# Geometry and spatial reasoning

Here is a list of all of the skills that cover geometry and spatial reasoning! These skills are organized by grade, and you can move your mouse over any skill name to view a sample question. To start practicing, just click on any link. IXL will track your score, and the questions will automatically increase in difficulty as you improve!

## Kindergarten skills

- K.1Inside and outside
- K.2Left, middle, and right
- K.3Top, middle, and bottom
- K.4Above and below
- K.5Location in a three-by-three grid
- S.1Identify shapes I
- S.2Identify shapes II
- S.3Same shape
- S.4Identify solid figures
- S.5Relate planar and solid figures
- S.6Count sides and corners
- S.7Compare sides and corners
- S.8Geometry of everyday objects
- S.9Symmetry I
- S.10Symmetry II

## First-grade skills

- K.1Identify 2-dimensional shapes
- K.2Identify 3-dimensional figures
- K.3Geometry of everyday objects
- K.4Relate planar and solid figures
- K.5Count sides and vertices
- K.6Count edges, vertices, and faces
- K.7Compare sides and vertices
- K.8Compare edges, vertices, and faces
- K.9Open and closed shapes
- K.10Flip, turn, and slide
- K.11Symmetry
- K.12Same shape
- L.1Left, middle, and right
- L.2Above, below, top, middle, and bottom
- L.3Location in a three-by-three grid

## Second-grade skills

- T.1Identify 2-dimensional shapes
- T.2Identify 3-dimensional shapes
- T.3Identify 2-dimensional and 3-dimensional shapes
- T.4Count sides and angles
- T.5Count edges, vertices, and faces
- T.6Compare sides and angles
- T.7Compare edges, vertices, and faces
- T.8Symmetry
- T.9Congruent
- T.10Flip, turn, and slide
- T.11Perimeter
- T.12Perimeter - word problems
- T.13Area
- T.14Create figures with a given area

## Third-grade skills

- R.1Identify 2-dimensional and 3-dimensional shapes
- R.2Which 2-dimensional shape is being described?
- R.3Count and compare sides and angles
- R.4Count and compare edges, vertices, and faces
- R.5Symmetry
- R.6Similar and congruent
- R.7Reflection, rotation, and translation
- R.8Perimeter
- R.9Perimeter: find the missing side length
- R.10Area of figures made of unit squares
- R.11Create figures with a given area
- R.12Area of rectangles
- R.13Area of squares and rectangles: word problems
- R.14Area of complex figures (with all right angles)
- R.15Area between two rectangles
- R.16Compare area and perimeter of two figures
- R.17Use area and perimeter to determine cost
- R.18Relationship between area and perimeter
- R.19Volume
- R.20Angles: greater than, less than, or equal to a right angle
- R.21Lines, line segments, and rays
- R.22Parallel, perpendicular, intersecting
- R.23Is it a polygon?
- R.24Triangles: equilateral, isosceles, and scalene
- R.25Triangles: acute, right, and obtuse
- R.26Classify quadrilaterals

## Fourth-grade skills

- P.1Identify 2-dimensional and 3-dimensional shapes
- P.2Classify triangles by side lengths
- P.3Classify triangles by angles
- P.4Classify triangles
- P.5Open and closed shapes and qualities of polygons
- P.6Which 2-dimensional shape is being described?
- P.7Which 3-dimensional figure is being described?
- P.8Classify quadrilaterals
- P.9Number of sides in polygons
- P.10Count and compare sides, edges, faces, and vertices
- P.11Similar and congruent
- P.12Nets of 3-dimensional figures
- P.13Acute, right, obtuse, and straight angles
- P.14Angles of 90, 180, 270, and 360 degrees
- P.15Measure angles with a protractor
- P.16Estimate angle measurements
- P.17Adjacent angles
- P.18Parts of a circle
- P.19Perimeter
- P.20Create figures with a given area
- P.21Area of squares and rectangles
- P.22Area of complex figures (with all right angles)
- P.23Area between two rectangles
- P.24Compare area and perimeter of two figures
- P.25Relationship between area and perimeter
- P.26Volume
- P.27Use area and perimeter to determine cost
- P.28Rotational symmetry
- P.29Lines of symmetry
- P.30Lines, line segments, and rays
- P.31Parallel, perpendicular, intersecting
- P.32Calculate radius, diameter and circumference

## Fifth-grade skills

- B.1Identify 2-dimensional and 3-dimensional shapes
- B.2Types of triangles
- B.3Open and closed shapes and qualities of polygons
- B.4Regular and irregular polygons
- B.5Number of sides in polygons
- B.6Which figure is being described?
- B.7Classify quadrilaterals
- B.8Reflection, rotation, and translation
- B.9Count and compare sides, edges, faces, and vertices
- B.10Similar and congruent
- B.11Nets of 3-dimensional figures
- B.12Types of angles
- B.13Measure angles with a protractor
- B.14Parts of a circle
- B.15Perimeter
- B.16Area of squares and rectangles
- B.17Area of triangles
- B.18Area of parallelograms and trapezoids
- B.19Area of compound figures
- B.20Area between two rectangles
- B.21Area and perimeter of irregular figures
- B.22Area and perimeter: word problems
- B.23Volume of rectangular prisms made of unit cubes
- B.24Volume of irregular figures made of unit cubes
- B.25Volume of cubes and rectangular prisms
- B.26Surface area
- B.27Three-dimensional figures viewed from different perspectives
- B.28Lines of symmetry
- B.29Rotational symmetry
- B.30Lines, line segments, and rays
- B.31Parallel, perpendicular, intersecting
- B.32Radius, diameter, circumference, and area of a circle
- B.33Find the unknown angle in triangles and quadrilaterals

## Sixth-grade skills

- Q.4Distance between two points
- Z.1Lines, line segments, and rays
- Z.2Review of 2- and 3-dimensional figures
- Z.3Angle review: measure and classify
- Z.4Estimate angle measurements
- Z.5Name angles
- Z.6Complementary and supplementary angles
- Z.7Transversal of parallel lines
- Z.8Triangle review
- Z.9Classify quadrilaterals
- Z.10Find missing angles and side lengths in triangles and quadrilaterals
- Z.11Sums of angles in polygons
- Z.12Parts of a circle
- Z.13Central angles of circles
- Z.14Similar and congruent figures
- Z.15Find side lengths of similar figures
- Z.16Reflection, rotation, and translation
- Z.17Translations: graph the image
- Z.18Reflections: graph the image
- Z.19Rotations: graph the image
- Z.20Symmetry
- Z.21Find lengths and measures of bisected lines and angles
- Z.22Perimeter
- Z.23Area
- Z.24Area of compound figures
- Z.25Area between two rectangles
- Z.26Area between two triangles
- Z.27Rectangles: relationship between perimeter and area
- Z.28Compare area and perimeter of two figures
- Z.29Circles: calculate area, circumference, radius, and diameter
- Z.30Circles: word problems
- Z.31Area between two circles
- Z.32Identify polyhedra and count faces, edges, and vertices
- Z.33Which figure is being described?
- Z.34Front, side, and top view
- Z.35Nets of 3-dimensional figures
- Z.36Volume of cubes and rectangular prisms
- Z.37Surface area of cubes and rectangular prisms
- Z.38Volume and surface area of triangular prisms
- Z.39Volume and surface area of cylinders
- Z.40Relate volume and surface area
- Z.41Semicircles: calculate area, perimeter, radius, and diameter
- Z.42Quarter circles: calculate area, perimeter, and radius
- Z.43Area of compound figures with triangles, semicircles, and quarter circles

## Seventh-grade skills

- P.1Lines, line segments, and rays
- P.2Parallel, perpendicular, intersecting
- P.3Name, measure, and classify angles
- P.4Identify complementary, supplementary, vertical, adjacent, and congruent angles
- P.5Find measures of complementary, supplementary, vertical, and adjacent angles
- P.6Transversal of parallel lines
- P.7Classify triangles
- P.8Classify quadrilaterals
- P.9Find missing angles in triangles and quadrilaterals
- P.10Identify and classify polygons
- P.11Interior angles of polygons
- P.12Similar and congruent figures
- P.13Similar figures: side lengths and angle measures
- P.14Similar figures and indirect measurement
- P.15Congruent figures: side lengths and angle measures
- P.16Congruence statements and corresponding parts
- P.17Perimeter
- P.18Area of rectangles and parallelograms
- P.19Area of triangles and trapezoids
- P.20Area and perimeter: word problems
- P.21Parts of a circle
- P.22Circles: calculate area, circumference, radius, and diameter
- P.23Circles: word problems
- P.24Find lengths and measures of bisected lines and angles
- P.25Front, side, and top view
- P.26Names and bases of 3-dimensional figures
- P.27Nets of 3-dimensional figures
- P.28Surface area
- P.29Volume
- P.30Perimeter, area, and volume: changes in scale
- P.31Semicircles: calculate area, perimeter, radius, and diameter
- P.32Quarter circles: calculate area, perimeter, and radius
- P.33Area of compound figures with triangles, semicircles, and quarter circles
- P.34Area between two shapes
- Q.1Identify reflections, rotations, and translations
- Q.2Translations: graph the image
- Q.3Translations: find the coordinates
- Q.4Reflections: graph the image
- Q.5Reflections: find the coordinates
- Q.6Rotations: graph the image
- Q.7Rotations: find the coordinates
- Q.8Symmetry
- R.1Pythagorean theorem: find the length of the hypotenuse
- R.2Pythagorean theorem: find the missing leg length
- R.3Pythagorean theorem: word problems
- R.4Converse of the Pythagorean theorem: is it a right triangle?

## Eighth-grade skills

- O.1Pythagorean theorem: find the length of the hypotenuse
- O.2Pythagorean theorem: find the missing leg length
- O.3Pythagorean theorem: find the perimeter
- O.4Pythagorean theorem: word problems
- O.5Converse of the Pythagorean theorem: is it a right triangle?
- P.4Distance between two points
- Q.1Identify complementary, supplementary, vertical, adjacent, and congruent angles
- Q.2Find measures of complementary, supplementary, vertical, and adjacent angles
- Q.3Transversal of parallel lines
- Q.4Classify triangles
- Q.5Classify quadrilaterals
- Q.6Find missing angles in triangles and quadrilaterals
- Q.7Identify and classify polygons
- Q.8Interior angles of polygons
- Q.9Similar and congruent figures
- Q.10Similar figures: side lengths and angle measures
- Q.11Congruent figures: side lengths and angle measures
- Q.12Congruence statements and corresponding parts
- Q.13Congruent triangles: SSS, SAS, and ASA
- Q.14Perimeter
- Q.15Area
- Q.16Area and perimeter: word problems
- Q.17Area between two shapes
- Q.18Parts of a circle
- Q.19Circles, semicircles, and quarter circles
- Q.20Circles: word problems
- Q.21Find lengths and measures of bisected lines and angles
- Q.22Front, side, and top view
- Q.23Base plans
- Q.24Names and parts of 3-dimensional figures
- Q.25Nets of 3-dimensional figures
- Q.26Surface area of prisms and cylinders
- Q.27Surface area of pyramids and cones
- Q.28Volume of prisms and cylinders
- Q.29Volume of pyramids and cones
- Q.30Volume and surface area of spheres
- Q.31Similar solids
- Q.32Volume and surface area of similar solids
- Q.33Perimeter, area, and volume: changes in scale
- R.1Identify reflections, rotations, and translations
- R.2Translations: graph the image
- R.3Translations: find the coordinates
- R.4Reflections: graph the image
- R.5Reflections: find the coordinates
- R.6Rotations: graph the image
- R.7Rotations: find the coordinates
- R.8Dilations: graph the image
- R.9Dilations: find the coordinates
- R.10Dilations: scale factor and classification
- R.11Symmetry
- Z.5Add polynomials to find perimeter
- Z.13Multiply polynomials to find area

## Algebra 1 skills

- E.6Minimum and maximum area and volume
- E.8Percent error: area and volume
- F.1Perimeter
- F.2Area
- F.3Area and perimeter: word problems
- F.4Volume
- F.5Surface area
- F.6Similar figures: side lengths and angle measures
- F.7Similar triangles and indirect measurement
- F.8Dilations and scale factors
- F.9Area and perimeter of similar figures
- F.10Area between two shapes
- F.11Similar solids
- F.12Volume and surface area of similar solids
- F.13Perimeter, area, and volume: changes in scale
- F.14Pythagorean theorem
- F.15Pythagorean theorem: word problems
- F.16Converse of the Pythagorean theorem: is it a right triangle?
- F.17Special right triangles
- G.2Midpoints
- G.3Distance between two points
- Z.5Add polynomials to find perimeter
- HH.1Trigonometric ratios: sin, cos, and tan
- HH.2Find trigonometric ratios using a calculator
- HH.3Inverses of trigonometric functions
- HH.4Trigonometric ratios: find a side length
- HH.5Trigonometric ratios: find an angle measure
- HH.6Solve a right triangle

## Geometry skills

- B.1Lines, line segments, and rays
- B.2Lengths of segments on number lines
- B.3Additive property of length
- B.4Midpoints
- B.5Congruent line segments
- B.6Perpendicular Bisector Theorem
- B.7Midpoint formula
- B.8Distance formula
- B.9Construct the midpoint or perpendicular bisector of a segment
- C.1Angle vocabulary
- C.2Angle measures
- C.3Identify complementary, supplementary, vertical, adjacent, and congruent angles
- C.4Find measures of complementary, supplementary, vertical, and adjacent angles
- C.5Angle bisectors
- C.6Construct an angle bisector
- C.7Construct a congruent angle
- C.8Proofs involving angles
- D.1Identify parallel, perpendicular, and skew lines and planes
- D.2Construct a perpendicular line
- D.3Transversals: name angle pairs
- D.4Transversals of parallel lines: find angle measures
- D.5Construct parallel lines
- D.6Proofs involving parallel lines I
- D.7Proofs involving parallel lines II
- E.7Find the distance between a point and a line
- E.8Find the distance between two parallel lines
- F.1Classify triangles
- F.2Triangle Angle-Sum Theorem
- F.3Exterior Angle Theorem
- G.1Polygon vocabulary
- G.2Interior angles of polygons
- G.3Exterior angles of polygons
- G.4Review: interior and exterior angles of polygons
- G.5Construct an equilateral triangle or regular hexagon
- G.6Construct a square
- H.1Three-dimensional figure vocabulary
- H.2Parts of three-dimensional figures
- H.3Nets and drawings of three-dimensional figures
- H.4Cross-sections of three-dimensional figures
- H.5Solids of revolution
- J.1Congruence statements and corresponding parts
- J.2Solve problems involving corresponding parts
- J.3Identify congruent figures
- K.1SSS and SAS Theorems
- K.2Proving triangles congruent by SSS and SAS
- K.3ASA and AAS Theorems
- K.4Proving triangles congruent by ASA and AAS
- K.5SSS, SAS, ASA, and AAS Theorems
- K.6SSS Theorem in the coordinate plane
- K.7Proving triangles congruent by SSS, SAS, ASA, and AAS
- K.8Proofs involving corresponding parts of congruent triangles
- K.9Congruency in isosceles and equilateral triangles
- K.10Proofs involving isosceles triangles
- K.11Hypotenuse-Leg Theorem
- L.1Translations: graph the image
- L.2Translations: find the coordinates
- L.3Translations: write the rule
- L.4Reflections: graph the image
- L.5Reflections: find the coordinates
- L.6Rotate polygons about a point
- L.7Rotations: graph the image
- L.8Rotations: find the coordinates
- L.9Classify congruence transformations
- L.10Compositions of congruence transformations: graph the image
- L.11Transformations that carry a polygon onto itself
- L.12Congruence transformations: mixed review
- L.13Dilations: graph the image
- L.14Dilations: find the coordinates
- L.15Dilations: scale factor and classification
- L.16Dilations and parallel lines
- M.1Midsegments of triangles
- M.2Triangles and bisectors
- M.3Identify medians, altitudes, angle bisectors, and perpendicular bisectors
- M.4Angle-side relationships in triangles
- M.5Triangle Inequality Theorem
- M.6Construct the circumcenter or incenter of a triangle
- M.7Construct the centroid or orthocenter of a triangle
- M.8Proofs involving triangles I
- M.9Proofs involving triangles II
- N.1Classify quadrilaterals
- N.2Properties of parallelograms
- N.3Proving a quadrilateral is a parallelogram
- N.4Properties of rhombuses
- N.5Properties of squares and rectangles
- N.6Properties of trapezoids
- N.7Properties of kites
- N.8Review: properties of quadrilaterals
- N.9Proofs involving quadrilaterals I
- N.10Proofs involving quadrilaterals II
- O.1Line symmetry
- O.2Rotational symmetry
- O.3Count lines of symmetry
- O.4Draw lines of symmetry
- P.1Identify similar figures
- P.2Similarity ratios
- P.3Similarity statements
- P.4Side lengths and angle measures in similar figures
- P.5Similar triangles and indirect measurement
- P.6Perimeters of similar figures
- P.7Similarity rules for triangles
- P.8Similar triangles and similarity transformations
- P.9Similarity of circles
- P.10Triangle Proportionality Theorem
- P.11Areas of similar figures
- Q.1Pythagorean Theorem
- Q.2Converse of the Pythagorean theorem
- Q.3Pythagorean Inequality Theorems
- Q.4Special right triangles
- R.1Trigonometric ratios: sin, cos, and tan
- R.2Trigonometric ratios: csc, sec, and cot
- R.3Trigonometric ratios in similar right triangles
- R.4Complementary angles
- R.5Find trigonometric functions of special angles
- R.6Find trigonometric functions using a calculator
- R.7Inverses of trigonometric functions
- R.8Trigonometric ratios: find a side length
- R.9Trigonometric ratios: find an angle measure
- R.10Solve a right triangle
- R.11Law of Sines
- R.12Law of Cosines
- R.13Solve a triangle
- S.1Perimeter
- S.2Area of rectangles and squares
- S.3Area of parallelograms and triangles
- S.4Area of trapezoids
- S.5Area and perimeter in the coordinate plane I
- S.6Area and perimeter in the coordinate plane II
- S.7Area and circumference of circles
- S.8Area of compound figures
- S.9Area between two shapes
- S.10Area and perimeter of similar figures
- S.11Heron's formula
- S.12Area and perimeter mixed review
- T.1Introduction to surface area and volume
- T.2Surface area of prisms and cylinders
- T.3Surface area of pyramids and cones
- T.4Volume of prisms and cylinders
- T.5Volume of pyramids and cones
- T.6Surface area and volume of spheres
- T.7Introduction to similar solids
- T.8Surface area and volume of similar solids
- T.9Surface area and volume review
- U.1Parts of a circle
- U.2Central angles
- U.3Arc measure and arc length
- U.4Area of sectors
- U.5Circle measurements: mixed review
- U.6Arcs and chords
- U.7Tangent lines
- U.8Perimeter of polygons with an inscribed circle
- U.9Inscribed angles
- U.10Angles in inscribed right triangles
- U.11Angles in inscribed quadrilaterals
- U.12Construct a tangent line to a circle
- U.13Construct an equilateral triangle inscribed in a circle
- U.14Construct a square inscribed in a circle
- U.15Construct a regular hexagon inscribed in a circle
- U.16Construct the inscribed or circumscribed circle of a triangle
- V.1Find the center of a circle
- V.2Find the radius or diameter of a circle
- V.3Write equations of circles in standard form from graphs
- V.4Write equations of circles in standard form using properties
- V.5Graph circles
- W.6Minimum and maximum area and volume
- W.8Percent error: area and volume
- X.7Geometric probability
- Y.1Compass directions and vectors
- Y.2Find the magnitude of a vector
- Y.3Find the component form of a vector
- Y.4Find the component form of a vector given its magnitude and direction angle
- Y.5Add and subtract vectors
- Y.6Graph a resultant vector using the triangle method
- Y.7Graph a resultant vector using the parallelogram method

## Algebra 2 skills

- T.1Identify the direction a parabola opens
- T.2Find the vertex of a parabola
- T.3Find the focus or directrix of a parabola
- T.4Find the axis of symmetry of a parabola
- T.5Write equations of parabolas in vertex form from graphs
- T.6Write equations of parabolas in vertex form using properties
- T.7Convert equations of parabolas from general to vertex form
- T.8Find properties of a parabola from equations in general form
- T.9Graph parabolas
- U.1Find the center of a circle
- U.2Find the radius or diameter of a circle
- U.3Write equations of circles in standard form from graphs
- U.4Write equations of circles in standard form using properties
- U.5Convert equations of circles from general to standard form
- U.6Find properties of circles from equations in general form
- U.7Graph circles
- V.1Find the center, vertices, or co-vertices of an ellipse
- V.2Find the length of the major or minor axes of an ellipse
- V.3Find the foci of an ellipse
- V.4Write equations of ellipses in standard form from graphs
- V.5Write equations of ellipses in standard form using properties
- V.6Convert equations of ellipses from general to standard form
- V.7Find properties of ellipses from equations in general form
- W.1Find the center of a hyperbola
- W.2Find the vertices of a hyperbola
- W.3Find the length of the transverse or conjugate axes of a hyperbola
- W.4Find the equations for the asymptotes of a hyperbola
- W.5Find the foci of a hyperbola
- W.6Write equations of hyperbolas in standard form from graphs
- W.7Write equations of hyperbolas in standard form using properties
- W.8Convert equations of hyperbolas from general to standard form
- W.9Find properties of hyperbolas from equations in general form
- X.1Convert between radians and degrees
- X.2Radians and arc length
- X.5Coterminal angles
- X.6Reference angles
- Y.1Pythagorean Theorem and its converse
- Y.2Special right triangles
- Y.3Trigonometric ratios: sin, cos, and tan
- Y.4Trigonometric ratios: csc, sec, and cot
- Y.5Trigonometric ratios in similar right triangles
- Y.6Find trigonometric ratios using the unit circle
- Y.7Sin, cos, and tan of special angles
- Y.9Find trigonometric functions using a calculator
- Y.10Inverses of sin, cos, and tan
- Y.12Solve trigonometric equations I
- Y.13Solve trigonometric equations II
- Y.14Trigonometric ratios: find a side length
- Y.15Trigonometric ratios: find an angle measure
- Y.16Solve a right triangle
- Y.17Law of Sines
- Y.18Law of Cosines
- Y.19Solve a triangle
- Y.20Area of a triangle: sine formula
- Y.21Area of a triangle: Law of Sines
- AA.1Complementary angles
- AA.2Symmetry and periodicity of trigonometric functions

## Precalculus skills

- L.1Convert between radians and degrees
- L.2Radians and arc length
- L.4Coterminal and reference angles
- L.5Find trigonometric ratios using right triangles
- L.6Find trigonometric ratios using the unit circle
- L.8Inverses of trigonometric functions
- L.9Solve trigonometric equations
- L.10Trigonometric ratios: find a side length
- L.11Trigonometric ratios: find an angle measure
- L.12Solve a right triangle
- L.13Law of Sines
- L.14Law of Cosines
- L.15Solve a triangle
- L.16Area of a triangle: sine formula
- L.17Area of a triangle: Heron's formula
- N.1Complementary angles
- N.2Symmetry and periodicity of trigonometric functions
- O.1Find properties of parabolas
- O.2Write equations of parabolas in vertex form
- O.3Graph parabolas
- O.4Find properties of circles
- O.5Write equations of circles in standard form
- O.6Graph circles
- O.7Find properties of ellipses
- O.8Find the eccentricity of an ellipse
- O.9Write equations of ellipses in standard form
- O.10Find properties of hyperbolas
- O.11Find the eccentricity of a hyperbola
- O.12Write equations of hyperbolas in standard form
- O.13Convert equations of conic sections from general to standard form
- Q.8Distance in the complex plane
- S.1Find the magnitude of a vector
- S.2Find the direction angle of a vector
- S.3Find the component form of a vector
- S.4Find the component form of a vector from its magnitude and direction angle
- S.5Find a unit vector
- S.6Add and subtract vectors
- S.7Multiply a vector by a scalar
- S.8Find the magnitude or direction of a vector scalar multiple
- S.9Find the magnitude and direction of a vector sum
- S.10Linear combinations of vectors
- S.11Graph a resultant vector using the triangle method
- S.12Graph a resultant vector using the parallelogram method
- T.1Find the magnitude of a three-dimensional vector
- T.2Find the component form of a three-dimensional vector
- T.3Find a three-dimensional unit vector
- T.4Add and subtract three-dimensional vectors
- T.5Scalar multiples of three-dimensional vectors
- T.6Linear combinations of three-dimensional vectors