Unit 14: Basic Geometry
Angles, shapes, perimeter, area, and volume, built up piece by piece until a circle unrolls into a triangle.
19 topics · 76 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.
Points, Lines, Rays, and Angles
- How it worksWalk me through how two rays sharing one starting point form an angle.
- Worked exampleFind the size of each angle when 4 rays split a full 360 degree turn into equal parts.
- Going furtherWhat happens to an angle's size if you make one of its rays twice as long?
- Ties togetherHow does a ray starting at zero on a number line connect to a ray forming an angle?
Your turn1 of 2
A cake is cut by 9 straight cuts through the center, splitting the whole cake evenly like a full 360 degree turn. What is the angle of each slice?Measuring Angles with a Protractor
- How it worksWalk me through lining up a protractor's center and zero line to read an angle's degrees.
- Going furtherWhat changes about reading a protractor if the angle opens toward the left instead of the right?
- Ties togetherHow does a protractor's 360 degree circle connect to the twelve numbers on a clock face?
Your turn1 of 2
To measure an angle with a protractor, where do you line up the protractor's center point?Types of Angles
- How it worksWatch a ray sweep around a point to show how angles go from acute to right to obtuse to straight.
- Common mistakeWhy isn't a right angle only a right angle when one of its sides points straight up?
- Worked exampleFind the type of angle formed by a clock's hands at 4 o'clock.
- Worked exampleFind the type of angle made when a book cover opens away from its pages to 200 degrees.
- Going furtherWhat happens to an angle's type if you keep opening it past 180 degrees?
Your turn1 of 2
An angle measures 59 degrees. Is it acute, right, obtuse, or straight?Classifying Triangles
- How it worksWalk me through sorting triangles into equilateral, isosceles, and scalene by comparing their side lengths.
- Worked exampleFind whether a triangle with sides 5 centimeters, 5 centimeters, and 8 centimeters is equilateral, isosceles, or scalene.
- Common mistakeWhy is a triangle balanced on one pointy corner still the exact same triangle as one sitting flat on a side?
- Ties togetherHow does classifying a triangle by its angles connect to the types of angles you already know?
Your turn1 of 2
A triangle has sides 6 centimeters, 6 centimeters, and 13 centimeters. Is it equilateral, isosceles, or scalene?Classifying Quadrilaterals
- Worked exampleFind whether a shape with four equal 4 inch sides and four right angles is a square or a rhombus.
- Common mistakeWhy isn't every shape with four equal sides automatically a square?
- Going furtherWhat happens to a rectangle's shape if you push its corners so the angles are no longer square?
Your turn1 of 2
A shape has four equal 8 inch sides and four right angles. Is it a square or a rhombus?Polygons and Their Names
- How it worksWatch shapes gain one side at a time to show how a triangle becomes a pentagon, then a hexagon, then an octagon.
- Common mistakeWhy isn't a circle considered a polygon no matter how many straight sides you add to a shape?
- Worked exampleFind the name for a polygon that has 7 straight sides and 7 corners.
- Going furtherWhat happens to a polygon's name if you add one side to a hexagon, and then add another side after that?
- Ties togetherHow does knowing a shape's number of sides help you find its perimeter later on?
Your turn1 of 2
A tabletop is shaped like a polygon with 9 sides. What is the sum of its interior angles?Perimeter: Walking Around the Outside
- How it worksWalk me through tracing your finger around a rectangle's four sides and adding up each length to find its perimeter.
- How it worksHow does adding twice the length and twice the width give you the same answer as adding all four sides?
- Worked exampleFind the perimeter of a rectangular garden that is 6 feet long and 4 feet wide.
- Common mistakeWhy doesn't a shape with a bigger area always have a bigger perimeter?
Your turn1 of 2
Priya's rectangular garden is 21 feet long and 20 feet wide. What is its perimeter?Area of a Rectangle
- How it worksWatch a rectangle fill up with rows of unit squares to show why area equals length times width.
- Worked exampleFind the area of a rectangular rug that is 5 feet long and 3 feet wide.
- Ties togetherHow does multiplying length times width for area connect to multiplication as making equal rows?
Your turn1 of 2
Ben's rectangular rug is 21 feet long and 20 feet wide. What is its area?Area of a Triangle as Half a Rectangle
- How it worksWatch a rectangle get cut along its diagonal to show that a triangle is exactly half of it.
- Common mistakeWhy isn't a triangle's area just base times height, without cutting anything in half?
- Common mistakeWhy can't you use any two sides of a triangle as the base and height if they don't meet at a right angle?
- Worked exampleFind the area of a triangle with a base of 8 centimeters and a height of 5 centimeters.
- Going furtherWhat happens to a triangle's area if you double its height but keep the base the same?
Your turn1 of 2
A triangular flag has a base of 21 centimetres and a height of 20 centimetres. What is its area?Area of a Parallelogram
- How it worksWatch a triangle slide off one end of a parallelogram and reattach on the other side to make a rectangle.
- Worked exampleFind the area of a parallelogram with a base of 10 meters and a height of 4 meters.
- Common mistakeWhy can't you use a parallelogram's slanted side length as its height when finding the area?
- Going furtherWhat happens to a parallelogram's area if you push its top sideways to make it lean over more?
Your turn1 of 2
A parallelogram-shaped garden bed has a base of 21 metres and a height of 20 metres. What is its area?Area of a Circle by Unrolling It
- How it worksWatch a circle get sliced into thin pizza wedges that rearrange into a shape that's almost a rectangle.
- Worked exampleFind the area of a pizza with a 3 inch radius, using 3 point 14 for π.
- Common mistakeWhy isn't a circle's radius the same as the distance you'd measure straight across it?
Your turn1 of 2
A round pizza has a 11 inch radius. Using 3.14 for π, what is its area?Circumference of a Circle
- How it worksWatch a circle's edge get unrolled flat into a straight line to show what circumference means.
- How it worksWalk me through multiplying a circle's diameter by 3 point 14 to find its circumference.
- Worked exampleFind the circumference of a circle with a diameter of 10 centimeters, using 3 point 14 for π.
- Common mistakeWhy isn't a circle's circumference just π times its radius, the same way you'd use the diameter?
- Going furtherWhat happens to a circle's circumference if you double its radius?
Your turn1 of 2
A wheel has a diameter of 21 centimetres. Using 3.14 for π, what is its circumference?Nets: Unfolding a 3D Shape Flat
- How it worksWatch a cube unfold along its edges into a flat cross-shaped net of six squares.
- Common mistakeWhy doesn't every arrangement of six squares fold up into a cube?
- Worked exampleFind the number of faces, edges, and corners you get when a six-square net folds into a cube.
- Going furtherWhat happens to a cube's net if you unfold it along a different set of edges instead?
Your turn1 of 2
A toy company unfolds 7 identical cube-shaped boxes into flat nets. How many square faces are there across all of them?Volume of a Rectangular Box
- How it worksWatch a rectangular box fill up layer by layer with unit cubes to show why volume equals length times width times height.
- Common mistakeWhy isn't a box's volume just its length plus its width plus its height added together?
- Going furtherWhat happens to a box's volume if you double its height but keep the length and width the same?
Your turn1 of 2
A shipping box is 11 by 10 by 14 inches. What is its volume?Surface Area of a Rectangular Box
- How it worksWatch a box's six faces peel off and lay flat so you can add up each rectangle's area.
- Worked exampleFind the surface area of a cube-shaped box with 4 inch sides.
- Worked exampleFind the surface area of a rectangular box that is 5 centimeters long, 3 centimeters wide, and 2 centimeters tall.
- Common mistakeWhy isn't a box's surface area the same as its volume just because both use the same three measurements?
- Going furtherWhat happens to a box's surface area if you double every one of its side lengths?
Your turn1 of 2
A gift box is a cube with 11 inch sides. What is its surface area?Volume of a Cylinder
- How it worksWatch a cylinder get filled with flat circular layers stacked up to show why volume equals the circle's area times height.
- Worked exampleFind the volume of a cylinder with a radius of 2 inches and a height of 5 inches, using 3 point 14 for π.
- Going furtherWhat happens to a cylinder's volume if you double its radius but keep the height the same?
- Ties togetherHow does stacking circles to find a cylinder's volume connect to finding a circle's area by unrolling it?
Your turn1 of 2
A can of soup has a radius of 7 inches and a height of 13 inches. Using 3.14 for π, what is its volume?Surface Area of a Cylinder
- How it worksWatch a cylinder's curved side peel off into a flat rectangle while its top and bottom stay as two circles.
- Going furtherWhat happens to a cylinder's surface area if you make it twice as tall but keep the same radius?
- Ties togetherHow does unrolling a cylinder's side into a rectangle connect to finding a circle's circumference?
Your turn1 of 2
A drum has a radius of 7 and a height of 13. Using 3.14 for π, what is its surface area?Line Symmetry
- How it worksWatch a shape fold along a line so one half lands exactly on top of the other half.
- Common mistakeWhy doesn't a rectangle have just one line of symmetry, the way a house shape usually does?
- Worked exampleFind how many lines of symmetry a square has.
- Worked exampleFind how many lines of symmetry an equilateral triangle has.
- Going furtherWhat happens to a shape's lines of symmetry if you stretch it taller without stretching it wider?
Your turn1 of 2
A stained-glass window is shaped like a regular polygon with 9 sides. How many lines of symmetry does it have?Transformations: Slides, Flips, and Turns
- How it worksWatch a triangle slide across the page, then flip over a line, then turn around a point, without changing its size.
- Worked exampleFind where a triangle's top corner ends up after you turn the whole triangle 90 degrees around its bottom-left corner.
- Common mistakeWhy doesn't flipping a shape over a line just mean sliding it to the other side?
- Ties togetherHow does turning a shape 90 degrees connect to the angles you measure with a protractor?
Your turn1 of 2
A ceiling fan blade is turned 247 degrees around its center. How many more degrees does it need to turn to complete a full 360 degree spin?