Session 9: Geometry — Area, Perimeter & Volume

Explanatory SVG

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2. Perimeter

The perimeter of a shape is the total distance around the outside. Think of it as the length of a fence you would need to enclose a yard — you are measuring the boundary.

Square: All 4 sides equal; side length = \(s\)
\[ P = 4s \]
Example: A square has a side of 7 cm. \( P = 4 \times 7 = 28 \text{ cm} \)

Rectangle: Opposite sides are equal; length = \(l\), width = \(w\)
\[ P = 2(l + w) \]
Example: A rectangle is 12 m long and 5 m wide. \( P = 2(12 + 5) = 2 \times 17 = 34 \text{ m} \)

Triangle: Three sides \(a\), \(b\), \(c\)
\[ P = a + b + c \]
Example: A triangle has sides 6, 8, and 10. \( P = 6 + 8 + 10 = 24 \)

Real-life connection: A homeowner wants to put a fence around a rectangular backyard that is 40 feet long and 25 feet wide. How many feet of fencing is needed?
\[ P = 2(40 + 25) = 2 \times 65 = 130 \text{ feet} \]

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3. Area

The area of a shape is the amount of flat surface it covers. Area is measured in square units (square feet, square meters, etc.). Think of it as how many 1×1 tiles you would need to cover the shape.

Rectangle: \(A = l \times w\)
Example: A room is 15 ft long and 10 ft wide. \( A = 15 \times 10 = 150 \text{ ft}^2 \)
Real-life: Buying carpet — you need to cover 150 square feet.

Square: \(A = s^2\)
Example: A square tile has a side of 4 in. \( A = 4^2 = 16 \text{ in}^2 \)

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

The base (\(b\)) is any side of the triangle, and the height (\(h\)) is the perpendicular distance from that base to the opposite vertex (not necessarily a side).

Example 1: Base = 10, height = 6.
\[ A = \frac{1}{2} \times 10 \times 6 = \frac{60}{2} = 30 \]

Example 2: A triangular garden has base 14 m and height 5 m.
\[ A = \frac{1}{2} \times 14 \times 5 = 35 \text{ m}^2 \]

Parallelogram: \(A = b \times h\)

A parallelogram looks like a slanted rectangle. The height is the perpendicular distance between the two parallel bases — not the slant side.

Example: Base = 9, height = 4. \( A = 9 \times 4 = 36 \)

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

A trapezoid has two parallel sides (called bases \(b_1\) and \(b_2\)) and a height.

Example: \(b_1 = 6\), \(b_2 = 10\), \(h = 4\).
\[ A = \frac{1}{2}(6 + 10) \times 4 = \frac{1}{2} \times 16 \times 4 = 32 \]

Real-life: A wall that is trapezoidal in shape — you need the area to know how much paint to buy.

Watch Out! For the triangle and trapezoid, students often forget the \(\frac{1}{2}\). Associate "triangle" and "trapezoid" both starting with a shape that is "half" of a rectangle or parallelogram — that is your reminder to halve!

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4. Circles

Circles require their own section because they involve the special number π (pi) ≈ 3.14 (or the fraction \(\frac{22}{7}\) when working with fractions).

Key Parts of a Circle:

• Radius (\(r\)): distance from the center to the edge (any point on the circle).
• Diameter (\(d\)): distance all the way across the circle through the center. \(d = 2r\).
• Circumference (\(C\)): the distance around the circle (like its perimeter).
• Area (\(A\)): the flat space inside the circle.

Circumference Formulas:
\[ C = 2\pi r \quad \text{or equivalently} \quad C = \pi d \]

Example 1: A circle has radius 5 cm.
\[ C = 2 \times 3.14 \times 5 = 31.4 \text{ cm} \]

Example 2: A circle has diameter 14 in.
\[ C = \pi \times 14 = 3.14 \times 14 = 43.96 \text{ in} \]

Area Formula:
\[ A = \pi r^2 \]

Example 1: Radius = 5 cm.
\[ A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ cm}^2 \]

Example 2: Diameter = 10 ft. First, find the radius: \(r = 10 \div 2 = 5\) ft.
\[ A = 3.14 \times 5^2 = 78.5 \text{ ft}^2 \]

Watch Out! Always use the radius in both formulas, not the diameter. If a problem gives you the diameter, divide by 2 first. This is the most common circle mistake.

GED® Tip: The formula sheet provided on the GED® includes \(C = 2\pi r\) and \(A = \pi r^2\). You will not need to memorize them, but you must recognize which formula to use and correctly substitute the given values.

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5. Composite Shapes

A composite shape is a figure made up of two or more simpler shapes joined together. The strategy is always the same:

1. Identify the simple shapes that make up the figure.
2. Calculate the area (or perimeter) of each piece.
3. Add (or subtract) as needed.

Example 1 — L-Shaped Room:

Imagine an L-shaped room. You can split it into two rectangles.

L-Shaped Room (split into two rectangles):

[ Rectangle R1: 8 units wide × 4 units tall ]
[ Rectangle R2: 4 units wide × 6 units tall ]

Area of R1 = 8 × 4 = 32 square units
Area of R2 = 4 × 6 = 24 square units
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Total Area = 32 + 24 = 56 square units ✓

The key: always break the L-shape into rectangles, calculate each separately, then add.

Example 2 — Semicircle on top of a Rectangle:

A shape consists of a rectangle 10 units wide and 6 units tall, with a semicircle on top whose diameter equals the rectangle's width (10 units, so \(r = 5\)).

• Rectangle area: \(10 \times 6 = 60\)
• Semicircle area: \(\frac{1}{2} \times \pi r^2 = \frac{1}{2} \times 3.14 \times 25 = 39.25\)
• Total area: \(60 + 39.25 = 99.25\) square units ✓

GED® Tip: Composite shape problems are common on the GED®. Draw the figure, label the parts, and work step by step. Never try to do it all in your head at once.

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What You Learned in Session 3

• Angles: acute (<90°), right (=90°), obtuse (>90°), straight (=180°); complementary pair = 90°; supplementary pair = 180°.
• Triangle angles always sum to 180°; types: equilateral, isosceles, scalene, right.
• Perimeter = total distance around a shape; formulas for square, rectangle, triangle.
• Area formulas: rectangle (\(lw\)), square (\(s^2\)), triangle (\(\frac{1}{2}bh\)), parallelogram (\(bh\)), trapezoid (\(\frac{1}{2}(b_1+b_2)h\)).
• Circles: radius vs. diameter; circumference = \(2\pi r\); area = \(\pi r^2\).
• Composite shapes: break into simple parts and add areas.

Excellent work making it through Session 3! Geometry can feel abstract at first, but notice how every formula we used connects to something you can picture. Practice sketching the shapes and labeling the parts as you work through the questions below. That habit will serve you extremely well on test day!

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1. Volume

Volume is the measure of how much three-dimensional space a solid shape occupies. Think of it as how much water you could pour into a container, or how much air fills a room. Volume is measured in cubic units — cm³, ft³, m³, etc.

The GED® formula sheet includes volume formulas, so focus on knowing when and how to apply each one.

Rectangular Prism (Box): \(V = l \times w \times h\)

A rectangular prism is the classic box shape — like a shoebox or a fish tank.

Example 1 (Fish tank): A fish tank is 20 in long, 10 in wide, and 12 in tall. How much water can it hold?
\[ V = 20 \times 10 \times 12 = 2{,}400 \text{ in}^3 \]

Example 2 (Shipping box): A box is 4 ft × 3 ft × 5 ft.
\[ V = 4 \times 3 \times 5 = 60 \text{ ft}^3 \]

Cube: A cube is a special rectangular prism where all three dimensions are equal.
\[ V = s^3 \]
Example: A cube has a side length of 6 cm. \( V = 6^3 = 6 \times 6 \times 6 = 216 \text{ cm}^3 \)

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

A cylinder is like a can — it has two circular bases and a curved side. The formula multiplies the area of the circular base (\(\pi r^2\)) by the height (\(h\)).

Example 1: A can has radius 4 cm and height 10 cm.
\[ V = 3.14 \times 4^2 \times 10 = 3.14 \times 16 \times 10 = 502.4 \text{ cm}^3 \]

Example 2: A cylinder has radius 3 and height 10.
\[ V = 3.14 \times 9 \times 10 = 3.14 \times 90 = 282.6 \]

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

A cone is like a cylinder, but it tapers to a point at the top. Its volume is exactly one-third of the cylinder with the same base and height.

Example: A cone has radius 3 and height 4.
\[ V = \frac{1}{3} \times 3.14 \times 3^2 \times 4 = \frac{1}{3} \times 3.14 \times 9 \times 4 = \frac{1}{3} \times 113.04 = 37.68 \]

Watch Out! Students often forget the \(\frac{1}{3}\) in the cone formula. Remember: a cone is like a "pointy cylinder" — and that point removes two-thirds of the volume.

GED® Tip: On the GED®, the formula sheet is your friend. But you must correctly identify the shape and plug in the right values. Always re-read the problem to confirm which dimension is the radius and which is the height.

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2. Surface Area

Surface area (SA) is the total area of all the faces of a 3D solid. Imagine you are wrapping a gift — the total amount of wrapping paper needed equals the surface area of the box.

Rectangular Prism: \(SA = 2(lw + lh + wh)\)

A rectangular box has 6 faces in 3 matched pairs:

• Two faces of \(l \times w\) (top and bottom)
• Two faces of \(l \times h\) (front and back)
• Two faces of \(w \times h\) (left and right side)

Worked Example: A box is 5 cm long, 4 cm wide, and 3 cm tall.

• \(lw = 5 \times 4 = 20\)
• \(lh = 5 \times 3 = 15\)
• \(wh = 4 \times 3 = 12\)
• \(SA = 2(20 + 15 + 12) = 2 \times 47 = 94 \text{ cm}^2\) ✓

Cube: \(SA = 6s^2\)

A cube has 6 identical square faces.

Example: A cube has side length 3 in.
\[ SA = 6 \times 3^2 = 6 \times 9 = 54 \text{ in}^2 \]

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

Think of a cylinder as two circular lids (top and bottom) plus a rectangular label wrapped around the side.

• Two circles: \(2\pi r^2\)
• Curved side (when unrolled, it's a rectangle with width = circumference = \(2\pi r\), height = \(h\)): \(2\pi r h\)

Example: Radius = 4, height = 10.
\[ SA = 2 \times 3.14 \times 4^2 + 2 \times 3.14 \times 4 \times 10 \]
\[ = 2 \times 3.14 \times 16 + 2 \times 3.14 \times 40 \]
\[ = 100.48 + 251.2 = 351.68 \]

Real-life: You want to paint all surfaces of a box. You need to know the surface area to buy the right amount of paint.

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