Geometry (HS pathway)
Area, Perimeter, Volume, and Cross-Sections
Geometry
- ✓By the end of this lesson students will be able to calculate the area and perimeter of two-dimensional shapes, including composite figures.
- ✓By the end of this lesson students will be able to calculate the volume of common three-dimensional solids.
- ✓By the end of this lesson students will be able to describe and sketch cross-sections of three-dimensional figures.
- ✓By the end of this lesson students will be able to apply appropriate units for area, perimeter, and volume measurements.
Key concepts
Area is the measure of the two-dimensional space a shape occupies. It is expressed in square units (e.g., square inches, square centimeters).
Perimeter is the total distance around the boundary of a two-dimensional shape. It is expressed in linear units (e.g., inches, centimeters). For a circle, the perimeter is called the circumference.
Volume is the measure of the three-dimensional space an object occupies. It is expressed in cubic units (e.g., cubic inches, cubic centimeters).
A cross-section is the two-dimensional shape formed when a three-dimensional object is sliced by a plane. The shape of the cross-section depends on the orientation of the plane relative to the solid.
Key facts to remember
- 1Area is measured in square units (e.g., cm²).
- 2Perimeter is measured in linear units (e.g., cm).
- 3Volume is measured in cubic units (e.g., cm³).
- 4The perimeter of a circle is called its circumference.
- 5A cross-section is a 2D shape resulting from slicing a 3D object.
- 6The formula for the volume of any prism or cylinder is V = Bh, where B is the area of the base.
- 7The formula for the volume of any pyramid or cone is V = (1/3)Bh, where B is the area of the base.
Worked examples
Example 1
A composite shape consists of a rectangle with length 12 cm and width 8 cm, topped by a semicircle whose diameter is the width of the rectangle. Find the total area and perimeter of the shape. Use π ≈ 3.14 and round to two decimal places.
Answer
Total Area = 121.12 cm², Total Perimeter = 44.56 cm
Remember to exclude any internal lines when calculating the perimeter of a composite shape.
Example 2
A cylindrical water tank has a radius of 4 meters and a height of 10 meters. Calculate the volume of the tank. Use π ≈ 3.14 and round to two decimal places.
Answer
The volume of the cylindrical tank is 502.40 m³.
Volume is always measured in cubic units.
Example 3
Describe the shape of the cross-section formed when a right circular cone is cut by a plane in the following ways:
Answer
c) When a right circular cone is cut by a plane perpendicular to its base but not passing through its apex (and not touching the base), the cross-section is a parabola.
b) When a right circular cone is cut by a plane perpendicular to its base and passing through its apex, the cross-section is an isosceles triangle.
Common mistakes
- ✗Confusing area and perimeter, or using the wrong units for each.
- ✗Forgetting to exclude internal lines when calculating the perimeter of composite shapes.
- ✗Using the radius as the diameter (or vice versa) in formulas for circles, cylinders, cones, or spheres.
- ✗Incorrectly identifying the base (B) for volume calculations of prisms, pyramids, cones, or cylinders, especially when the object is oriented differently.
- ✗Misidentifying the shape of a cross-section, particularly for non-standard cuts.
Exam tips
- ★Always draw a clear diagram for the problem, labeling all given dimensions.
- ★Write down the correct formula before substituting values to avoid errors and earn partial credit.
- ★Pay close attention to the units given in the problem and ensure your final answer uses the correct units (linear, square, or cubic).
- ★Show all steps of your work clearly. This helps you catch mistakes and allows the grader to follow your reasoning.
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