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Geometry - Khan Academy

Geometry - Khan Academy

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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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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 ...

Before learning any new concept (mathematical or otherwise), it's important we learn and use a common language and label concepts consistently.

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 right now: ...

Formal definition of a circle. Tangent and secant lines. Diameters and radii. major and minor arcs

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.

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 :)

Let's use our virtual protractor to measure some angles together, shall we?

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 right now: ...

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 ...

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 ...

Proving that vertical angles are equal Practice this lesson yourself on right now: ...

Parallel lines, transversal lines, corresponding angles Practice this lesson yourself on right now: ...

Proof that the sum of the measures of the angles in a triangle are 180

Figuring out angles in a triangle. A little about exterior angles being the sum of the remote interior angles

Multiple ways to solve for the angles of multiple triangles

Interesting problem finding the sums of particular exterior angles of an irregular pentagon

Proof by contradiction that corresponding angle equivalence implies parallel lines

Example of angle hunting! Practice this lesson yourself on right now: ...

Showing a generalized way to find the sum of the interior angles of any polygon Practice this lesson yourself on right now: ...

More on the difference between a theorem and axiom. Proving a cool result using SSS

SSS, SAS, ASA and AAS postulates for congruent triangles. Showing AAA is only good for similarity and SSA is good for neither

Using triangle congruency postulates to show that two intersecting segments are perpendicular

More elegant way to find the sum of the exterior angles of a convex polygon

Using the SSS, ASA, SAS, and AAS postulates to find congruent triangles Practice this lesson yourself on right now: ...

Proving that a point is the midpoint via triangle congruency

Showing that congruent legs imply equal base angles and vice versa Practice this lesson yourself on right now: ...

Showing that all of the angles of an equilateral triangle are 60 degrees Practice this lesson yourself on right now: ...

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

Integrating what we know about isosceles triangles and parallel lines

Another example of using congruency to figure out a bunch of angles

Great geometry primer on perimeter and area. Listen carefully. Sal explains the concepts so you'll understand them. Practice this lesson yourself on right now: https://www.khanacade ...

Proving that the area of any triangle is 1/2 b x h Practice this lesson yourself on right now: ...

A shape that has an infinite perimeter but finite area

Finding the formula for the area of an equilateral triangle with side s

Starting to figure out the area of a Koch Snowflake (which has an infinite perimeter)

Summing an infinite geometric series to finally find the finite area of a Koch Snowflake

Perimeter of rectangle covered by 9 non-overlapping squares. From 2000 American Invitational Math Exam

Introduction to what it means for triangles to be similar Practice this lesson yourself on right now: ...

Thinking about what we need to know to determine whether two triangles are similar Practice this lesson yourself on right now: ...

Multiple examples looking for similarity of triangles Practice this lesson yourself on right now: ...

Two example problems involving similarity Practice this lesson yourself on right now: ...

Interesting similarity problem where we don't have a lot of information to work with Practice this lesson yourself on right now: ...

The same side not corresponding to itself in two similar triangles Practice this lesson yourself on right now: ...

Example of using similarity and congruence to find the area of a triangle Practice this lesson yourself on right now: ...

Proof of the Pythagorean Theorem using similarity Practice this lesson yourself on right now: ...

Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2)

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 right now: ...

An introduction to one of the most amazing ideas/numbers in mathematics

What an angle is. Angles in a circle. Complementary and supplementary angles.

More on complementary and supplementary angles. Introduction to opposite angles.

Angles formed when a transversal intersects parallel lines.

Using what we know to solve for angles in the Angle Game.

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 right now: ...

Area of rectangles and triangles. Perimeter of rectangles.

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 right now: https://www.k ...

Area of a circle and how it relates to radius and diameter

Introduction to the Pythagorean Theorem Practice this lesson yourself on right now: ...

More Pythagorean Theorem examples. Introduction to 45-45-90 triangles.

A few more 45-45-90 examples and an introduction to 30-60-90 triangles.

More examples using 30-60-90 triangles. More free lessons at:

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 right now: ...

Finding the volume and surface area of a cylinder Practice this lesson yourself on right now: ...

Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides

Showing that the expression in part 1 is identical to Heron's Formula

Showing that an inscribed angle is half of a central angle that subtends the same arc Practice this lesson yourself on right now: ...

Problem that requires us to figure out the area of an equilateral triangle inscribed in a circle (A little trigonometry used)

Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle

Area of Diagonal Generated Triangles of Rectangle are Equal

Showing that the centroid is 2/3 of the way along a median

Proof that the diagonals of a rhombus are perpendicular bisectors of each other

Simple proof using RSH postulate to show that a radius perpendicular to a chord bisects it

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 ...

Showing that the midpoint of the hypotenuse is the circumcenter

Three points uniquely define a circle. The center of a circle is the circumcenter of any triangle the circle is circumscribed about.

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

Using angle bisectors to find the incenter and incircle of a triangle

Using the angle bisector theorem to solve for sides of a triangle Practice this lesson yourself on right now: ...

Showing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as well

Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median)

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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