Practice
Geometry
Courses
Take a guided, problem-solving based approach to learning Geometry. These compilations provide unique perspectives and applications you won't find anywhere else.
Geometry Fundamentals
What's inside
- Introduction
- Angles and Lengths
- Area
- Pythagoras' Geometry
- Scaling and Volume
Beautiful Geometry
What's inside
- Introduction
- Tessellations and Reptiles
- Polyominos
- Folding Puzzles
- Guarding Galleries
- Pick's Theorem
Geometry I
What's inside
- Introduction
- Angles
- Triangles
- Composite Polygons
- Regular Polygons
- When Geometry Gets Tough
Additional Practice
Sharpen your skills with these quizzes designed to check your understanding of the fundamentals.
Geometric Measurement
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Geometry Warmups
Mathematics is filled with shapes that are kaleidoscopic in variety. Wielded since ancient times, the power of geometry helps us examine and measure these shapes.
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Angles and Lines
Forgive us for being obtuse, but this is a cute concept, and we think it’s right for you.
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Calculating Areas
If the area of a square is 144, what is the perimeter of the square? If a square and a circle have the same perimeter, which of them will have a greater area?
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Composite Figures
Overlap, inscribe, and circumscribe a collection of simple geometric shapes to make a complex, composite figure.
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Volume
What is the volume of a sphere of radius 5? If we make a cylinder two times taller, and its radius three times bigger, how many times larger would it become in volume?
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Surface Area
A spherical balloon is inflated until its volume becomes 27 times its original volume. How much does its surface area increase?
Triangles
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Properties of Triangles
Given a triangle with side lengths of 5, 12, and 14, is the largest angle in the triangle acute, right, or obtuse? Geometric knowledge helps us deduce much about triangles from limited information.
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Classification of Triangles
Can a scalene triangle be obtuse? Is an equilateral triangle always acute?
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Congruent and Similar Triangles
If you want to find similar triangles, use only SSS, SAS and AAA. Don't make an ASS of yourself.
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Area of Triangles
You know that Area = base x height / 2, but what other ways are there to find the area of a triangle? Brace yourself for some potent formulae.
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Triangle Centers
What do the orthocenter, centroid, and circumcenter share in common? They lie on the Euler line!
Polygons
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Quadrilaterals
These shapes have four sides and 360° of interior angle goodness.
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Similar Polygons
Similar polygons have the same shape, but not necessarily the same size. Are rectangles with dimensions 8 x 5 and 25 x 40 similar?
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Regular Polygons
With equal angles and equal side lengths, what more could you want from a polygon?
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General Polygons
Convex and concave, regular and irregular, similar and congruent – learn every variety of polygon you can encounter.
Circles
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Properties of Circles
What is pi? If the area of a circle is 49π, what is its circumference? If a circle is inscribed in a square, what is the ratio of the areas of the circle and the square?
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Tangent and Secant Lines
A tangent to a circle is a line intersecting the circle at exactly one point. Can you prove that the line from the center of the circle to the point of tangency is perpendicular to the tangent line?
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Inscribed and Circumscribed Figures
Learn about packing shapes neatly into others, and the resulting geometric properties.
Coordinate Geometry
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Equation of a Line
Explore how simple equations have graphical implications starting with the equation of a line.
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2D Coordinate Geometry
In the 1600s, René Descartes married algebra and geometry to create the Cartesian plane.
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Conic Sections
Parabolas, ellipses, and hyperbolas, oh my! Learn about this eccentric bunch of shapes.
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3D Coordinate Geometry
Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.
Trigonometry
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Trigonometric Functions
SOH-CAH-TOA! Have you heard the call of the trig function? Use them to navigate the trickier sides (and angles) of geometric shapes.
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Graphs of Trigonometric Functions
Plot the six different trig functions, discover their illuminating interactions, and ride the wave!
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Trigonometric Equations
Equations with trigonometry in them can look scary, but that's nothing that a quick little substitution can't fix.
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Solving Triangles
Trigonometric problem solving culminates in this chapter. Leave no side and no angle unmeasured!
Trigonometric Identities
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Pythagorean Identities
Trigonometric identities bring new life to the Pythagorean theorem by re-envisioning the legs of a right triangle as sine and cosine.
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Fundamental Trigonometric Identities
These are your basic building blocks for solving trigonometric equations and understanding how the pieces fit together. Using these identities can make sense of even the scariest looking trig.
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Sum and Difference Trigonometric Formulas
These formulas explain how to add and subtract trigonometric functions (and their arguments). If you've got sum time, see what a difference these formulas will make for your trig toolkit.
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Proving Trigonometric Identities
So, you've memorized your fundamental identities, but how can you prove the more obscure ones? See how to apply the basic building blocks of trig to understand deeper relationships.
Vectors
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Properties of a Vector
Vectors allow you to represent quantities with both size and direction, such as the velocity of an airplane. Better yet, they do so in a mathematically-useful way. Dive in to see how!
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Dot Product of Vectors
The dot product (also known as the scalar product) is an operation on vectors that can tell you the angle between the vectors.
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Cross Product of Vectors
The cross product is a fundamental operation on vectors. It acts on vectors in three dimensions and results in another vector in three dimensions which is perpendicular to both of the other vectors!
3D Geometry
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Volume
What is the volume of a sphere of radius 5? If we make a cylinder two times taller, and its radius three times bigger, how many times larger would it become in volume?
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Surface Area
A spherical balloon is inflated until its volume becomes 27 times its original volume. How much does its surface area increase?
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Polyhedra
When shapes get three-dimensional, things get weird. Dive in to learn about the cube, tetrahedron, octahedron, and more.
Euclidean Geometry
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Euclidean Geometry Triangles
They say triangles are the simplest polygon, but they're still not all that simple. Dive into this advanced treatment of triangles, and learn beautiful results from Euler's line to Routh's Theorem.
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Euclidean Geometry Circles
Think circles are straightforward? Well, they're clearly curved, not straight. And they aren't easy, either. Dive into this challenging chapter full of advanced theorems related to circles.
Geometric Transformations
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Isometries
Symmetries are one of the most beautiful part of Geometry. Slide into reflections, rotations, translations, and more, and you'll be on your way to geometric artistry.
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Common Misconceptions (Geometry)
Geometry is tricky -- are squares rectangles, or are rectangles squares? There are 10 centimeters in a meter, so how many square centimeters are in a square meter? Get your facts straight here.
Community Wiki
Browse through thousands of Geometry wikis written by our community of experts.
Geometric Measurement
- Angles
- Parallel Lines (Geometry)
- Area of a Triangle
- Area of a Rectangle
- Circles
- Length and Area Problem Solving
- Composite Figures
- Volume of a Cuboid
- Volume of a Cylinder
- Volume of a Sphere
- Volume of a Pyramid
- Volume Problem Solving
- Calculating the Size of Earth
- Sphere Packing
- Surface Area
- Surface Area of a Cuboid
- Surface Area of a Sphere
- Surface Area of a Cylinder
Triangles
- Triangles
- Pythagorean Theorem
- Triangle Inequality
- Symmedian
- Classification of Triangles
- Properties of Equilateral Triangles
- Properties of Isosceles Triangles
- Congruent and Similar Triangles
- Heron's Formula
- Pascal's Theorem
- Centroid of a Triangle
- Circumcenter
- Incenter
- Orthocenter
- Euler Line
- Ceva's Theorem
- Apollonius's Theorem
Polygons
Circles
Coordinate Geometry
- Linear Equations
- Equations of Parallel and Perpendicular Lines
- Midpoint of a Line Segment
- Distance Formula
- Distance between Point and Line
- Section Formula
- Equation of Locus
- Determining Coordinates
- Coordinate Geometry - Angle Bisector
- Conic Sections
- Equation of a Circle
- Parabola
- Ellipse
- Hyperbola
- Discriminant of a Conic Section
- 3D Coordinate Geometry - Distance
- 3D Coordinate Geometry - Equation of a Line
- 3D Coordinate Geometry - Skew Lines
- 3D Coordinate Geometry - Equation of a Plane
- 3D Coordinate Geometry - Parallel Planes
- 3D Coordinate Geometry - Perpendicular Planes
- 3D Coordinate Geometry - Intersection of Planes
Trigonometry
- Trigonometry
- Converting Degrees and Radians
- Basic Trigonometric Functions
- Reciprocal Trigonometric Functions
- Inverse Trigonometric Functions
- Bearing - Word Problems
- Hyperbolic Trigonometric Functions
- Local Linearization
- Sine and Cosine Graphs
- Tangent and Cotangent Graphs
- Cosecant and Secant Graphs
- Inverse Trigonometric Graphs
- Symmetry in Trigonometric Graphs
- Trigonometric Graphs - Amplitude and Periodicity
- Graphical Transformation of Trigonometric Functions
- Graphs of Trigonometric Functions - Problem Solving
- Trigonometric Equations
- Trigonometric Equations - Double Angle Formula
- Lengths in Right Triangles
- Sine Rule (Law of Sines)
- Cosine Rule (Law of Cosines)
- Solving Triangles
Trigonometric Identities
- Pythagorean Identities
- Ratio of Trigonometric Functions
- Trigonometric Even-odd Functions
- Trigonometric Periodicity Identities
- Trigonometric Co-function Identities
- Trigonometric R method
- Fundamental Trigonometric Identities - Problem Solving (Easy)
- Double Angle Identities
- Sum and Difference Formulas
- Triple Angle Identities
- Expansions of sin(nx) and cos(nx)
- Half Angle Tangent Substitution
- Trigonometric Power Reduction Identities
- Product-to-Sum Trigonometric Formulas
- Sum to Product Trigonometric Identities
- Proving Trigonometric Identities
- Chebyshev Polynomials - Definition and Properties