High School Geometry Curriculum

Below are skills needed, with links to resources to help with that skill. We also encourage plenty of exercises and book work. Curriculum Home

Important: this is a guide only.
Check with your local education authority to find out their requirements.

High School Geometry | Measurement
☐ Define radian measure
Radians
Definition of Radian
☐ Convert between radian and degree measures
Degrees (Angles)
Radians
☐ Define a Steradian and know its relationship to square degrees.
Steradian
High School Geometry | Geometry (Plane)
☐ Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
Polygons
Circle
Area of Circle, Triangle, Square, Rectangle, Parallelogram, Trapezium, Ellipse and Sector
Area Calculator
Double Hearts Ratio
Perimeter
Activity: Garden Area
Interactive Polygons
☐ Determine the length of an arc of a circle, given its radius and the measure of its central angle
Circle Sector and Segment
Definition of Arc
Radians
Definition of Arc Length
☐ Construct a bisector of a given angle, using a straightedge and compass, and justify the construction
Definition of Contruction (Geometry)
Definition of Compass (Drawing)
Angle Bisector Construction
Bisect
☐ Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction
Definition of Contruction (Geometry)
Definition of Compass (Drawing)
Line Bisector Construction
Bisect
☐ Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction
Definition of Contruction (Geometry)
Definition of Compass (Drawing)
Parallel Line through a Point Construction
Perpendicular to a Point on a Line Construction
Perpendicular to a Point NOT on a Line Construction
☐ Construct an equilateral triangle, using a straightedge and compass, and justify the construction
Definition of Contruction (Geometry)
Definition of Compass (Drawing)
Equilateral Triangle OR 60 degree angle Construction
☐ Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles
Angle Bisector Construction
Perpendicular to a Point on a Line Construction
Bisect
Triangle Centers
☐ Solve problems using compound loci
Definition of Locus
Set of All Points
☐ Identify corresponding parts of congruent triangles and other figures
Congruent
Congruent Triangles
☐ Investigate, justify, and apply the isosceles triangle theorem and its converse
Definition of Isosceles Triangle
Triangles - Equilateral, Isosceles and Scalene
☐ Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem
Triangles Contain 180 Degrees
Exterior Angle
☐ Based on the measure of given angles formed by the transversal and the lines, determine whether two or more lines cut by a transversal are parallel.
Alternate Exterior Angles
Alternate Interior Angles
Consecutive Interior Angles
Corresponding Angles
Parallel Lines, and Pairs of Angles
Transversals
Definition of Parallel Lines
☐ Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons
Exterior Angles of Polygons
Interior Angles of Polygons
Regular Polygons - Properties
Polygons
Interactive Polygons
☐ Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons
Exterior Angles of Polygons
Interior Angles of Polygons
Regular Polygons - Properties
Polygons
☐ Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals
Interactive Quadrilaterals
Definition of Parallelogram
Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram
Parallelogram
☐ Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares, kites) involving their angles, sides, and diagonals
Interactive Quadrilaterals
Definition of Rectangle
Definition of Rhombus
Definition of Square
Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram
Square (Geometry)
Rectangle
Rhombus
Kite
☐ Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals
Interactive Quadrilaterals
Definition of Trapezoid
Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram
Trapezoid
☐ Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, kites, or trapezoids
Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram
Interactive Quadrilaterals
Rectangle
Rhombus
Square (Geometry)
Parallelogram
Trapezoid
Kite
☐ Investigate, justify, and apply theorems about similar triangles
Similar Triangles
Theorems about Similar Triangles
☐ Given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle, investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle.
Theorems about Similar Triangles
Similar Triangles
☐ Investigate, justify, and apply theorems about mean proportionality: * the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse * the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg
Mean Proportional
☐ Investigate, justify, and apply theorems regarding chords of a circle: * perpendicular bisectors of chords * the relative lengths of chords as compared to their distance from the center of the circle
Circle
Definition of Chord
Bisect
☐ Investigate, justify, and apply theorems about tangent lines to a circle: * a perpendicular to the tangent at the point of tangency * two tangents to a circle from the same external point * common tangents of two non-intersecting or tangent circles
Definition of Tangent (line)
Point to Tangents on a Circle Construction
Circle Theorems
☐ Investigate, justify, and apply theorems about two lines intersecting a circle when the vertex is inside the circle (two chords) or on the circle (tangent and chord).
Circle Theorems
☐ Investigate, justify, and apply theorems regarding segments intersected by a circle: * along two tangents from the same external point * along two secants from the same external point * along a tangent and a secant from the same external point * along two intersecting chords of a given circle
Circle Theorems
☐ Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
Reflection Symmetry
Geometry Rotation
Geometry - Reflection
Point Symmetry
Geometry Translation
Rotational Symmetry
Transformations
Symmetry - Reflection and Rotation
☐ Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections
Geometry Rotation
Transformations
Symmetry - Reflection and Rotation
Rotational Symmetry
Geometry - Reflection
Reflection Symmetry
Point Symmetry
Geometry Translation
☐ Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)
Congruent
Transformations
Rotational Symmetry
Definition of Perpendicular
Definition of Parallel
Definition of Congruent
Geometry Rotation
Geometry Translation
☐ Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries)
Similar
Definition of Similar
Geometry Resizing
Transformations
☐ Investigate, justify, and apply the properties that remain invariant under similarities
Similar
☐ Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism
Similar
☐ Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90 degrees and 180 degrees, reflections over the lines x=0, y=0, and y=x, and dilations centered at the origin
Geometry - Reflection
Geometry Rotation
Transformations
Geometry Translation
Geometry Resizing
☐ Construct the center of a circle using a straight edge and compass.
Center of Circle Construction
☐ Calculate the area of a segment of a circle, given the measure of a central angle and the radius of the circle
Circle Sector and Segment
Definition of Segment
☐ Construct a circle touching three points using a straight edge and compass.
Circle touching 3 Points Construction
Circumscribe a Circle in a Triangle Construction
☐ Circumscribe a circle on a triangle using a straight edge and compass.
Circumscribe a Circle in a Triangle Construction
Circle touching 3 Points Construction
☐ Construct a triangle with three known sides using a ruler and compass, and justify the construction
Constructing A Triangle With 3 Known Sides
3, 4, 5 Triangle
☐ Cut a line into n equal segments using a straightedge and compass, and justify the construction
Cut a line into N segments Construction
☐ Construct a circle inscribed within a triangle (incircle) using a ruler and compass, and justify the construction.
Angle Bisector Construction
Inscribe a Circle in a Triangle Construction
☐ Construct a pentagon using a ruler and compass, and justify the construction.
Pentagon Construction
☐ Construct a tangent from a point to a circle using a ruler and compass, and justify the construction.
Point to Tangents on a Circle Construction
Circle Theorems
☐ Know that the apothem of a regular polygon is the radius of its incircle, and know its relationship to the radius of the circumcircle of the polygon or the length of side of the polygon.
Regular Polygons - Properties
Definition of Apothem
☐ Calculation of the area of a regular polygon from the number of sides and either the length of side, radius of the circumcircle or length of apothem.
Regular Polygons - Properties
☐ Investigate, justify, and apply theorems about the number of diagonals of regular polygons.
Regular Polygons - Properties
Diagonals of Polygons
Interactive Polygons
☐ Investigate the properties of the pentagram, and its relationship to the golden ratio.
The Pentagram
☐ Use a ruler and drafting triangle to construct a line parallel to a given line and passing through a given point, or to construct a line perpendicular to a given line at a given point.
Using a Ruler and Drafting Triangle
☐ Understand that a plane is a flat surface with no thickness that goes on forever.
What is a Plane?
☐ Know how to find the ratio of the areas of similar shapes given the ratio of their lengths.
Theorems about Similar Triangles
Ratios
Similar Triangles
Similar
☐ Investigate and understand circle theorems including the Angle at the Center Theorem, the Angles Subtended by Same Arc Theorem and The Angle in the Semicircle Theorem.
Circle
Circle Theorems
☐ Investigate cyclic quadrilaterals and know that opposite angles of a cyclic quadrilateral are supplementary.
Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram
Circle Theorems
☐ Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle using a straightedge and compass, and justify the constructions.
Constructing A Triangle With 3 Known Sides
Equilateral Triangle OR 60 degree angle Construction
☐ Prove that all circles are similar.
Circle
☐ Calculate unknown lengths inside a circle using the Intersecting Chords Theorem.
Intersecting Chords Theorem
☐ Calculate unknown lengths outside a circle using the Intersecting Secants Theorem.
Intersecting Secants Theorem
☐ Investigate, justify, and apply theorems about two lines intersecting a circle when the vertex is outside the circle (two tangents, two secants, or tangent and secant).
Circle Theorems
Angle of Intersecting Secants Theorem
High School Geometry | Geometry (Solid)
☐ Use formulas to calculate volume and surface area of rectangular solids and cylinders
Equations and Formulas
Area of Circle, Triangle, Square, Rectangle, Parallelogram, Trapezium, Ellipse and Sector
Spinning Cylinder
Cuboids, Rectangular Prisms and Cubes
Volume of a Cuboid
Definition of Volume
Activity: Soup Can
Definition of Surface Area
☐ Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them
Perpendicular and Parallel
Parallel and Perpendicular Lines and Planes
☐ Know and apply that the lateral edges of a prism are congruent and parallel
Cuboids, Rectangular Prisms and Cubes
Prisms with Examples
Unfold the Prism Puzzle
☐ Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal
Volume of a Cuboid
Cuboids, Rectangular Prisms and Cubes
Prisms with Examples
Pouring Liquid
☐ Know and apply that the volume of a prism is the product of the area of the base and the altitude
Volume of a Cuboid
Cuboids, Rectangular Prisms and Cubes
Prisms with Examples
Pouring Liquid
☐ Apply the properties of a regular pyramid, including: # lateral edges are congruent # lateral faces are congruent isosceles triangles # volume of a pyramid equals one-third the product of the area of the base and the altitude
Spinning Pentagonal Pyramid
Spinning Square Pyramid
Spinning Tetrahedron
Pyramids
Spinning Triangular Pyramid
☐ Apply the properties of a cylinder, including: * bases are congruent * volume equals the product of the area of the base and the altitude * lateral area of a right circular cylinder equals the * product of an altitude and the circumference of the base
Pouring Liquid
Spinning Cylinder
☐ Apply the properties of a right circular cone, including: * lateral area equals one-half the product of the slant height and the circumference of its base * volume is one-third the product of the area of its base and its altitude
Spinning Cone
☐ Apply the properties of a sphere, including: * the intersection of a plane and a sphere is a circle * a great circle is the largest circle that can be drawn on a sphere * two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles * surface area is 4 pi r^2 * volume is (4/3) pi r^3
Sphere
☐ Know and apply that through a given point there passes one and only one plane perpendicular to a given line
Parallel and Perpendicular Lines and Planes
Perpendicular and Parallel
☐ Know and apply that through a given point there passes one and only one line perpendicular to a given plane
Parallel and Perpendicular Lines and Planes
Perpendicular and Parallel
☐ Know and apply that two lines perpendicular to the same plane are coplanar
Parallel and Perpendicular Lines and Planes
Perpendicular and Parallel
☐ Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane
Parallel and Perpendicular Lines and Planes
Perpendicular and Parallel
☐ Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane
Parallel and Perpendicular Lines and Planes
Perpendicular and Parallel
☐ Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane
Parallel and Perpendicular Lines and Planes
Perpendicular and Parallel
☐ Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines
Parallel and Perpendicular Lines and Planes
Perpendicular and Parallel
☐ Know and apply that if two planes are perpendicular to the same line, they are parallel
Perpendicular and Parallel
Parallel and Perpendicular Lines and Planes
☐ Understand what is meant by the cross section of a prism, cylinder, pyramid, sphere or torus and recognize the shape of the cross section.
Cross Sections
Prisms with Examples
Pyramids
Torus
Sphere
Spinning Cylinder
☐ Understand what is meant by the dihedral angle between two planes.
Dihedral Angle Calculator
☐ Understand Euler's Formula connecting the numbers of faces, vertices and edges of the Platonic solids and many other solids.
Euler's Formula
Platonic Solids
Polyhedrons
Vertices, Edges and Faces
Activity: Investigating Solids
☐ Understand why there are exactly five Platonic solids.
Platonic Solids - Why Five?
☐ Know the properties of a torus, including the formulas for surface area and volume.
Torus
☐ Use formulas to calculate the surface areas and volumes of the tetrahedron, the cube, the octahedron, the dodecahdron and the icosahedron.
Spinning Dodecahedron
Spinning Icosahedron
Spinning Octahedron
Spinning Tetrahedron
Spinning Cube (Hexahedron)
☐ Compare the volumes and surface areas of a cone (radius r, height 2r), a sphere (radius r) and a cylinder (radius r, height 2r).
Cone vs Sphere vs Cylinder
Spinning Cone
Sphere
Spinning Cylinder
☐ Calculate the cross-sectional area of a partially filled horizontal cylinder using the formula Area = cos-1((r - h)/r)r^2 - (r - h)sqrt(2rh - h^2) and hence calculate its volume.
Volume of Horizontal Cylinder
High School Geometry | Coordinates
☐ Understand Polar Coordinates, and how to convert from Cartesian coordinates to polar coordinates and vice versa.
Cartesian Coordinates
Polar and Cartesian Coordinates
Definition of Polar Coordinates
Sine, Cosine and Tangent in Four Quadrants
Activity: A Walk in the Desert 2
High School Geometry | Trigonometry
☐ Find the sine, cosine, and tangent ratios (or their reciprocals) of an angle of a right triangle, given the lengths of the sides
Sohcahtoa: Sine, Cosine, Tangent
Sine, Cosine, Tangent
Trigonometry
☐ Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle
Sohcahtoa: Sine, Cosine, Tangent
Finding an Angle in a Right Angled Triangle
Random Trigonometry
Trigonometry
Sine, Cosine, Tangent
☐ Find the measure of a side of a right triangle, given an acute angle and the length of another side
Sohcahtoa: Sine, Cosine, Tangent
Finding a Side in a Right Angled Triangle
Trigonometry
Random Trigonometry
Sine, Cosine, Tangent
☐ Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides
Pythagoras Theorem
Definition of Pythagoras Theorem
3, 4, 5 Triangle
Activity: A Walk in the Desert
Activity: Drawing Squares
Activity: Pythagoras' Theorem
☐ Express and apply the six trigonometric functions as ratios of the sides of a right triangle, and know the trigonometric identities: tan(x) = sin(x)/cos(x) etc
Sohcahtoa: Sine, Cosine, Tangent
Definition of Sine
Definition of Cosine
Definition of Tangent (function)
Definition of Cosecant
Definition of Secant (function)
Definition of Cotangent
Trigonometry
Sine, Cosine, Tangent
☐ Know the exact and approximate values of the sine, cosine, and tangent of 0, 30, 45, 60, 90, 180, and 270 degree angles
Unit Circle
Sine, Cosine, Tangent
Sohcahtoa: Sine, Cosine, Tangent
Solving Triangles by Reflection
☐ Sketch and use the reference angle for angles in standard position
Sine, Cosine and Tangent in Four Quadrants
Unit Circle
☐ Know and apply the co-function and reciprocal relationships between trigonometric ratios
Trigonometry
Sine, Cosine, Tangent
Trigonometric Identities
Definition of Cotangent
Definition of Secant (function)
Definition of Cosecant
Magic Hexagon for Trig Identities
☐ Use the reciprocal and co-function relationships to find the values of the secant, cosecant, and cotangent of 0, 30, 45, 60, 90, 180, and 270 degree angles
Unit Circle
Trigonometric Identities
☐ Sketch the unit circle and represent angles in standard position
Unit Circle
Trigonometry
Interactive Unit Circle
☐ Find the value of trigonometric functions, if given a point on the terminal side of angle (theta)
Polar and Cartesian Coordinates
Sine, Cosine and Tangent in Four Quadrants
Activity: A Walk in the Desert 2
☐ Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function
Domain, Range and Codomain
Graphs of Sine, Cosine and Tangent
Inverse Functions
Inverse Sine, Cosine, Tangent
☐ Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent
Finding an Angle in a Right Angled Triangle
Inverse Functions
Inverse Sine, Cosine, Tangent
☐ Sketch the graphs of the inverses of the sine, cosine, and tangent functions
Graphs of Sine, Cosine and Tangent
Equation Grapher
Inverse Sine, Cosine, Tangent
☐ Determine the trigonometric functions of any angle, using technology
Scientific Calculator
☐ Justify the Pythagorean identities
Pythagoras Theorem
Trigonometric Identities
Unit Circle
☐ Solve simple trigonometric equations for all values of the variable from 0 degrees to 360 degrees (four quadrants)
Sine, Cosine and Tangent in Four Quadrants
Inverse Sine, Cosine, Tangent
☐ Determine amplitude, period, frequency, phase shift and vertical shift given the graph or equation of a periodic function
Graphs of Sine, Cosine and Tangent
Definition of Frequency
Amplitude, Period, Phase Shift and Frequency
Introduction to Waves
☐ Sketch and recognize one cycle of a function of the form y = A sin(Bx) or y = A cos(Bx)
Function Grapher and Calculator
☐ Sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x)
Graphs of Sine, Cosine and Tangent
Function Grapher and Calculator
☐ Write the trigonometric function that is represented by a given periodic graph
Graphs of Sine, Cosine and Tangent
☐ Solve for an unknown side or angle, using the Law of Sines
Triangle Identities
The Law of Sines
☐ Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle
The Law of Sines
Area of Triangles Without Right Angles
☐ Determine the solution(s) of triangles from the SSA situation (ambiguous case)
Solving SSA Triangles
☐ Apply the angle sum and difference formulas for trigonometric functions
Trigonometric Identities
☐ Apply the double-angle and half-angle formulas for trigonometric functions
Trigonometric Identities
☐ Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles
Congruent Triangles
How To Find if Triangles are Congruent
☐ Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle
Triangles Contain 180 Degrees
Interactive Triangles
Triangles - Equilateral, Isosceles and Scalene
☐ Investigate, justify, and apply the triangle inequality theorem
Definition of Triangle Inequality Theorem
☐ Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle
The Law of Cosines
Solving SSS Triangles
☐ Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1
Centroid
Triangle Centers
☐ Establish similarity of triangles, using the following theorems: AA, SAS, and SSS
Similar Triangles
How To Find if Triangles are Similar
☐ Investigate, justify, apply, and prove the Pythagorean theorem and its converse. Include proof of the Pythagorean Theorem using triangle similarity.
Right Angled Triangles
Pythagorean Theorem Proof
Pythagoras Theorem
Definition of Pythagoras Theorem
3, 4, 5 Triangle
Activity: A Walk in the Desert
☐ Sketch and recognize the graphs of the functions y=sin(x), y=cos(x) and y=tan(x)
Graphs of Sine, Cosine and Tangent
Sine Function - Graph Exercise
Trigonometry
☐ Find the area of a triangle given the lengths of its three sides, using Heron's formula.
Heron's Formula
☐ Recognize that an AAA triangle is impossible to solve.
Solving AAA Triangles
☐ Use the symmetric properties of an equilateral triangle to solve triangles by reflection.
Solving Triangles by Reflection
☐ Be familiar with the triangle identities that are true for all triangles: The Law of Sines, The Law of Cosines and the Law of Tangents.
Triangle Identities
The Law of Sines
The Law of Cosines
☐ Know and apply the opposite angle identities: sin(-A) = -sin(A), cos(-A) = cos(A) and tan(-A) = -tan(A)
Trigonometric Identities
☐ Know how to find the values of sine, cosine and tangent in each of the four quadrants; including determining the correct sign.
Sine, Cosine, Tangent
Sine, Cosine and Tangent in Four Quadrants
Unit Circle
☐ Solve for an unknown side or angle, using the Law of Cosines
Triangle Identities
The Law of Cosines
☐ Solve a triangle using the Law of Sines and the Law of Cosines
Triangle Identities
The Law of Cosines
The Law of Sines
Solving Triangles
Solving AAS Triangles
Solving ASA Triangles
Solving SAS Triangles
Solving SSA Triangles
Solving SSS Triangles
Solving AAA Triangles
☐ Use the magic hexagon to remember trigonometric identities
Magic Hexagon for Trig Identities
Trigonometric Identities
☐ Use the Pythagorean Theorem in three dimensions, including calculating the length of a space diagonal of a cuboid given the length, width and height.
Pythagoras Theorem
Pythagoras Theorem in 3D
☐ Know how to express a bearing using three-figure bearings, and how to convert between three-figure bearings and the principal compass bearings.
Compass: North South East and West