Unit 12: Area, Circumference, and Volume
Circles, combined shapes, and the leap from flat area into the volume and surface area of solids you can hold.
6 topics · 22 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.
Area and Circumference of Circles
- How it worksHow do the formulas for circumference and area both come from the same radius, using π in two different ways?
- Worked exampleFind the circumference and the area of a circle with a radius of 7.
- Common mistakeWhy doesn't doubling a circle's radius just double its area?
Your turn1 of 2
A circular pool has a radius of 14 feet. Using π, what is its circumference?Composite Area
- How it worksHow do you split a composite figure into rectangles, triangles, or circles and add or subtract their areas?
- Worked exampleFind the area of a shape made from a 10 by 6 rectangle with a semicircle of radius 3 removed from one side.
- Going furtherWhat happens to your strategy for composite area when a piece is missing from the middle of the shape instead of the edge?
Your turn1 of 2
A rectangular tabletop is 16 by 12 inches, but a semicircle of radius 4 inches is cut out of one side for a sink. Using π, what is the remaining area?Volume and Surface Area of Prisms and Cylinders
- How it worksHow do you find a prism's or cylinder's volume by multiplying its base area by its height, and its surface area by unfolding all its faces?
- Worked exampleFind the volume and surface area of a rectangular prism that is 4 by 3 by 5.
- Worked exampleFind the volume of a cylinder with radius 3 and height 10.
- Common mistakeWhy isn't a shape's surface area just its volume measured a different way?
- Going furtherWhat happens to a prism's volume when you double just its height, keeping the base the same?
Your turn1 of 2
A shipping crate is 11 by 10 by 14 feet. What is its volume?Volume of Cones and Pyramids
- How it worksHow does the one-third factor connect a cone's or pyramid's volume back to the volume of its matching cylinder or prism?
- Worked exampleFind the volume of a cone with radius 3 and height 9.
- Worked exampleFind the volume of a pyramid with a rectangular base of 6 by 4 and a height of 9.
- Common mistakeWhy doesn't a cone with the same base and height as a cylinder hold half the cylinder's volume?
Your turn1 of 2
An ice cream cone has a radius of 9 centimetres and a height of 13 centimetres. Using π, what is its volume?Surface Area of Pyramids, Cones, and Spheres
- How it worksHow do you find the surface area of a cone, a pyramid, and a sphere by adding up their flat and curved parts?
- Worked exampleFind the surface area of a cone with radius 4 and slant height 9.
- Worked exampleFind the surface area of a sphere with radius 5.
- Common mistakeWhy can't you find a cone's surface area using its height in place of its slant height?
Your turn1 of 2
A party hat shaped like a cone has a radius of 4 inches and a slant height of 15 inches. Using π, what is its total surface area?Volume of Spheres
- How it worksHow does the formula four-thirds π r cubed measure the space packed inside a perfectly round ball?
- Worked exampleFind the volume of a sphere with a radius of 6.
- Ties togetherHow does the sphere volume formula relate to the volume formulas you already learned for cylinders and cones?
Your turn1 of 2
A globe has a radius of 9 inches. Using π, what is its volume?