Geometry: Polygons, Circles and Solids
Area and perimeter of a polygon, area and circumference of a circle, and the volume of a cube, a box and a cylinder, all with no calculator.
10 topics · 37 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Perimeter of a Polygon
- How it worksHow does an L-shaped figure still give up its perimeter when two of its sides carry no labels?
- Worked exampleFind the perimeter of a rectangle 14 long and 9 wide.
- Worked exampleFind the side length of a square whose perimeter comes to 68.
Your turneasy1 of 3
Write an expression, in terms of w, for the perimeter of a rectangle 24 long and w wide.Area of Rectangles, Squares and Parallelograms
- How it worksHow does sliding the triangle off one end of a parallelogram turn it into a plain rectangle?
- Worked exampleFind the area of a parallelogram with base 12 and height 7.
- Worked exampleFind the area of a rectangle whose perimeter is 40 and whose length is twice its width.
- Common mistakeWhy isn't the slanted side of a parallelogram the height you are supposed to multiply by?
Your turneasy1 of 3
What is the area of a parallelogram with base 19 and height 11?Area of Triangles and Trapezoids
- How it worksHow do two copies of the same trapezoid lock together into one parallelogram?
- Worked exampleFind the area of a triangle with base 15 and height 8.
- Worked exampleFind the area of a trapezoid whose parallel sides are 6 and 10 with a height of 5.
- Going furtherWhat happens to a triangle's area when its top vertex slides sideways but stays at the same height?
Your turneasy1 of 3
What is the area of a triangle with base 23 and height 12?Interior and Exterior Angles of Polygons
- How it worksHow does slicing a polygon into triangles from one corner produce the angle sum formula?
- Worked exampleFind the interior angle sum of an octagon and the size of one angle if it is regular.
- Common mistakeWhy doesn't the sum of the exterior angles change when the polygon grows more sides?
Your turneasy1 of 3
5 identical polygons each have n sides. Write an expression, in terms of n, for the combined sum of all their interior angles, in degrees.Circumference of a Circle
- How it worksHow does a circle unroll into a straight segment measuring exactly 2 π r?
- Worked exampleFind the circumference of a circle of radius 7, leaving π in the answer.
- Worked exampleFind the radius of a circle whose circumference measures 31.4.
- Common mistakeWhy isn't π exactly equal to the fraction 22/7?
- Going furtherWhat distance does a wheel of radius 14 inches roll through in one complete turn?
Your turneasy1 of 3
What is the circumference of a circle of radius 15? Use π.Area of a Circle
- How it worksHow does slicing a circle into wedges and interleaving them build a rectangle of area π r squared?
- Common mistakeWhy doesn't the area formula take the diameter the way the circumference formula happily can?
- Worked exampleFind the area of a circle whose diameter measures 10.
- Worked exampleFind the area of the ring left between two circles of radius 5 and radius 3.
Your turneasy1 of 3
What is the area of a circle whose diameter measures 23? Use π.Shaded Regions and Composite Figures
- How it worksHow do you break an odd figure into rectangles, triangles and slices of circles you already know?
- Worked exampleFind the shaded area when a circle of radius 3 is punched out of a 10 by 8 rectangle.
- Worked exampleFind the area of a track made of a 20 by 10 rectangle with a semicircle on each short end.
- Going furtherWhat happens to the shaded area when the inner shape is moved around but never resized?
Your turneasy1 of 3
A circle of radius 3 is punched out of a 23 by 8 rectangle. What area is left shaded?Volume of a Cube and a Rectangular Box
- How it worksHow does stacking layer after layer of unit cubes build the length times width times height rule?
- Worked exampleFind the volume of a box measuring 8 by 5 by 3.
- Worked exampleFind the edge length of a cube whose volume comes to 216.
- Common mistakeWhy doesn't doubling every edge of a cube simply double the volume inside?
Your turneasy1 of 3
What is the volume of a box measuring 15 by 5 by 3?Volume of a Cylinder
- How it worksHow does the area of the circular base times the height give the cylinder formula?
- Worked exampleFind the volume of a cylinder of radius 3 and height 10, leaving π in the answer.
- Common mistakeWhy doesn't a cylinder hold only twice as much when you double its radius alone?
Your turneasy1 of 3
Write an expression, in terms of h, for the volume of a cylinder of radius 6 and height h, leaving π in your answer.Surface Area of a Box and a Cube
- How it worksHow does unfolding a box into a flat net turn surface area into six rectangles you can add up?
- Worked exampleFind the surface area of a box measuring 4 by 6 by 10.
- Worked exampleFind the surface area of a cube whose volume is 125.
Your turneasy1 of 3
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