Area & Volume – SAT Math

1. Area of Plane Figures

Rectangle

\(A = l \cdot w\)

Length × Width

Square

\(A = s^2\)

Side²

Triangle

\(A = \frac{1}{2} b h\)

½ × Base × Height

Parallelogram

\(A = b h\)

Base × Height

Trapezoid

\(A = \frac{1}{2} h (b_1 + b_2)\)

½ × Height × (Base₁ + Base₂)

Circle

\(A = \pi r^2\)

π × Radius²

📌 RULE: Area Formulas

Key Points:
• For a triangle, the height must be perpendicular to the base.
• For a circle, the radius is half the diameter: \(r = d/2\).
• For composite shapes, break them into simpler shapes and add areas.

📝 SOLVED EXAMPLE 1 — Area of a Triangle

Find the area of a triangle with base 10 cm and height 6 cm.

Step 1: \(A = \frac{1}{2} b h\)

Step 2: \(A = \frac{1}{2} \times 10 \times 6\)

Step 3: \(A = \frac{1}{2} \times 60 = \color{var(--math)}{30}\)

✅ Area = \(30\) cm²

💡 Tip: The height must be perpendicular to the base.

📝 SOLVED EXAMPLE 2 — Area of a Circle

Find the area of a circle with radius 7 cm. (Use \(\pi \approx 3.14\))

Step 1: \(A = \pi r^2\)

Step 2: \(A = 3.14 \times 7^2\)

Step 3: \(A = 3.14 \times 49 = \color{var(--math)}{153.86}\)

✅ Area = \(153.86\) cm²

💡 Tip: Remember \(r^2 = r \times r\).

2. Volume of 3D Solids

Rectangular Prism

\(V = l \cdot w \cdot h\)

Length × Width × Height

Cube

\(V = s^3\)

Side³

Cylinder

\(V = \pi r^2 h\)

π × Radius² × Height

Cone

\(V = \frac{1}{3} \pi r^2 h\)

⅓ × π × Radius² × Height

Sphere

\(V = \frac{4}{3} \pi r^3\)

⁴⁄₃ × π × Radius³

Pyramid

\(V = \frac{1}{3} \cdot A_{base} \cdot h\)

⅓ × Base Area × Height

📌 RULE: Volume Formulas

Key Points:
• A prism has two parallel, congruent bases.
• A cylinder is a circular prism.
• A cone is ⅓ of a cylinder with the same base and height.
• A pyramid is ⅓ of a prism with the same base and height.
• A sphere has radius \(r\).

📝 SOLVED EXAMPLE 3 — Volume of a Rectangular Prism

Find the volume of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

Step 1: \(V = l \cdot w \cdot h\)

Step 2: \(V = 8 \times 5 \times 3\)

Step 3: \(V = \color{var(--math)}{120}\)

✅ Volume = \(120\) cm³

💡 Tip: Volume is measured in cubic units (cm³, m³, etc.).

📝 SOLVED EXAMPLE 4 — Volume of a Cylinder

Find the volume of a cylinder with radius 4 cm and height 10 cm. (Use \(\pi \approx 3.14\))

Step 1: \(V = \pi r^2 h\)

Step 2: \(V = 3.14 \times 4^2 \times 10\)

Step 3: \(V = 3.14 \times 16 \times 10 = \color{var(--math)}{502.4}\)

✅ Volume = \(502.4\) cm³

💡 Tip: For a cylinder, the base is a circle, so use \(\pi r^2\) for the base area.

3. Surface Area

Rectangular Prism

\(SA = 2lw + 2lh + 2wh\)

Sum of all 6 faces

Cube

\(SA = 6s^2\)

6 × Side²

Cylinder

\(SA = 2\pi r^2 + 2\pi r h\)

2 circles + rectangle

Sphere

\(SA = 4\pi r^2\)

4 × π × Radius²

📝 SOLVED EXAMPLE 5 — Surface Area

Find the surface area of a cube with side length 5 cm.

Step 1: \(SA = 6s^2\)

Step 2: \(SA = 6 \times 5^2\)

Step 3: \(SA = 6 \times 25 = \color{var(--math)}{150}\)

✅ Surface Area = \(150\) cm²

💡 Tip: Surface area is measured in square units (cm², m², etc.).

🧪 Practice Questions

Solve each problem using the formulas above. Click "Show Answer" to see the full solution.

Question 1

Find the area of a rectangle with length 12 cm and width 5 cm.

A) 30 cm²

B) 40 cm²

C) 50 cm²

D) 60 cm²

✓ Answer: D

\(A = l \times w = 12 \times 5 = 60\) cm²

📝 Solution: \(12 \times 5 = 60\)

Question 2

Find the area of a triangle with base 8 cm and height 6 cm.

A) 20 cm²

B) 24 cm²

C) 28 cm²

D) 48 cm²

✓ Answer: B

\(A = \frac{1}{2} \times 8 \times 6 = 24\) cm²

📝 Solution: \(0.5 \times 8 \times 6 = 24\)

Question 3

Find the volume of a rectangular prism with length 10 cm, width 4 cm, and height 3 cm.

A) 100 cm³

B) 110 cm³

C) 120 cm³

D) 130 cm³

✓ Answer: C

\(V = 10 \times 4 \times 3 = 120\) cm³

📝 Solution: \(10 \times 4 \times 3 = 120\)

Question 4

Find the area of a circle with radius 5 cm. (Use \(\pi \approx 3.14\))

A) 31.4 cm²

B) 50 cm²

C) 65 cm²

D) 78.5 cm²

✓ Answer: D

\(A = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5\)

📝 Solution: \(3.14 \times 25 = 78.5\)

Question 5

Find the volume of a cylinder with radius 3 cm and height 8 cm. (Use \(\pi \approx 3.14\))

A) 150.72 cm³

B) 200.96 cm³

C) 226.08 cm³

D) 250.72 cm³

✓ Answer: C

\(V = \pi r^2 h = 3.14 \times 3^2 \times 8 = 3.14 \times 9 \times 8 = 226.08\)

📝 Solution: \(3.14 \times 9 \times 8 = 226.08\)

Question 6

Find the surface area of a cube with side length 4 cm.

A) 48 cm²

B) 64 cm²

C) 96 cm²

D) 128 cm²

✓ Answer: C

\(SA = 6s^2 = 6 \times 4^2 = 6 \times 16 = 96\) cm²

📝 Solution: \(6 \times 16 = 96\)

Question 7

Find the area of a trapezoid with bases 8 cm and 12 cm, and height 5 cm.

A) 40 cm²

B) 50 cm²

C) 60 cm²

D) 70 cm²

✓ Answer: B

\(A = \frac{1}{2}h(b_1 + b_2) = \frac{1}{2} \times 5 \times (8 + 12) = 2.5 \times 20 = 50\) cm²

📝 Solution: \(0.5 \times 5 \times 20 = 50\)

Question 8

Find the volume of a sphere with radius 6 cm. (Use \(\pi \approx 3.14\))

A) 678.24 cm³

B) 723.46 cm³

C) 800 cm³

D) 904.32 cm³

✓ Answer: D

\(V = \frac{4}{3}\pi r^3 = \frac{4}{3} \times 3.14 \times 6^3 = \frac{4}{3} \times 3.14 \times 216 = 904.32\)

📝 Solution: \(4/3 \times 3.14 \times 216 = 904.32\)

Question 9

Find the surface area of a cylinder with radius 4 cm and height 6 cm. (Use \(\pi \approx 3.14\))

A) 200.96 cm²

B) 251.2 cm²

C) 301.44 cm²

D) 351.68 cm²

✓ Answer: B

\(SA = 2\pi r^2 + 2\pi r h = 2\pi(4^2) + 2\pi(4)(6) = 2\pi(16) + 2\pi(24) = 32\pi + 48\pi = 80\pi \approx 251.2\)

📝 Solution: \(80 \times 3.14 = 251.2\)

Question 10

Find the volume of a cone with radius 4 cm and height 9 cm. (Use \(\pi \approx 3.14\))

A) 120.56 cm³

B) 140.67 cm³

C) 150.72 cm³

D) 160.84 cm³

✓ Answer: C

\(V = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times 3.14 \times 4^2 \times 9 = \frac{1}{3} \times 3.14 \times 16 \times 9 = 150.72\)

📝 Solution: \(1/3 \times 3.14 \times 16 \times 9 = 150.72\)

🎉 WELL DONE!

You've completed the Area & Volume lesson. You now know the formulas for area of plane figures, volume and surface area of 3D solids.

← Back to Topic List 📖 Previous: Evaluating Statistical Claims 📖 Next: Lines, Angles & Triangles →