Surface Area and Volume of 3D Solids

Master the formulas for cubes, prisms, pyramids, cylinders, cones, and spheres

CAPS Grade 10 Mathematics

This topic focuses on calculating surface area and volume for various three-dimensional shapes. Each section includes interactive games and quizzes to test your understanding.

3D Solids Gallery

a
Cube
V = a³ | SA = 6a²
l,w,h
Rectangular Prism
V = lwh | SA = 2(lw+lh+wh)
r h
Cylinder
V = πr²h | SA = 2πr² + 2πrh
r
Sphere
V = 4/3πr³ | SA = 4πr²
r h
Cone
V = 1/3πr²h | SA = πr² + πrl
b h
Square Pyramid
V = 1/3b²h | SA = b² + 2bl

Learning Outcomes

  • Identify and classify different 3D solids
  • Understand and calculate surface area of various solids
  • Calculate volume of prisms, pyramids, cylinders, cones, and spheres
  • Apply formulas to solve real-world problems
  • Distinguish between slant height and perpendicular height

1. Prisms

C

Cube

SA = 6a²
V = a³

Side = 4 cm

SA = 6 × 4² = 96 cm²
V = 4³ = 64 cm³
R

Rectangular Prism

SA = 2(lw + lh + wh)
V = lwh

l=10 cm, w=5 cm, h=3 cm

SA = 2(10×5 + 10×3 + 5×3) = 190 cm²
V = 10×5×3 = 150 cm³
T

Triangular Prism

SA = 2A + ph
V = A × h

A = area of triangle base
p = perimeter of triangle

Quiz 1 - Cube

Cube side = 5 cm. What is its volume

A) 25 cm³
B) 125 cm³
C) 150 cm³
D) 100 cm³

2. Pyramids

S

Square-Based Pyramid

SA = b² + 2bl
V = (1/3)b²h

b = base side, l = slant height, h = perpendicular height

T

Triangular Pyramid

SA = sum of 4 triangles
V = (1/3)Ah
H

Height vs Slant Height

  • Perpendicular height: Inside pyramid
  • Slant height: Along face
  • Slant height for SA, perpendicular height for volume

3. Cylinders, Cones & Spheres

C

Cylinder

SA = 2πr² + 2πrh
V = πr²h

r=4 cm, h=12 cm

SA = 2π(16) + 2π(4)(12) = 32π + 96π = 128π cm²
V = π(16)(12) = 192π cm³
C

Cone

SA = πr² + πrl
V = (1/3)πr²h

l = slant height

S

Sphere

SA = 4πr²
V = (4/3)πr³

Radius = 7 cm

SA = 4π(49) = 196π cm²
V = (4/3)π(343) = 1372π/3 cm³

Quiz 2 - Cylinder

Cylinder radius = 3 cm, height = 10 cm. What is volume (π ≈ 3.14)

A) 282.6 cm³
B) 94.2 cm³
C) 188.4 cm³
D) 376.8 cm³

Example Problems

Example 1: Rectangular Prism

Problem

Rectangular box: length 10 cm, width 5 cm, height 3 cm. Find SA and V.

Solution
SA = 2(10×5 + 10×3 + 5×3) = 190 cm²
V = 10×5×3 = 150 cm³

Example 2: Real-World Application

Problem

Cylindrical silo holds 500 m³ grain, height 10 m. Find radius.

Solution
500 = πr²(10)
r² = 500/(10π) = 50/π
r = √(50/π) ≈ 3.99 m

Formulas Summary

Surface Area

  • Cube: 6a²
  • Rectangular prism: 2(lw + lh + wh)
  • Cylinder: 2πr² + 2πrh
  • Sphere: 4πr²
  • Cone: πr² + πrl
  • Square pyramid: b² + 2bl

Volume

  • Cube: a³
  • Rectangular prism: lwh
  • Triangular prism: Ah
  • Cylinder: πr²h
  • Sphere: (4/3)πr³
  • Cone: (1/3)πr²h
  • Pyramid: (1/3)Ah

Key Variables

  • a = side length
  • l = length, w = width
  • h = height
  • r = radius
  • A = area of base
  • l (slant) = slant height

Practice & Assess

Test your knowledge with these interactive games.

Match - Shape to Formula

Cube
V = a³
Cylinder
V = πr²h
Sphere
V = 4/3πr³
Cone
V = 1/3πr²h

Fill - Sphere Surface Area

SA of sphere = 4 × ___ × r²

Practice Questions

Q1

Cube side 6 cm. Find SA and V.

Q2

Cylinder r=3 cm, h=10 cm. Find volume.

Q3

Sphere radius 5 cm. Calculate SA.

Common Mistakes to Avoid

Mistake 1

Confusing SA and volume

Mistake 2

Using wrong height (slant vs perpendicular)

Mistake 3

Forgetting π in formulas

Summary of Key Concepts

Prisms: V = Area of base × height
Pyramids & Cones: V = 1/3 × base area × height
Sphere: V = 4/3πr³, SA = 4πr²
Surface Area: Sum of areas of all faces

Key Terms

Cube Prism Pyramid Cylinder Cone Sphere Surface Area Volume Slant Height Perpendicular Net Apex
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