Surface Area and Volume

Grade 10 · Math · Study Guide

Calculate and compare surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres.

Surface area is the total area of all faces or curved surfaces of a solid, measured in square units. Volume is the amount of space enclosed by a solid, measured in cubic units.

For prisms and cylinders, volume = base area × height. The lateral surface area of a cylinder is 2πrh, and the total surface area adds two circular bases: 2πrh + 2πr². For a rectangular prism, SA = 2(lw + lh + wh).

Cones and pyramids have volume = ⅓ × base area × height. The lateral surface area of a cone is πrl, where l is the slant height (l = √(r² + h²)). The slant height of a pyramid face must be calculated using the Pythagorean theorem applied to the apothem.

A sphere has volume = (4/3)πr³ and surface area = 4πr². These formulas arise from calculus but can be confirmed by Archimedes' argument that a sphere inscribed in a cylinder has 2/3 the volume and surface area of the cylinder.

Composite solids combine two or more basic shapes. Find the total volume by addition, and for surface area, subtract any faces that are hidden (internal) where shapes join.

Key terms

Slant Height
The distance from the apex of a cone or pyramid to the midpoint of a base edge, along the lateral face.
Lateral Surface Area
The area of all faces of a solid except the base(s).
Apex
The single vertex at the top of a cone or pyramid.
Apothem
The perpendicular distance from the centre of a regular polygon to the midpoint of one of its sides.
Composite Solid
A three-dimensional figure made by combining two or more basic geometric solids.
Cross-Section
The shape formed when a plane cuts through a solid, used to identify the base for volume calculations.

Flashcards

Volume of Prism/Cylinder
V = base area × height.
Volume of Pyramid/Cone
V = ⅓ × base area × height.
Volume of Sphere
V = (4/3)πr³.
SA of Cylinder
SA = 2πr² + 2πrh.
SA of Cone
SA = πr² + πrl, where l = √(r²+h²) is the slant height.
SA of Sphere
SA = 4πr².
SA of Rectangular Prism
SA = 2(lw + lh + wh).
Slant Height of Cone
l = √(r²+h²) by the Pythagorean theorem.

Questions

  1. What is the volume of a cone with radius 3 cm and height 4 cm? (Use π ≈ 3.14)

    1. 37.68 cm³
    2. 113.04 cm³
    3. 12.56 cm³
    4. 50.24 cm³
    Show answer

    Answer: 37.68 cm³

    V = ⅓πr²h = ⅓ × 3.14 × 9 × 4 = ⅓ × 113.04 = 37.68 cm³.

  2. The surface area of a sphere with radius r is ______πr².

    Show answer

    Answer: 4

    The surface area formula for a sphere is SA = 4πr².

  3. A cylinder has radius 5 cm and height 10 cm. What is the total surface area? (Leave answer in terms of π)

    1. 150π cm²
    2. 100π cm²
    3. 50π cm²
    4. 200π cm²
    Show answer

    Answer: 150π cm²

    SA = 2πr² + 2πrh = 2π(25) + 2π(5)(10) = 50π + 100π = 150π cm².

  4. Doubling the radius of a sphere doubles its volume.

    Show answer

    Answer: False

    V = (4/3)πr³, so doubling r multiplies volume by 2³ = 8, not 2.

  5. A square pyramid has a base side of 6 cm and a slant height of 5 cm. What is the total surface area?

    1. 96 cm²
    2. 60 cm²
    3. 36 cm²
    4. 120 cm²
    Show answer

    Answer: 96 cm²

    Base area = 6² = 36 cm². Lateral area = ½ × perimeter × slant height = ½ × 24 × 5 = 60 cm². Total = 36 + 60 = 96 cm².

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