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
Convert between radian and degree measures
Define a Steradian and know its relationship to square degrees.
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).
Determine the length of an arc of a circle, given its radius and the measure of its central angle
Construct a bisector of a given angle, using a straightedge and compass, and justify the construction
Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction
Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction
Construct an equilateral triangle, using a straightedge and compass, and justify the construction
Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles
Solve problems using compound loci
Identify corresponding parts of congruent triangles and other figures
Investigate, justify, and apply the isosceles triangle theorem and its converse
Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem
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.
Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons
Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons
Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals
Investigate, justify, and apply theorems about special quadrilaterals (rectangles, rhombuses, squares, kites) involving their angles, sides, diagonals and symmetry
Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals
Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, kites, or trapezoids
Investigate, justify, and apply 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.
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
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
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
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).
Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines
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
Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections
Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism
Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)
Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries)
Investigate, justify, and apply the properties that remain invariant under similarities
Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism
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
Construct the center of a circle using a straight edge and compass.
Calculate the area of a segment of a circle, given the measure of a central angle and the radius of the circle
Construct a circle touching three points using a straight edge and compass.
Circumscribe a circle on a triangle using a straight edge and compass.
Construct a triangle with three known sides using a ruler and compass, and justify the construction
Cut a line into n equal segments using a straightedge and compass, and justify the construction
Construct a circle inscribed within a triangle (incircle) using a ruler and compass, and justify the construction.
Construct a pentagon using a ruler and compass, and justify the construction.
Construct a tangent from a point to a circle using a ruler and compass, and justify the construction.
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.
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.
Investigate, justify, and apply theorems about the number of diagonals of regular polygons.
Investigate the properties of the pentagram, and its relationship to the golden ratio.
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.
Understand that a plane is a flat surface with no thickness that goes on forever.
Know how to find the ratio of the areas of similar shapes given the ratio of their lengths.
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.
Investigate cyclic quadrilaterals and know that opposite angles of a cyclic quadrilateral are supplementary.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle using a straightedge and compass, and justify the constructions.
Prove that all circles are similar.
Calculate unknown lengths inside a circle using the Intersecting Chords Theorem.
Calculate unknown lengths outside a circle using the 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).
High School Geometry | Geometry (Solid)
Use formulas to calculate volume and surface area of rectangular solids and cylinders
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
Know and apply that the lateral edges of a prism are congruent and parallel
Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal
Know and apply that the volume of a prism is the product of the area of the base and the altitude
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
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
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
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
Know and apply that through a given point there passes one and only one plane perpendicular to a given line
Know and apply that through a given point there passes one and only one line perpendicular to a given plane
Know and apply that two lines perpendicular to the same plane are coplanar
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
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
Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane
Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines
Know and apply that if two planes are perpendicular to the same line, they are parallel
Understand what is meant by the cross section of a prism, cylinder, pyramid, sphere or torus and recognize the shape of the cross section.
Understand what is meant by the dihedral angle between two planes.
Understand Euler's Formula connecting the numbers of faces, vertices and edges of the Platonic solids and many other solids.
Understand why there are exactly five Platonic solids.
Know the properties of a torus, including the formulas for surface area and volume.
Use formulas to calculate the surface areas and volumes of the tetrahedron, the cube, the octahedron, the dodecahdron and the icosahedron.
Identify three-dimensional objects generated by rotations of two-dimensional objects.
Compare the volumes and surface areas of a cone (radius r, height 2r), a sphere (radius r) and a cylinder (radius r, height 2r).
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.
High School Geometry | Coordinates
Understand Polar Coordinates, and how to convert from Cartesian coordinates to polar coordinates and vice versa.
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
Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle
Find the measure of a side of a right triangle, given an acute angle and the length of another side
Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides
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
Know the exact and approximate values of the sine, cosine, and tangent of 0, 30, 45, 60, 90, 180, and 270 degree angles
Sketch and use the reference angle for angles in standard position
Know and apply the co-function and reciprocal relationships between trigonometric ratios
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
Sketch the unit circle and represent angles in standard position
Find the value of trigonometric functions, if given a point on the terminal side of angle (theta)
Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function
Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent
Sketch the graphs of the inverses of the sine, cosine, and tangent functions
Determine the trigonometric functions of any angle, using technology
Justify the Pythagorean identities
Solve simple trigonometric equations for all values of the variable from 0 degrees to 360 degrees (four quadrants)
Determine amplitude, period, frequency, phase shift and vertical shift given the graph or equation of a periodic function
Sketch and recognize one cycle of a function of the form y = A sin(Bx) or y = A cos(Bx)
Sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x)
Write the trigonometric function that is represented by a given periodic graph
Solve for an unknown side or angle, using the Law of Sines
Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle
Determine the solution(s) of triangles from the SSA situation (ambiguous case)
Apply the angle sum and difference formulas for trigonometric functions
Apply the double-angle and half-angle formulas for trigonometric functions
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
Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle
Investigate, justify, and apply the 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
Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1
Establish similarity of triangles, using the following theorems: AA, SAS, and SSS
Investigate, justify, apply, and prove the Pythagorean theorem and its converse.
Include proof of the Pythagorean Theorem using triangle similarity.
Sketch and recognize the graphs of the functions y=sin(x), y=cos(x) and y=tan(x)
Find the area of a triangle given the lengths of its three sides, using Heron's formula.
Recognize that an AAA triangle is impossible to solve.
Use the symmetric properties of an equilateral triangle to solve 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.
Know and apply the opposite angle identities: sin(-A) = -sin(A), cos(-A) = cos(A) and
tan(-A) = -tan(A)
Know how to find the values of sine, cosine and tangent in each of the four quadrants; including determining the correct sign.
Solve for an unknown side or angle, using the Law of Cosines
Solve a triangle using the Law of Sines and the Law of Cosines
Use the magic hexagon to remember 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.
Know how to express a bearing using three-figure bearings, and how to convert between three-figure bearings and the principal compass bearings.