You have learned how to calculate the perimeter, area, and volume of many different figures. Recall that you can apply perimeter and area to two-dimensional shapes while volume is a three-dimensional measurement. You also learned how to measure the surface area of a three-dimensional shape by creating its net. A net is the composite polygon created when you unfold a three-dimensional figure and lay it flat. Once the net is created, you can decompose it into triangles, quadrilaterals, and other polygons to calculate the surface area. Some formulas you will need to work with nets are shown.
Triangle Rectangle Square
A=(b×h)/2 A=l×w A=s^2
The net of a square pyramid is shown. It has four equilateral triangles and square base. Use the net to calculate the surface area of the square pyramid by answering the following questions.
Give the measurements that represent the length of the base and the height of each triangle in the net. (2 points)
Using the base and height from Question 1, calculate the area of one triangle. Show your work. (2 points)
Calculate the total area of all four triangles shown in the net. Explain your thinking. (3 points)
What is the measurement for one side length of the square shown in the net? (1 point)
Calculate the area of the square shown in the net. (2 points)
Calculate the total surface area of the pyramid shown in the net. (2 points)
Liam and his brothers are building the shed pictured. Use what you know about surface area to help them determine the total number of square feet of plywood they will need to cover the entire structure, including a ceiling and floor.
Hint: There will be only one ceiling.
What two three-dimensional shapes make up the shed? (1 point)
Calculate the surface area of the rectangular prism including the floor and ceiling. (4 points)
Calculate the surface area of the triangular prism excluding one of the rectangular faces (ceiling). (3 points)
Using the areas calculated in Questions 8 and 9, how many square feet of plywood are needed to cover the shed including a ceiling and floor? (2 points)
If one piece of plywood has an area of 32 square feet, how many pieces of plywood will Liam and his brothers need to buy? (2 points)
Archimedes was a famous Greek mathematician. One day he was asked to figure out the volume of gold that was in the king’s crown. Since a crown has an irregular shape, he needed to come up with an inventive way to measure this volume. One day, while Archimedes was taking a bath, he noticed the water level changed when he entered the water. This led him to use water displacement to calculate the volume of the crown. Luckily for you, the shapes that you will explore have formulas to calculate the volume. These are shown below.
Cube Rectangular Prism Triangular Prism
V=s^3 V=l×w×h V=Bh
Use the formulas and what you have learned about volume to answer the questions below.
Calculate the volume of a cube with a side length of 6 meters. Show your work. (2 points)
Look again at the illustration of the shed Liam and his brothers are building. What is the volume of the rectangular prism? Show your work. (2 points)
Look again at the illustration of the shed Liam and his brothers are building. What is the volume of the triangular prism? Show your work. (2 points)
An in-ground rectangular swimming pool is 16 feet wide, 36 feet long, and 5.5 feet deep. A gallon of water has a volume of about 0.13 cubic feet.
Calculate the volume of the swimming pool. Show your work. (2 points)
How many gallons of water are needed to fill the entire pool? Round to the nearest gallon. (2 points)
