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    • ▾High school geometry
      • ▸Basics of geometry
        • •Proofs in this course
        • •Geometric definitions
        • •Linear pair theorem
        • •Area of a regular polygon
        • •Diagonal length of a right rectangular prism
        • •Distance and midpoint in 3D
        • •Circumcircles and incircles
      • ▸Transformations
        • •Preserved properties of rigid transformations
        • •Find measures using rigid transformations
        • •Find the angle of rotation
        • •Find the sequence of rigid transformations
        • •Reflecting figures
        • •Reflecting a point over any non-horizontal line
      • ▸Congruence
        • •Congruent polygons
        • •Congruent supplements and complements theorems
        • •ASA and AAS congruence
        • •Isosceles triangle theorem and its converse
        • •SSS congruence
        • •SAS congruence
        • •Converse of the Pythagorean theorem
        • •Proving isosceles trapezoids
      • ▾Similarity
        • •Intro to similarity
        • •All circles are similar
        • •Finding angles and side lengths in similar polygons
        • •Finding the scale factor for similar polygons
        • •Third angle theorem
        • •Similar triangles
        • •Ratios of area, surface area, and volume, for similar shapes
        • •Crossed ladders theorem
      • ▸Polygons
        • •Perimeter of composite figures
        • •Finding the volume and surface area of composite figures
        • •Sum of the interior angles of a triangle
        • •Sum of interior angles for simple polygons
        • •Sum of exterior angles
        • •Pentagonal tilings
      • ▸Incenter and circumcenter of a triangle
        • •Intro to the incenter and circumcenter of a triangle
        • •Perpendicular bisector theorem and its converse
        • •Circumcenter theorem
        • •Angle bisector theorem
        • •Incenter theorem
      • ▸Triangles
        • •Euler line
        • •Geometric mean theorems
        • •AM-GM inequality
        • •Triangle angle bisector theorem
        • •Ceva's theorem
        • •Proving the altitude of an isosceles triangle cuts two congruent right triangles
        • •Centroid and orthocenter
        • •Ordering triangle sides and angles
        • •Midsegment and proportionality theorems
        • •Menelaus's theorem
        • •Side splitter theorem
        • •Hinge theorem and its converse
        • •Hypotenuse-leg congruence theorem
        • •Hypotenuse-angle congruence theorem
        • •Areas of triangles and quadrilaterals on grids
        • •Exterior angle theorem
      • ▸Quadrilaterals
        • •Diagonals of a kite are perpendicular
        • •Diagonals of a parallelogram bisect each other
        • •Diagonals of a rhombus are perpendicular bisectors
        • •Area of a rhombus from diagonals
        • •Diagonals of a rectangle are congruent
        • •Opposite sides of a parallelogram are congruent
        • •Opposite angles of a parallelogram are congruent
        • •British flag theorem
        • •Cyclic quadrilaterals
        • •Fuhrmann's theorem
        • •Classifying quadrilaterals from four points
      • ▸Circle theorems
        • •Thales's theorem and its converse
        • •Central angle theorem
        • •Angles standing on the same arc are congruent
        • •Chord properties
        • •Congruent tangents theorem
        • •Radius and tangent theorem
        • •Radii inside right triangles
        • •Alternate segments theorem
        • •Power of a point
        • •Three squares puzzle
      • ▸Volume and surface area
        • •Cavalieri's principle
        • •Volume of prisms and cylinders
        • •Volume and surface area of a sphere
      • ▸Analytic geometry
        • •A square within a square
     › High school geometry › Similarity

    Ratios of area, surface area, and volume, for similar shapes

    Students will learn how scaling a 2D figure affects its area. Then students will learn how scaling a 3D figure affects its surface area and volume.

    Conclude by giving your students these challenges:

    • Triangular Teaser by NRICH
    • One O Five by NRICH
    • 2016 AMC 8, Problem 4

    Mentally determine what fraction is shaded.

    Answer: \((3/16)\pi\)

    Lessons and practice problems