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Grade 10

What Math Does a 10th Grader Learn? All 34 Topics, in Order

By Knomi Learning · Updated

The short answer

A 10th grader learns 34 math topics, in 6 units: Congruence and Proof, Similarity and Trigonometry, Three Dimensions, Coordinate Geometry, Circles and Probability. That is 124 smaller skills in all, taught in the order below, each building on earlier ones.

  • The earlier topics 10th grade leans on most: Area of Polygons (Grade 6), Pythagorean Theorem (Grade 8), Angle Relationships (Grade 7), Circles (Grade 7) and Multi-Step Equations (Grade 8).
  • All 124 skills are matched to a Common Core standard.
  • Every one of them has a full lesson on Knomi.

Unit 1: Congruence and Proof

1

Geometry Foundations

The words the whole course is built from: point, line and plane, then exact definitions of angle, circle, parallel and perpendicular. Every proof later rests on these, so the tutor insists on the exact wording, not just a picture.

What it covers

  • Points, Lines and Planes
  • Precise Definitions
  • Segments and Midpoints
  • Angle Pairs

Builds on

  • Grade 8: Variables on Both Sides
  • Grade 7: Vertical and Adjacent Angles
  • Grade 7: Slicing Prisms
  • Grade 7: Parts of a Circle and Pi
  • Grade 7: Complementary and Supplementary Angles
  • Grade 7: Equations for Unknown Angles
  • and 2 more
Geometry Foundations on the 10th grade map
2

Reasoning and Proof

What a proof is before any proof about triangles: if-then statements and their converses, the reasons a step is allowed to cite, and how a short proof is laid out. Measuring one picture is never a reason, and the tutor says so.

What it covers

  • Conditional Statements
  • Properties of Equality and Congruence
  • Writing Proofs

Builds on

  • Grade 9: Reasoning Step by Step
  • Grade 8: Converse of the Theorem
Reasoning and Proof on the 10th grade map
3

Rigid Motions

Grade 8's slides, flips and turns become functions of the whole plane with exact definitions: reflect across any line, rotate about any point, translate along any vector. A kid who can only work with the axes and the origin is sent back to Grade 8.

What it covers

  • Transformations as Functions
  • Translations by Vectors
  • Reflections Across Any Line
  • Rotations About Any Point
  • Defining Rigid Motions

Builds on

  • Grade 8: Properties of Rigid Motions
  • Grade 8: Function Rules
  • Grade 8: Translations
  • Grade 8: Distance Between Two Points
  • Grade 8: Reflections
  • Grade 8: Rotations
  • and 2 more
Rigid Motions on the 10th grade map
4

Symmetry and Sequences

The rigid motions that carry a figure onto itself (its symmetries) and the chains of motions that carry one figure onto another. A kid who sees a symmetry that is not there is asked to test it with one point.

What it covers

  • Reflective Symmetry
  • Rotational Symmetry
  • Symmetries of Quadrilaterals
  • Sequences of Rigid Motions

Builds on

  • Grade 8: Angle Sums of Polygons
  • Grade 8: Sequences of Transformations
  • Grade 6: Trapezoids and Other Quadrilaterals
Symmetry and Sequences on the 10th grade map
5

Congruence by Rigid Motions

Congruent now means one thing: a sequence of rigid motions carries one figure onto the other. From that the kid explains why congruent triangles match part for part, and why SSS, SAS and ASA work while SSA and AAA do not.

What it covers

  • Congruence as Rigid Motion
  • Corresponding Parts
  • Why SSS, SAS and ASA Work

Builds on

  • Grade 8: Congruent Figures
  • Grade 8: Properties of Rigid Motions
  • Grade 8: Variables on Both Sides
  • Grade 8: Finding a Leg
  • Grade 7: One Triangle, Many or None
Congruence by Rigid Motions on the 10th grade map
6

Triangle Congruence

With the criteria justified, the kid uses them: pick the criterion a marked diagram supports, find missing measures, and prove triangles congruent and then their parts. A shared side or a pair of vertical angles is information too.

What it covers

  • Choosing a Criterion
  • Measures in Congruent Triangles
  • Proving Triangles Congruent
  • Using CPCTC

Builds on

  • Grade 8: Triangle Angle Sum
  • Grade 8: Parallel Lines and Transversals
  • Grade 7: Equations for Unknown Angles
Triangle Congruence on the 10th grade map
7

Line and Angle Theorems

The angle facts used since Grades 7 and 8 get proved: vertical angles, the angles a transversal makes with parallel lines, and the converses that prove two lines parallel. The facts become reasons the kid can cite.

What it covers

  • Vertical Angles Theorem
  • Angles with Parallel Lines
  • Proving Lines Parallel

Builds on

  • Grade 8: Parallel Lines and Transversals
  • Grade 7: Equations for Unknown Angles
Line and Angle Theorems on the 10th grade map
8

Triangle Theorems

The facts about triangles get their proofs: the angle sum, the base angles of an isosceles triangle, the perpendicular bisector, the midsegment, and the points where a triangle's special lines meet.

What it covers

  • Angle Sum and Exterior Angles
  • Isosceles and Equilateral Triangles
  • Perpendicular Bisector Theorem
  • Midsegments
  • Centers of a Triangle

Builds on

  • Grade 8: Triangle Angle Sum
  • Grade 8: Exterior Angles of Triangles
  • Grade 8: Angle Sums of Polygons
  • Grade 7: Solving Proportions
Triangle Theorems on the 10th grade map
9

Quadrilateral Theorems

A diagonal cuts a parallelogram into two congruent triangles, and the kid proves its properties, the tests that show a quadrilateral is a parallelogram, and what rectangles, rhombuses, squares, trapezoids and kites add.

What it covers

  • Properties of Parallelograms
  • Proving a Parallelogram
  • Rectangles, Rhombuses and Squares
  • Trapezoids and Kites

Builds on

  • Grade 6: Trapezoids and Other Quadrilaterals
Quadrilateral Theorems on the 10th grade map
10

Compass Constructions

With a compass and a straightedge the kid copies, bisects, drops perpendiculars, draws parallels and inscribes regular polygons in a circle, and says why each construction works. The compass is what carries a length.

What it covers

  • Copying Segments and Angles
  • Bisecting Segments and Angles
  • Perpendicular and Parallel Lines
  • Inscribed Regular Polygons

Builds on

  • Grade 7: Triangle Side Lengths
  • Grade 7: Parts of a Circle and Pi
Compass Constructions on the 10th grade map

Unit 2: Similarity and Trigonometry

11

Dilations and Similarity

A dilation scales every distance from a center by one factor, keeps angles, and sends lines to parallel lines. Similar is then defined the way congruent was: rigid motions and a dilation carry one figure onto the other.

What it covers

  • Dilations from Any Center
  • Properties of Dilations
  • Similarity Transformations
  • Similar Figures and Scale Factor

Builds on

  • Grade 8: Scale Factor and Center
  • Grade 8: Dilations with Coordinates
  • Grade 8: Similar Figures
  • Grade 8: Side Lengths in Similar Figures
  • Grade 7: Scale and Area
  • Grade 7: Solving Proportions
Dilations and Similarity on the 10th grade map
12

Triangle Similarity

Two pairs of equal angles are enough to make triangles similar, and so are matching side ratios. The kid decides when triangles are similar, solves for missing sides, and writes similarity proofs.

What it covers

  • The AA Criterion
  • SSS and SAS Similarity
  • Solving Similar Triangles
  • Proving Triangles Similar

Builds on

  • Grade 8: Angle-Angle Similarity
  • Grade 8: Variables on Both Sides
  • Grade 6: Ratio Tables
Triangle Similarity on the 10th grade map
13

Similarity Theorems

What similar triangles prove: a parallel line splits two sides in proportion, an angle bisector splits the opposite side in the ratio of its neighbours, and the altitude in a right triangle proves the Pythagorean theorem.

What it covers

  • Side-Splitter Theorem
  • Angle Bisector Theorem
  • Pythagorean Theorem by Similarity
  • Proving with Similarity

Builds on

  • Grade 8: Squares on the Sides
  • Grade 8: Equations with Squares and Cubes
  • Grade 8: Slope Triangles
  • Grade 7: Solving Proportions
Similarity Theorems on the 10th grade map
14

Right Triangles

The Pythagorean theorem with exact answers in simplest radical form, its proofs, and the two special right triangles whose sides come in the ratios 1 : 1 : √2 and 1 : √3 : 2.

What it covers

  • Proofs of the Pythagorean Theorem
  • Pythagorean Theorem and Radicals
  • 45-45-90 Triangles
  • 30-60-90 Triangles

Builds on

  • Grade 9: Special Products
  • Grade 9: Simplifying Square Roots
  • Grade 8: Converse of the Theorem
  • Grade 8: Pythagorean Theorem in 3D
  • Grade 8: Finding a Leg
  • Grade 7: Expanding Expressions
  • and 1 more
Right Triangles on the 10th grade map
15

Trigonometric Ratios

All right triangles with the same acute angle are similar, so each ratio of two sides depends only on the angle. Those ratios are sine, cosine and tangent, built from similar triangles before the calculator takes over.

What it covers

  • Ratios in Right Triangles
  • Sine, Cosine and Tangent
  • Complementary Angles

Builds on

  • Grade 8: Finding a Leg
  • Grade 8: Variables on Both Sides
  • Grade 6: Part and Whole Ratios
Trigonometric Ratios on the 10th grade map
16

Solving Right Triangles

Sine, cosine and tangent find a missing side, the inverse ratios find a missing angle, and both go to work on ladders, ramps, towers and bearings. The kid draws the triangle out of the story before writing any number.

What it covers

  • Finding a Side
  • Finding an Angle
  • Elevation and Depression
  • Modeling with Right Triangles

Builds on

  • Grade 8: Finding the Hypotenuse
  • Grade 8: Right Triangle Word Problems
  • Grade 8: Substitution
  • Grade 6: Multiplication and Division Equations
Solving Right Triangles on the 10th grade map
17

Laws of Sines and Cosines

Trigonometry for any acute triangle, not just right ones: the area ½ab sin C, the Law of Sines and the Law of Cosines. These are (+) standards a school may skip, kept because all three cross-check curricula expect them at this age.

What it covers

  • Area with Sine
  • Law of Sines
  • Law of Cosines
  • Choosing a Law

Builds on

  • Grade 8: Rearranging Formulas
  • Grade 6: Area of Triangles
  • Grade 6: Order with Exponents
Laws of Sines and Cosines on the 10th grade map

Unit 3: Three Dimensions

18

Visualizing Solids

Seeing in three dimensions: the parts of a solid, the shape a plane cuts from it, and the solid a flat shape sweeps out when it spins about a line. These pictures are where the volume formulas come from.

What it covers

  • Solids and Their Parts
  • Cross-Sections
  • Solids of Revolution

Builds on

  • Grade 8: Volume of Cones
  • Grade 8: Volume of Cylinders
  • Grade 7: Volume of Right Prisms
  • Grade 7: Slicing Prisms
  • Grade 7: Slicing Pyramids
  • Grade 6: Nets of Solid Figures
Visualizing Solids on the 10th grade map
19

Why the Formulas Work

Grade 7 and 8 handed over the formulas; here the kid sees why they are true. Sectors rearranged give πr², equal slices give equal volumes (Cavalieri), and a cube cut into three pyramids shows where the 1/3 comes from.

What it covers

  • Circumference and Area of a Circle
  • Cavalieri's Principle
  • Pyramids and Cones
  • Volume of a Sphere

Builds on

  • Grade 8: Volume of Cylinders
  • Grade 8: Volume of Cones
  • Grade 8: Volume of Spheres
  • Grade 7: Circumference
  • Grade 7: Area of a Circle
  • Grade 7: Volume of Right Prisms
  • and 1 more
Why the Formulas Work on the 10th grade map
20

Volume and Surface Area

The formulas become tools: volumes and surface areas of every solid, backwards to a missing dimension, composite solids, and what scaling every length by k does (areas × k², volumes × k³).

What it covers

  • Prisms and Pyramids
  • Cylinders, Cones and Spheres
  • Surface Area of Solids
  • Composite Solids
  • Scaling Area and Volume

Builds on

  • Grade 8: Volume Word Problems
  • Grade 8: Cube Roots
  • Grade 8: Surface Area of Cylinders
  • Grade 7: Surface Area of Prisms
  • Grade 6: Trapezoids and Other Quadrilaterals
  • Grade 6: Composite Figures
  • and 1 more
Volume and Surface Area on the 10th grade map
21

Modeling with Geometry

Real objects are not perfect solids, but a tree trunk is close enough to a cylinder to estimate its wood. The kid models objects with shapes, works with density, and solves design problems with a cost or a constraint.

What it covers

  • Modeling Objects with Shapes
  • Density
  • Design Problems

Builds on

  • Grade 8: Rearranging Formulas
  • Grade 7: Circle Problems
  • Grade 6: Rates and Unit Rates
  • Grade 6: Converting Units
Modeling with Geometry on the 10th grade map

Unit 4: Coordinate Geometry

22

Distance and Midpoints

The distance formula is the Pythagorean theorem with the legs read off the coordinates. With it come the midpoint, the point that splits a segment in a given ratio, and perimeters and areas of shapes on the grid.

What it covers

  • Distance Formula
  • Midpoint Formula
  • Partitioning a Segment
  • Perimeter and Area with Coordinates

Builds on

  • Grade 8: Distance Between Two Points
  • Grade 7: Signs of Products and Powers
  • Grade 7: Adding Signed Fractions and Decimals
  • Grade 6: Mean
  • Grade 6: Sharing in a Ratio
  • Grade 6: Perimeter and Area on the Plane
  • and 1 more
Distance and Midpoints on the 10th grade map
23

Parallel and Perpendicular

Grade 8 noticed it; here the kid proves it: parallel lines have equal slopes and perpendicular slopes multiply to −1. Then the facts become tools for writing lines through a point.

What it covers

  • Slopes of Parallel Lines
  • Slopes of Perpendicular Lines
  • Lines Through a Given Point

Builds on

  • Grade 8: Slope Triangles
  • Grade 8: Number of Solutions of a System
  • Grade 8: Slope from Two Points
  • Grade 8: Solving for y
  • Grade 8: Equations from Two Points
Parallel and Perpendicular on the 10th grade map
24

Coordinate Proofs

Coordinates turn a geometry question into arithmetic: slopes, distances and midpoints decide what four points make, whether a point is on a circle, and, with letters for coordinates, prove theorems for every case.

What it covers

  • Classifying Figures by Coordinates
  • Points on a Circle
  • Proving Theorems with Coordinates

Builds on

  • Grade 6: Combining Like Terms
Coordinate Proofs on the 10th grade map
25

Focus and Directrix

A parabola is every point the same distance from a point (the focus) and a line (the directrix). Setting the two distances equal and squaring gives the familiar equation; reading the focus back from an equation closes the loop.

What it covers

  • Parabola as a Locus
  • Equation from Focus and Directrix
  • Focus and Directrix from an Equation

Builds on

  • Grade 9: Features of a Parabola
  • Grade 9: Special Products
  • Grade 9: Graphs in Vertex Form
  • Grade 8: Rearranging Formulas
  • Grade 6: Reciprocals and Mixed Numbers
Focus and Directrix on the 10th grade map

Unit 5: Circles

26

Arcs and Central Angles

The circle's parts, arcs measured by their central angles, and the inscribed angle theorem: an inscribed angle is half its arc. From it come the right angle in a semicircle and the inscribed quadrilateral's supplementary opposite angles.

What it covers

  • Parts of a Circle
  • Arc Measure
  • Inscribed Angles
  • Inscribed Quadrilaterals

Builds on

  • Grade 7: Parts of a Circle and Pi
  • Grade 7: Equations for Unknown Angles
  • Grade 7: Complementary and Supplementary Angles
Arcs and Central Angles on the 10th grade map
27

Tangents and Chords

Congruence and similarity applied to circles: a tangent meets the radius at a right angle, tangents from one point are equal, a radius perpendicular to a chord bisects it, and crossing chords and secants obey angle and product rules.

What it covers

  • Radius and Tangent
  • Chords and Radii
  • Angles from Chords and Secants
  • Segment Products

Builds on

  • Grade 9: Solving by Completing the Square
  • Grade 8: Converse of the Theorem
  • Grade 8: Finding a Leg
Tangents and Chords on the 10th grade map
28

Arc Length and Sectors

An arc is a fraction of the circumference and a sector the same fraction of the area, the fraction being the central angle over 360°. Because all circles are similar, arc length over radius depends only on the angle: that ratio is the radian.

What it covers

  • Arc Length
  • Radians
  • Area of a Sector

Builds on

  • Grade 7: Circumference
  • Grade 7: Constant of Proportionality
  • Grade 7: Area of a Circle
  • Grade 6: Percent of a Quantity
  • Grade 6: Converting Units
Arc Length and Sectors on the 10th grade map
29

Circles of a Triangle

Every triangle has one circle through its three corners and one circle touching its three sides. The kid constructs both, says why each works, and constructs a tangent from a point outside a circle.

What it covers

  • Circumscribed Circle
  • Inscribed Circle
  • Tangent from an Outside Point
Circles of a Triangle on the 10th grade map
30

Equations of Circles

The equation (x − h)² + (y − k)² = r² is the Pythagorean theorem for every point on the circle. The kid writes it, reads it back (completing the square when it is expanded), and finds where a line meets a circle.

What it covers

  • Equation from Center and Radius
  • Completing the Square for Circles
  • Lines and Circles Intersecting

Builds on

  • Grade 9: Solving by Completing the Square
  • Grade 9: Special Products
  • Grade 9: Substituting into the Quadratic
Equations of Circles on the 10th grade map

Unit 6: Probability

31

Events and Probability

Grade 7 probability with the language of sets: events as parts of a sample space, joined with 'and', 'or' and 'not', drawn in Venn diagrams and two-way tables, and added without counting the overlap twice.

What it covers

  • Sample Spaces and Events
  • Unions, Intersections and Complements
  • The Addition Rule
  • Geometric Probability

Builds on

  • Grade 8: Reading and Building Two-Way Tables
  • Grade 8: Relative Frequencies
  • Grade 7: Sample Spaces
  • Grade 7: Theoretical Probability
  • Grade 7: Relative Frequency
  • Grade 7: Likelihood from 0 to 1
  • and 2 more
Events and Probability on the 10th grade map
32

Independent Events

Two events are independent when one happening does not change the chance of the other, and then P(A and B) = P(A) · P(B). Drawing without replacement breaks that, and the second probability changes.

What it covers

  • Compound Events in Stages
  • Testing for Independence
  • Drawing Without Replacement

Builds on

  • Grade 9: Conditional Relative Frequencies
  • Grade 8: Association in Two-Way Tables
  • Grade 7: Probability of Compound Events
  • Grade 7: Multiplying Rational Numbers
  • Grade 7: Sample Spaces
Independent Events on the 10th grade map
33

Conditional Probability

'Given that' shrinks the sample space: P(A | B) is the share of B's outcomes that are also in A. The kid reads it from tables and trees, uses the formula, and explains it in everyday situations.

What it covers

  • Conditional Probability from Tables
  • The Conditional Probability Formula
  • Probability in Everyday Life

Builds on

  • Grade 9: Conditional Relative Frequencies
  • Grade 8: Relative Frequencies
  • Grade 7: Dividing Rational Numbers
  • Grade 7: Probability Models
Conditional Probability on the 10th grade map
34

Counting Outcomes

When there are too many outcomes to list, count them: the counting principle, permutations when order matters, combinations when it does not, and probabilities as one count over another. A (+) standard, kept because Grade 12's probability leans on it.

What it covers

  • Counting Principle and Permutations
  • Combinations
  • Probability by Counting

Builds on

  • Grade 7: Sample Spaces
  • Grade 6: Meaning of Exponents
  • Grade 6: Factors and Prime Factorization
Counting Outcomes on the 10th grade map

Questions parents ask

How many math topics are in 10th grade?

On Knomi's map, 10th grade math has 34 topics in 6 units, made of 124 smaller skills.

What should a student know before starting 10th grade math?

The earlier topics 10th grade leans on most are Area of Polygons (Grade 6), Pythagorean Theorem (Grade 8), Angle Relationships (Grade 7), Circles (Grade 7) and Multi-Step Equations (Grade 8). A student who is quick and sure with those is ready. Each topic below lists exactly what it builds on.

Is this 10th grade math list aligned to Common Core?

Common Core is the spine. All 124 skills carry a Common Core reference, and the order follows the usual US course.

Does Knomi teach 10th grade math?

Every one of them has a full lesson on Knomi. The 10th grade map is open to anyone, with no account needed to look around.

See 10th grade math as a map.

Every topic above, with lines to what it needs first. Open it and look around, no account needed.

Open the Grade 10 map

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