Geometry## Let's come up with an exact way to measure an angle using a protractor. We'll also learn about acute angles. Not to be confused with augly angles :) |

## 7) Measuring angles using a protractor | Angles and intersecting lines | Geometry## Let's use our virtual protractor to measure some angles together, shall we? |

## 8) More angle measurements using a protractor | Angles and intersecting lines | Geometry## We're practicing measuring angles using the virtual protractor. Do this a few times with us and you'll use a real protractor like a pro! Practice this lesson yourself on KhanAcademy.org right now: ... |

## 9) Complementary and supplementary angles | Angles and intersecting lines | Geometry## Let's add some tools to our geometry tool belt. Here's a thorough explanation of complementary and supplementary angles, as well as definitions of adjacent and straight angles. Practice this lesson y ... |

## 10) Introduction to vertical angles | Angles and intersecting lines | Geometry## By using our knowledge of supplementary, adjacent, and vertical angles, we can solve problems involving the intersection of two lines. Including this one! Practice this lesson yourself on KhanAcademy ... |

## 11) Proof: Vertical angles are equal | Angles and intersecting lines | Geometry## Proving that vertical angles are equal Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 12) Angles formed by parallel lines and transversals | Geometry## Parallel lines, transversal lines, corresponding angles Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 13) Proof - Sum of Measures of Angles in a Triangle are 180## Proof that the sum of the measures of the angles in a triangle are 180 |

## 14) Triangle angle example 1 | Angles and intersecting lines | Geometry## Figuring out angles in a triangle. A little about exterior angles being the sum of the remote interior angles |

## 15) Triangle angle example 2 | Angles and intersecting lines | Geometry## Another example finding angles in triangles |

## 16) Triangle angle example 3 | Angles and intersecting lines | Geometry## Multiple ways to solve for the angles of multiple triangles |

## 17) Challenging triangle angle problem | Angles and intersecting lines | Geometry## Interesting problem finding the sums of particular exterior angles of an irregular pentagon |

## 18) Proof: Corresponding angle equivalence implies parallel lines | Geometry## Proof by contradiction that corresponding angle equivalence implies parallel lines |

## 19) Finding more angles | Angles and intersecting lines | Geometry## Example of angle hunting! Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 20) Sum of interior angles of a polygon | Angles and intersecting lines | Geometry## Showing a generalized way to find the sum of the interior angles of any polygon Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 21) Congruent triangles and SSS | Congruence | Geometry## What it means for triangles to be congruent |

## 22) SSS to show a radius is perpendicular to a chord that it bisects | Geometry## More on the difference between a theorem and axiom. Proving a cool result using SSS |

## 23) Other triangle congruence postulates | Congruence | Geometry## SSS, SAS, ASA and AAS postulates for congruent triangles. Showing AAA is only good for similarity and SSA is good for neither |

## 24) Two column proof showing segments are perpendicular | Congruence | Geometry## Using triangle congruency postulates to show that two intersecting segments are perpendicular |

## 25) Sum of the exterior angles of convex polygon | Geometry## More elegant way to find the sum of the exterior angles of a convex polygon |

## 26) Finding congruent triangles | Congruence | Geometry## Using the SSS, ASA, SAS, and AAS postulates to find congruent triangles Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 27) More on why SSA is not a postulate | Congruence | Geometry## SSA special cases including RSH |

## 28) Congruent triangle proof example | Congruence | Geometry## Proving that a point is the midpoint via triangle congruency |

## 29) Congruent triangle example 2 | Congruence | Geometry## Showing that segments have the same length |

## 30) Congruent legs and base angles of isosceles triangles | Congruence | Geometry## Showing that congruent legs imply equal base angles and vice versa Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 31) Equilateral triangle sides and angles congruent | Congruence | Geometry## Showing that all of the angles of an equilateral triangle are 60 degrees Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 32) Equilateral and isosceles example problems | Congruence | Geometry## Three example problems involving isosceles and equilateral triangles. 2 from Art of Problem Solving (by Richard Rusczyk) book Practice this lesson yourself on KhanAcademy.org right now: https://www ... |

## 33) Another isosceles example problem | Congruence | Geometry## Possible angles for an isosceles triangle |

## 34) Example involving an isosceles triangle and parallel lines | Congruence | Geometry## Integrating what we know about isosceles triangles and parallel lines |

## 35) Figuring out all the angles for congruent triangles example | Congruence | Geometry## Another example of using congruency to figure out a bunch of angles |

## 36) Perimeter and area: the basics | Perimeter, area, and volume | Geometry## Great geometry primer on perimeter and area. Listen carefully. Sal explains the concepts so you'll understand them. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacade ... |

## 37) Triangle area proofs | Perimeter, area, and volume | Geometry## Proving that the area of any triangle is 1/2 b x h Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 38) Interesting perimeter and area problems | Perimeter, area, and volume | Geometry## Three example problems involving perimeter and area |

## 39) Koch snowflake fractal | Perimeter, area, and volume | Geometry## A shape that has an infinite perimeter but finite area |

## 40) Area of an equilateral triangle | Perimeter, area, and volume | Geometry## Finding the formula for the area of an equilateral triangle with side s |

## 41) Area of Koch snowflake (part 1) - advanced | Perimeter, area, and volume | Geometry## Starting to figure out the area of a Koch Snowflake (which has an infinite perimeter) |

## 42) Area of Koch snowflake (part 2) - advanced | Perimeter, area, and volume | Geometry## Summing an infinite geometric series to finally find the finite area of a Koch Snowflake |

## 43) Challenging perimeter problem | Perimeter, area, and volume | Geometry## Perimeter of rectangle covered by 9 non-overlapping squares. From 2000 American Invitational Math Exam |

## 44) Similar triangle basics | Similarity | Geometry## Introduction to what it means for triangles to be similar Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 45) Similarity postulates | Similarity | Geometry## Thinking about what we need to know to determine whether two triangles are similar Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 46) Similar triangle example problems | Similarity | Geometry## Multiple examples looking for similarity of triangles Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 47) Similarity example problems | Similarity | Geometry## Two example problems involving similarity Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 48) Challenging similarity problem | Similarity | Geometry## Interesting similarity problem where we don't have a lot of information to work with Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 49) Similarity example where same side plays different roles | Similarity | Geometry## The same side not corresponding to itself in two similar triangles Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 50) Finding area using similarity and congruence | Similarity | Geometry## Example of using similarity and congruence to find the area of a triangle Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 51) Pythagorean theorem proof using similarity | Geometry## Proof of the Pythagorean Theorem using similarity Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 52) 30-60-90 triangle side ratios proof | Right triangles and trigonometry | Geometry## Proving the ratios between the sides of a 30-60-90 triangle |

## 53) 45-45-90 triangle side ratios | Right triangles and trigonometry | Geometry## Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2) |

## 54) 30-60-90 triangle example problem | Right triangles and trigonometry | Geometry## Using what we know about 30-60-90 triangles to solve what at first seems to be a challenging problem Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/g ... |

## 55) The golden ratio | Introduction to Euclidean geometry | Geometry## An introduction to one of the most amazing ideas/numbers in mathematics |

## 56) Introduction to angles (old) | Angles and intersecting lines | Geometry## What an angle is. Angles in a circle. Complementary and supplementary angles. |

## 57) Angles (part 2) | Angles and intersecting lines | Geometry## More on complementary and supplementary angles. Introduction to opposite angles. |

## 58) Angles (part 3) | Angles and intersecting lines | Geometry## Angles formed when a transversal intersects parallel lines. |

## 59) Angles formed between transversals and parallel lines | Geometry## Angles of parallel lines |

## 60) Angles of parallel lines 2 | Angles and intersecting lines | Geometry## Angles of parallel lines examples |

## 61) The angle game | Angles and intersecting lines | Geometry## Using what we know to solve for angles in the Angle Game. |

## 62) Similar triangles | Similarity | Geometry## Introduction to similar triangles |

## 63) Similar triangles (part 2) | Similarity | Geometry## More on similar triangles |

## 64) Angle game (part 2) | Angles and intersecting lines | Geometry## More examples of the Angle Game. |

## 65) Acute right and obtuse angles | Angles and intersecting lines | Geometry## We briefly discussed the types of angles in the last tutorial but we'll delve a little deeper into acute, obtuse, and right angles here. Practice this lesson yourself on KhanAcademy.org right now: ... |

## 66) Area and Perimeter## Area of rectangles and triangles. Perimeter of rectangles. |

## 67) Circles: radius, diameter, circumference and Pi | Geometry## A circle is at the foundation of geometry and how its parts relate to each other is both completely logical and a wonder. Practice this lesson yourself on KhanAcademy.org right now: https://www.k ... |

## 68) Area of a circle## Area of a circle and how it relates to radius and diameter |

## 69) The Pythagorean theorem intro | Right triangles and trigonometry | Geometry## Introduction to the Pythagorean Theorem Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 70) Pythagorean theorem II | Right triangles and trigonometry | Geometry## More Pythagorean Theorem examples. Introduction to 45-45-90 triangles. |

## 71) 45-45-90 triangles | Right triangles and trigonometry | Geometry## Introduction to 45-45-90 Triangles |

## 72) Intro to 30-60-90 triangles | Right triangles and trigonometry | Geometry## A few more 45-45-90 examples and an introduction to 30-60-90 triangles. |

## 73) 30-60-90 Triangles II## More examples using 30-60-90 triangles. More free lessons at: http://www.khanacademy.org/video?v=3mLUJ |

## 74) Find the volume of a triangular prism and cube | Geometry## In this tutorial we'll practice finding solid geometry volume. Keep paper and pencil nearby in case you want to remember the formulas. Practice this lesson yourself on KhanAcademy.org right now: ... |

## 75) Cylinder volume and surface area | Perimeter, area, and volume | Geometry## Finding the volume and surface area of a cylinder Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 76) Heron's formula | Perimeter, area, and volume | Geometry## Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides |

## 77) Part 1 of proof of Heron's formula | Perimeter, area, and volume | Geometry## Part 1 of the proof of Heron's Formula |

## 78) Part 2 of proof of Heron's formula | Perimeter, area, and volume | Geometry## Showing that the expression in part 1 is identical to Heron's Formula |

## 79) Inscribed and central angles | Circles | Geometry## Showing that an inscribed angle is half of a central angle that subtends the same arc Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 80) Area of inscribed equilateral triangle (some basic trig used) | Circles | Geometry## Problem that requires us to figure out the area of an equilateral triangle inscribed in a circle (A little trigonometry used) |

## 81) Right Triangles Inscribed in Circles (Proof)## Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle |

## 82) Area of diagonal generated triangles of rectangle are equal | Geometry## Area of Diagonal Generated Triangles of Rectangle are Equal |

## 83) Triangle medians and centroids | Special properties and parts of triangles | Geometry## Seeing that the centroid is 2/3 of the way along every median |

## 84) Triangle medians and centroids (2D proof) | Geometry## Showing that the centroid is 2/3 of the way along a median |

## 85) Rhombus diagonals | Quadrilaterals | Geometry## Proof that the diagonals of a rhombus are perpendicular bisectors of each other |

## 87) Perpendicular radius bisects chord | Congruence | Geometry## Simple proof using RSH postulate to show that a radius perpendicular to a chord bisects it |

## 88) Circumcenter of a triangle | Special properties and parts of triangles | Geometry## Multiple proofs showing that a point is on a perpendicular bisector of a segment if and only if it is equidistant from the endpoints. Using this to establish the circumcenter, circumradius, and circum ... |

## 89) Circumcenter of a right triangle | Geometry## Showing that the midpoint of the hypotenuse is the circumcenter |

## 90) Three points defining a circle | Special properties and parts of triangles | Geometry## Three points uniquely define a circle. The center of a circle is the circumcenter of any triangle the circle is circumscribed about. |

## 91) Point-line distance and angle bisectors | Geometry## Thinking about the distance between a point and a line. Proof that a point on an angle bisector is equidistant to the sides of the angle and a point equidistant to the sides is on an angle bisector |

## 92) Incenter and incircles of a triangle | Geometry## Using angle bisectors to find the incenter and incircle of a triangle |

## 93) Angle bisector theorem proof | Special properties and parts of triangles | Geometry## What the angle bisector theorem is and its proof |

## 94) Inradius, perimeter, and area | Special properties and parts of triangles | Geometry## Showing that area is equal to inradius times semiperimeter |

## 95) Angle bisector theorem examples | Geometry## Using the angle bisector theorem to solve for sides of a triangle Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/geome ... |

## 96) Medians divide into smaller triangles of equal area | Geometry## Showing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as well |

## 97) Exploring medial triangles | Special properties and parts of triangles | Geometry## What a medial triangle is and its properties |

## 98) Proving that the centroid is 2-3rds along the median | Geometry## Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median) |

## 99) Median centroid right triangle example | Geometry## Example involving properties of medians |

## 100) Proof: Triangle altitudes are concurrent (orthocenter) | Geometry## Showing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter). |

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