Math · Geometry

High school Geometry, unit by unit

Proofs, similarity, trigonometry, circles, coordinates, volume and probability: here is everything high school Geometry covers, with worked problems and the mistakes to avoid. Then practice each skill in Mathflow, from intuition to proof.

Start for free See the 10 units

Inscribed angle theorem: the central angle

\angle AOB

is twice the inscribed angle

\angle ACB

, here

80^\circ = 2 \times 40^\circ

What you’ll learn

The 10 units of Geometry

The course as it is organized in Mathflow, in order. Each unit is split into precise skills that you practice one at a time.

Unit 15 skills
Foundations & Logic
Work from precise definitions rather than appearances, describe translations, reflections and rotations exactly, find the symmetries of a figure, and compose rigid motions.
- Use precise geometric definitions
- Represent and compare plane transformations
- Identify the symmetries of a figure
- Specify translations, reflections and rotations
- Construct images and compose rigid motions
Unit 26 skills
Congruence
Show that figures are congruent with rigid motions, justify the SSS, SAS and ASA criteria, and prove theorems about lines, angles, triangles and parallelograms.
- Establish congruence through rigid motions
- Relate triangle congruence to corresponding parts
- Justify triangle congruence criteria
- Prove results about lines and angles
- Prove foundational triangle theorems
- Prove and use parallelogram theorems
Unit 32 skills
Constructions
Copy and bisect segments and angles with exact constructions, explain why they work, and construct a square and a regular hexagon in a circle.
- Make and justify fundamental constructions
- Construct regular figures in a circle
Unit 48 skills
Similarity & Trigonometry
Study dilations and similarity, use the AA criterion, prove the Pythagorean theorem with similar triangles, and build sine, cosine and tangent to solve right triangles.
- Investigate the geometry of dilations
- Establish similarity with transformations
- Explain and apply AA similarity
- Prove triangle proportionality and Pythagoras
- Solve geometric problems with similarity and congruence
- Build trigonometric ratios from similar right triangles
- Use complementary-angle trigonometric relationships
- Solve right triangles in context
Unit 54 skills
Circle Geometry
Show that all circles are similar, connect inscribed angles, chords, radii and tangents, construct the circles of a triangle, and use radians for arc length and sector area.
- Prove that circles are similar
- Connect circle angles, chords, radii and tangents
- Construct triangle circles and analyze cyclic quadrilaterals
- Relate radians, arc length and sector area
Unit 66 skills
Coordinate Geometry & Conics
Write the equation of a circle, derive a parabola from a focus and a directrix, and prove geometric claims with coordinates, slopes, ratios and areas.
- Derive and interpret circle equations
- Derive a parabola from a focus and directrix
- Prove geometric claims with coordinates
- Justify and apply parallel and perpendicular slope criteria
- Partition a directed segment in a given ratio
- Calculate coordinate perimeters and areas
Unit 76 skills
Modeling & 3D Figures
Explain and use the volume formulas of cylinders, pyramids, cones and spheres, picture cross-sections and solids of revolution, and model real objects, densities and designs.
- Explain circle and solid measurement formulas
- Solve volume problems for standard solids
- Visualize sections and solids of revolution
- Choose and evaluate geometric models
- Use area and volume densities
- Solve constrained geometric design problems
Unit 87 skills
Conditional Probability
Describe events in a sample space, compute conditional probabilities from two-way tables, test independence, and use the addition rule without counting the overlap twice.
- Represent events within a sample space
- Test independence using joint probabilities
- Interpret conditional probability and independence
- Analyze two-way frequency tables
- Explain conditional probability in everyday language
- Calculate conditional probabilities by restricting outcomes
- Apply the probability addition rule
Unit 96 skills
Further Geometry
Find the area of any triangle from two sides and an angle, prove and use the laws of sines and cosines, construct tangents, and derive the equations of ellipses and hyperbolas.
- Derive and use the trigonometric triangle-area formula
- Prove the sine and cosine laws
- Solve non-right triangles and handle ambiguous data
- Construct tangents from an external point
- Derive ellipse and hyperbola equations from focal conditions
- Explain sphere volume with Cavalieri reasoning
Unit 104 skills
Further Probability
Apply the general multiplication rule, count with permutations and combinations, design fair random procedures, and compare strategies with probability models.
- Use the general probability multiplication rule
- Choose counting methods for compound probabilities
- Design fair random decision procedures
- Evaluate strategies with probability models

Geometry at a glance

In the traditional US sequence, Geometry is the second of three courses: it comes after Algebra 1 and before Algebra 2. Many students take it in 10th grade.

Before you start. You need the algebra of Algebra 1, such as solving equations, slope and square roots, and the geometry of 8th grade math: the Pythagorean theorem, transformations and volume.

After Geometry. Trigonometry continues in Algebra 2 and Pre-Calculus, where angles move onto the unit circle and become functions.

In Mathflow. The course has $10$ units and $54$ skills, including two probability units. The last two units go beyond the core, with topics marked (+) in the Common Core, such as the laws of sines and cosines. Each skill is a series in three levels, and each unit ends with a Chapter Challenge.

Worked problems

Two problems, solved step by step

Right-triangle trigonometry and the equation of a circle: two classics of Geometry. Try each one on paper before you read the solution.

Problem 1
A ramp: Pythagoras and trigonometry
A ramp rises $2$ feet over a horizontal distance of $24$ feet.
1. How long is the ramp, to the nearest tenth of a foot?
2. What angle does the ramp make with the ground, to the nearest tenth of a degree?
Solution
The rise, the horizontal distance and the ramp form a right triangle. The ramp is the hypotenuse $L$ .
a. By the Pythagorean theorem:
$L = \sqrt{2^2 + 24^2} = \sqrt{580} \approx 24.1$
The ramp is about $24.1$ feet long.
b. Call $\theta$ the angle with the ground. The rise is the side opposite $\theta$ , and the horizontal distance is the adjacent side, so use the tangent:
$\tan\theta = \dfrac{2}{24} \quad\Rightarrow\quad \theta = \tan^{-1}\!\left(\dfrac{2}{24}\right) \approx 4.8^\circ$
Check: the tangent uses only the two given lengths, so it doesn’t depend on the rounded value of $L$ . With the sine, $\sin\theta = \dfrac{2}{\sqrt{580}}$ gives the same angle.
Problem 2
Center and radius of a circle
A circle has the equation $x^2 + y^2 - 6x + 4y - 12 = 0$ .
1. Find its center and its radius.
2. Is the point $(6, 2)$ on the circle?
Solution
a. Group the $x$ terms and the $y$ terms, then complete the square for each group. Half of $-6$ is $-3$ , and half of $4$ is $2$ :
$\begin{aligned} (x^2 - 6x + 9) + (y^2 + 4y + 4) &= 12 + 9 + 4 \\ (x - 3)^2 + (y + 2)^2 &= 25 \end{aligned}$
This is the form $(x - h)^2 + (y - k)^2 = r^2$ . The center is $(3, -2)$ and the radius is $r = \sqrt{25} = 5$ . Watch the signs: $y + 2$ means $k = -2$ .
b. Replace $x$ with $6$ and $y$ with $2$ :
$(6 - 3)^2 + (2 + 2)^2 = 9 + 16 = 25$
The equation holds, so $(6, 2)$ is on the circle: its distance to the center is exactly $5$ .

How Mathflow teaches it

Geometry in three levels

Every skill of the course is a series of short missions in three levels. Here is what you actually do in three of them.

Level 1 · Explore
Explore and play with the concepts to build your intuition.
Level 2 · Explain
Turn your intuitions into the right words and uncover hidden mistakes.
Level 3 · Solve
Reason it out, crunch the numbers, and find your way to the answer.

You don’t start with a theorem to memorize. In the first level, you move the figure: you drag vertices, turn sides, swing a ladder and watch the measures change. In the second level, you put words on what you saw: you complete a proof, decide which of two students is right, or tap the first line of an argument that isn’t justified. In the third level, you solve problems on your own, like the ones you meet in class and on tests.

If you get stuck, Alphi, the AI math tutor, gives you a hint instead of the answer, at any hour. When the skills of a unit are done, a Chapter Challenge checks the whole unit.

Unit 2 · Congruence
1Justify triangle congruence criteria
- Level 1 · Explore
  Turn sides $AC$ and $BC$ until their ends meet to close the triangle. Then swing side $BC$ to make a second triangle with the same three marked measures, and see why SSA is not enough.
- Level 2 · Explain
  Complete the rigid-motion argument that proves SAS, put the steps of the SSS argument in order, and explain why AAA is not a congruence criterion.
- Level 3 · Solve
  $AB \parallel DE$ and $C$ is the midpoint of $BD$ . You choose the criterion that proves $\triangle ABC \cong \triangle EDC$ , and you tap the first unjustified line of a classmate’s proof.
Unit 4 · Similarity & Trigonometry
2Solve right triangles in context
- Level 1 · Explore
  Slide the foot of a ladder until it makes a $75^\circ$ angle with the ground, and set the sun’s angle of elevation so that a tree’s shadow is $24$ feet long.
- Level 2 · Explain
  The angle of depression from a $45$ -foot cliff to a kayak is $28^\circ$ . You tap the first wrong line of Omar’s work, and you explain why the angle of depression equals the angle of elevation.
- Level 3 · Solve
  From the top of a lighthouse, the angle of depression to a boat is $14^\circ$ . You find the horizontal distance $d$ to the boat, to the nearest foot.
Unit 8 · Conditional Probability
3Analyze two-way frequency tables
- Level 1 · Explore
  Sort $10$ survey cards into a two-way table, then move a slider until $P(\text{recycles} \mid \text{south})$ equals $P(\text{recycles})$ .
- Level 2 · Explain
  Ava and Ben find the probability that a student who skips breakfast is late, and you decide who is right. You also complete the sentence that says which total each probability divides by.
- Level 3 · Solve
  In a survey sample, $24$ of the $32$ tenth graders favor science. The school has $400$ tenth graders, and you estimate how many of them favor science.

Practice Geometry for free

Common mistakes

Five mistakes to avoid

These slips come up again and again in Geometry. Learn to spot them, and you’ll catch them in your own work.

Using SSA or AAA to prove congruence
Two sides and an angle that is not between them (SSA) can give two different triangles. Three equal angles (AAA) fix the shape but not the size. Use SSS, SAS, ASA or AAS instead.
Mixing up opposite and adjacent
Opposite and adjacent depend on the angle you start from. In a right triangle with acute angles $A$ and $B$ , the side opposite $A$ is adjacent to $B$ . That is why $\sin A = \cos B$ when $A + B = 90^\circ$ .
Giving an inscribed angle the measure of its arc
An inscribed angle is half of the central angle that intercepts the same arc. If arc $AB$ measures $84^\circ$ , the inscribed angle $ACB$ measures $42^\circ$ , not $84^\circ$ .
Using degrees in $s = r\theta$
The arc length formula $s = r\theta$ needs $\theta$ in radians. For a $60^\circ$ arc on a circle of radius $6$ , $\theta = \dfrac{\pi}{3}$ and $s = 2\pi \approx 6.28$ , not $6 \times 60 = 360$ .
Dividing by the wrong total
In a two-way table, $P(\text{late} \mid \text{bus})$ divides by the number of bus riders, not by the grand total. If $12$ of $60$ bus riders are late, $P(\text{late} \mid \text{bus}) = \dfrac{12}{60} = 0.2$ , whatever the size of the school.

Frequently asked questions

What do you learn in high school Geometry?

Transformations and congruence, proofs about lines, angles, triangles and parallelograms, constructions, similarity and right-triangle trigonometry, circles, coordinate geometry, volume and modeling, and conditional probability. In Mathflow, that’s 10 units and 54 skills.

What grade is Geometry?

In the traditional US sequence, Geometry comes after Algebra 1 and before Algebra 2. Many students take it in 10th grade.

Do you need Algebra 1 for Geometry?

Yes. You solve equations, use slope and square roots, and work on the coordinate plane throughout the course, especially in coordinate geometry.

How do I get better at geometry proofs?

Start from what is given and mark it on the figure. Then justify every step with a definition, a postulate or a theorem. Finding the first unjustified line in someone else’s proof is a good way to train before you write your own.

Is probability part of Geometry?

Often, yes. In the traditional pathway described in Appendix A of the Common Core, conditional probability and the rules of probability are part of the Geometry course. Mathflow’s Geometry course has two probability units.

Is there an app to practice Geometry?

Mathflow covers the whole course with figures you move yourself, three levels per skill, and Alphi, an AI math tutor, for hints when you’re stuck. It’s free to start on iOS and Android.

Start for free

Standards

Geometry and the Common Core

The Common Core State Standards organize high school math by conceptual categories, not by course. The Geometry category has six domains: congruence; similarity, right triangles and trigonometry; circles; expressing geometric properties with equations; geometric measurement and dimension; and modeling with geometry.

Appendix A of the standards proposes a traditional US pathway of three courses, Algebra I, Geometry and Algebra II. Its Geometry course covers these six domains and adds conditional probability and the rules of probability from the Statistics and Probability category. Many states use their own version of the standards, so check your school’s course outline for the exact list.

High school Geometry, unit by unit

Foundations & Logic

Congruence

Constructions

Similarity & Trigonometry

Circle Geometry

Coordinate Geometry & Conics

Modeling & 3D Figures

Conditional Probability

Further Geometry

Further Probability

Geometry at a glance

A ramp: Pythagoras and trigonometry

Center and radius of a circle

Level 1 · Explore

Level 2 · Explain

Level 3 · Solve

1Justify triangle congruence criteria

2Solve right triangles in context

3Analyze two-way frequency tables

Using SSA or AAA to prove congruence

Mixing up opposite and adjacent

Giving an inscribed angle the measure of its arc

Using degrees in s=rθs = r\thetas=rθ

Dividing by the wrong total

Math is all about flow.

Using degrees in $s = r\theta$