Area and Volume — Geometry Unit 10 practice.
This unit covers area of polygons, surface area, volume of prisms and cylinders and volume of pyramids and cones — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers area of polygons, surface area, volume of prisms and cylinders and volume of pyramids and cones — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Area of Polygons
Students must know the area formulas for triangles, rectangles, parallelograms, trapezoids, and regular polygons. Exams frequently combine shapes into composite figures requiring students to add or subtract areas. Understanding how the apothem works for regular polygons is essential.
Key Points
- Triangle: A = ½bh; Parallelogram: A = bh; Trapezoid: A = ½(b₁ + b₂)h
- Regular polygon: A = ½ × apothem × perimeter
- For composite figures, split into known shapes, find each area, then add or subtract
- Height must always be perpendicular to the base — never a slant side
A trapezoid has bases of 8 cm and 14 cm, and a height of 5 cm. Find its area.
Use A = ½(b₁ + b₂)h = ½(8 + 14)(5). Add the bases first: 8 + 14 = 22, then multiply: ½ × 22 × 5 = 55. The area is 55 cm².
2 Surface Area
Surface area is the total area of all outer faces of a 3D figure. Students must distinguish between lateral surface area (sides only) and total surface area (sides plus bases). Formulas differ for prisms, cylinders, pyramids, and cones.
Key Points
- Prism: SA = 2B + Ph, where B = base area, P = base perimeter, h = height
- Cylinder: SA = 2πr² + 2πrh
- Pyramid: SA = B + ½Pℓ, where ℓ = slant height (not the vertical height)
- Cone: SA = πr² + πrℓ; slant height ℓ = √(r² + h²) if not given
Find the total surface area of a cylinder with radius 3 in and height 10 in. Use π ≈ 3.14.
The two circular bases contribute 2πr² = 2(3.14)(3²) = 2(3.14)(9) = 56.52 in². The lateral surface contributes 2πrh = 2(3.14)(3)(10) = 188.4 in². Total SA = 56.52 + 188.4 = 244.92 in².
3 Volume of Prisms and Cylinders
Volume of any prism or cylinder equals the area of the base times the height. Students must correctly identify and calculate the base shape's area before multiplying. Oblique prisms and cylinders use the same formula as long as height is measured perpendicularly.
Key Points
- Prism: V = Bh, where B is the area of the base polygon
- Cylinder: V = πr²h
- Height is always the perpendicular distance between the two bases
- Units for volume are always cubed (cm³, in³, etc.)
A triangular prism has a right triangle base with legs 6 m and 8 m, and the prism is 12 m long. Find its volume.
First find the base area: B = ½(6)(8) = 24 m². Then apply V = Bh = 24 × 12 = 288 m³. The key step is calculating the triangular base area correctly before multiplying by the length.
4 Volume of Pyramids and Cones
The volume of a pyramid or cone is exactly one-third the volume of the corresponding prism or cylinder with the same base and height. Students must use the vertical height, not the slant height, in these formulas. Exams often give slant height and require students to find vertical height using the Pythagorean theorem first.
Key Points
- Pyramid: V = ⅓Bh, where B = area of the base polygon
- Cone: V = ⅓πr²h
- Slant height ≠ vertical height; use a² + b² = c² to find the vertical height if needed
- Both formulas share the ⅓ factor — a common exam trap is forgetting it
A cone has a diameter of 8 cm and a slant height of 5 cm. Find its volume. Use π ≈ 3.14.
The radius is 4 cm. Find the vertical height using the Pythagorean theorem: h = √(5² − 4²) = √(25 − 16) = √9 = 3 cm. Now apply V = ⅓πr²h = ⅓(3.14)(16)(3) = ⅓(150.72) ≈ 50.24 cm³.
Questions, answered.
What is Area and Volume?
Area and Volume is Unit 10 of Geometry, covering area of polygons, surface area, volume of prisms and cylinders and volume of pyramids and cones.
How to study for Geometry Unit 10?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.