These problems practice standards G-GMD.B.4 and G-MG.A.1 to A.3: cross sections of solids, solids made by rotation, modeling objects with shapes, density, and design with constraints.
The 24 problems move from naming slices to rotation volumes and density, then to design questions about fences, cans, boxes and banners. Draw a sketch whenever it helps.
Problem 1: Slicing a cube
A plane cuts a cube parallel to its base. Name the shape of the cross section. If the cube has an edge of \(6\text{ cm}\), find the area of the cross section.
Problem 2: Rectangle from a cylinder
A cylinder has a radius of \(4\text{ cm}\) and a height of \(10\text{ cm}\). A plane passes through its axis perpendicular to the bases. Describe the cross section and find its area.
Problem 3: Circle from a cylinder
A plane cuts a cylinder with a radius of \(5\text{ cm}\) parallel to its bases. Name the cross section and find its area and circumference in terms of \(\pi\).
Problem 4: Cutting a cone
A cone has a radius of \(3\text{ in}\) and a height of \(4\text{ in}\). (a) What shape is the cross section when a plane cuts parallel to the base? (b) What shape is it when the plane passes through the apex and the center of the base? (c) Find the area of the shape in part (b).
Problem 5: Slicing a sphere
A sphere has a radius of \(13\text{ in}\). A plane cuts it at a distance of \(5\text{ in}\) from the center. Find the radius and area of the circular cross section.
Problem 6: Slicing a pyramid
A square pyramid has a base side of \(10\text{ cm}\) and a height of \(12\text{ cm}\). A plane parallel to the base cuts the pyramid halfway up. Find the side length and area of the cross section.
Problem 7: Slices of a prism
A rectangular prism measures \(6\text{ cm}\) by \(4\text{ cm}\) by \(3\text{ cm}\). Find the area of the cross section for a cut parallel to each pair of faces.
Problem 8: Rotating a rectangle
A rectangle \(3\text{ cm}\) wide and \(7\text{ cm}\) tall is rotated about its \(7\text{ cm}\) side. Name the solid, give its radius and height, and find its volume in terms of \(\pi\).
Problem 9: Rotating a right triangle
A right triangle with legs \(6\text{ in}\) and \(8\text{ in}\) is rotated about the \(8\text{ in}\) leg. Name the solid, find its slant height, and find its volume in terms of \(\pi\).
Problem 10: Rotating about the other leg
The same right triangle with legs \(6\text{ in}\) and \(8\text{ in}\) is now rotated about the \(6\text{ in}\) leg. Find the volume of the new cone in terms of \(\pi\), and tell how much more it holds than the cone in the previous problem.
Problem 11: Rotating a semicircle
A semicircle with a radius of \(5\text{ cm}\) is rotated about its diameter. Name the solid and find its volume in terms of \(\pi\) and to the nearest hundredth.
Problem 12: Rotating a triangle about its altitude
An isosceles triangle has a base of \(8\text{ in}\) and a height of \(5\text{ in}\). It is rotated about its altitude. Name the solid and find its volume in terms of \(\pi\) and to the nearest hundredth.
Problem 13: Modeling a tree trunk
Priya models a tree trunk as a cylinder with a diameter of \(2\text{ ft}\) and a height of \(30\text{ ft}\). Estimate its volume to the nearest hundredth.
Problem 14: Population density
A city has 45,000 people living in \(30\text{ mi}^{2}\). Find the population density.
Problem 15: Density of a county
A rectangular county measures \(12\text{ mi}\) by \(9\text{ mi}\) and has 54,000 residents. Find its area and its population density.
Problem 16: Heating a room
A furnace must supply 6 BTUs of heat for every cubic foot of a room. Ethan's room is \(12\text{ ft}\) by \(10\text{ ft}\) with an \(8\text{ ft}\) ceiling. How many BTUs are needed?
Problem 17: Density of a metal block
A metal block measures \(5\text{ cm}\) by \(4\text{ cm}\) by \(3\text{ cm}\) and has a mass of \(162\text{ g}\). Find its density in grams per cubic centimeter.
Problem 18: Comparing towns
Town A has 9,600 residents on \(12\text{ mi}^{2}\). Town B has 7,350 residents on \(7\text{ mi}^{2}\). Which town is more densely populated, and by how many people per square mile?
Problem 19: Fencing a pen
Liam has \(48\text{ ft}\) of fence for a rectangular dog pen and wants the largest area. Using whole-number side lengths, which dimensions should he use? How much more area is that than an \(8\text{ ft}\) by \(16\text{ ft}\) pen?
Problem 20: Two soup cans
A company compares two closed cans that each hold \(72\pi\text{ cm}^{3}\). Can A has a radius of \(3\text{ cm}\) and a height of \(8\text{ cm}\). Can B has a radius of \(4\text{ cm}\) and a height of \(4.5\text{ cm}\). Verify the volumes, then find the total surface area of each. Which can needs less metal?
Problem 21: Sizing a water tank
A cylindrical tank has a radius of \(3\text{ ft}\) and must hold at least \(1{,}000\) gallons. Use \(1\text{ ft}^{3}\approx 7.5\) gallons. What is the least height, to the nearest tenth of a foot, that works?
Problem 22: Choosing a shipping box
Maya can ship an item in Box 1 (\(10\text{ in}\) by \(8\text{ in}\) by \(6\text{ in}\)) or Box 2 (\(12\text{ in}\) by \(8\text{ in}\) by \(5\text{ in}\)). Cardboard costs \(\$0.02\) per square inch. Show that the volumes are equal, then find which box is cheaper to make and by how much.
Problem 23: Banner grid
A banner is \(36\text{ in}\) wide. A designer leaves a \(3\text{ in}\) margin on each side and uses three equal columns separated by two gutters, each \(3\text{ in}\) wide. Find the width of one column.
Problem 24: Crowded field
A concert field measures \(120\text{ yd}\) by \(80\text{ yd}\) and holds 14,400 people. (a) Find the density in people per square yard. (b) Safety rules allow at most 2 people per square yard. How many more people may be admitted?
See every chapter on the Geometry math page.