GED Mathematics · Chapter 7 · pp. 412–477

Geometry

Geometry is the maths of shapes and space: how far it is around a garden, how much carpet a room needs, how much water a pool holds. On the GED® Mathematical Reasoning test you get a formula sheet, so the skill is knowing which formula fits the shape, putting the right numbers into it, and giving the answer in the right units.

Learning objectives

By the end of this chapter you can:

Name four-sided figures and triangles by their sides and angles, and find a missing angle.
Use the Pythagorean theorem to find a missing side of a right triangle.
Find the perimeter and area of squares, rectangles, parallelograms, triangles and trapezoids.
Find the circumference and area of a circle.
Find the volume and surface area of prisms, cubes, cylinders, pyramids, cones and spheres.
Break a combined figure into simple parts, and do geometry calculations on the TI-30XS MultiView™.

In this chapter: Lesson 1 Plane figures · Lesson 2 Triangles · Lesson 3 Pythagorean relationship · Lesson 4 Perimeter and area · Lesson 5 Circles · Lesson 6 Volume · Lesson 7 Surface area · Lesson 8 Combined figures · Lesson 9 Geometry calculator skills

🎯 About the test

The test gives you the formula sheet below, and it says to use π ≈ 3.14. You still need to know which shape each formula belongs to and what every letter means, so learn the table before test day. Knowing the common ones by heart (rectangle, triangle, circle) also saves time.

The geometry formulas on the GED® formula sheet

Shape	Area or perimeter	Letters
Square	A = s² · P = 4s	s = side
Rectangle	A = lw · P = 2l + 2w	l = length, w = width
Parallelogram	A = bh	b = base, h = height
Triangle	A = bh · P = s₁ + s₂ + s₃	s₁, s₂, s₃ = the three sides
Trapezoid	A = h(b₁ + b₂)	b₁, b₂ = the two parallel sides
Circle	A = πr² · C = 2πr or C = πd	r = radius, d = diameter, π ≈ 3.14

Solid	Surface area	Volume
Rectangular / right prism	SA = ph + 2B	V = Bh
Cylinder	SA = 2πrh + 2πr²	V = πr²h
Pyramid	SA = ps + B	V = Bh
Cone	SA = πrs + πr²	V = πr²h
Sphere	SA = 4πr²	V = πr³

p = perimeter of the base, B = area of the base, h = height, s = slant height, π ≈ 3.14. The sheet also has the Pythagorean theorem, a² + b² = c².

✏️ How to use this page

Work through the lessons in order. Each one has worked examples and a short Try it. Lesson 3 has a Pythagorean solver and Lesson 9 has a shape calculator that shows its working, so you can check your own. The full practice sets are in this chapter's printable assignment, and there is a quiz at the end.

Next: Lessons 1 and 2, plane figures and triangles →

Lesson 1 · Plane figures

🔑 Key ideas

Four-sided figures are named by the properties of their sides and angles.
The four inside angles of any four-sided figure add up to 360°.
Use those facts and a little algebra to find a missing angle.

Angles

An angle is measured in degrees (°). A small square in the corner of a drawing means a right angle, exactly 90°. The symbol ∠ means “angle”, so ∠A is the angle at point A, and m∠A is its measure.

Four-sided figures

Small squares mark right angles, ticks mark equal sides, and arrowheads mark sides that are parallel.

Figure	Sides	Angles
Rectangle	Opposite sides are parallel and the same length.	Four right angles.
Square	All four sides are the same length; opposite sides are parallel.	Four right angles. (A square is a special rectangle.)
Parallelogram	Opposite sides are parallel and the same length.	Opposite angles are equal.
Rhombus	A parallelogram with all four sides the same length.	Opposite angles are equal.
Trapezoid	Exactly one pair of parallel sides.	No rule about the angles.

💡 Don't trust your eyes. Read which properties the question gives you (parallel sides, equal sides, right angles), and name the figure from those.

📘 Example 1 · A missing angle in a parallelogram

In parallelogram ABCD, two opposite angles each measure 110°. What do the other two measure?

Opposite angles of a parallelogram are equal, so the other two angles are equal too. Call each one x.
All four angles add up to 360°: 2x + 110° + 110° = 360°.
2x + 220° = 360°, so 2x = 140° and x = 70°.

📘 Example 2 · A trapezoid with two right angles

A trapezoid has two right angles. The other two angles are x and 2x. Find them.

90° + 90° + x + 2x = 360°, so 3x + 180° = 360°.
3x = 180°, so x = 60° and 2x = 120°. Check: 90 + 90 + 60 + 120 = 360. ✔

✏️ Try it

Which four-sided figures always have four right angles?
Three angles of a four-sided figure measure 85°, 95° and 100°. What is the fourth?
A four-sided figure has sides of 10, 15, 10 and 15. Its opposite angles are equal, but none is a right angle. What is it?

Answers

1. The rectangle and the square · 2. 360 − 85 − 95 − 100 = 80° · 3. A parallelogram

Lesson 2 · Triangles

🔑 Key ideas

A triangle is named by its three corners (vertices): ∆DEF has sides DE, EF and DF.
Triangles are classified two ways: by their sides and by their angles.
The three angles of any triangle add up to 180°.

Top row: by sides. Bottom row: by angles. Every triangle gets one name from each row, such as “right isosceles”.

In an equilateral triangle the angles are equal too (60° each).
In an isosceles triangle the two angles opposite the equal sides are equal.
In a scalene triangle no sides and no angles are equal.

📘 Example 1 · Classify ∆PQR

In ∆PQR, sides PQ and PR are the same length, and ∠P is a right angle.

By sides: two sides are equal, so it is isosceles.
By angles: one angle is 90°, so it is a right triangle.
∆PQR is a right isosceles triangle. Its other two angles are equal: (180 − 90) ÷ 2 = 45° each.

📘 Example 2 · In ∆ABC, ∠A = 55° and ∠B = 100°. Find ∠C.

55° + 100° + x = 180°, so 155° + x = 180° and x = 25°.

✏️ Try it

Two angles of a scalene triangle measure 38° and 56°. What is the third angle?
A triangle has two angles of 30° and 60°. Classify it by its angles.
Every side of a triangle is 7 cm. Name it both ways.

Answers

1. 180 − 38 − 56 = 86° · 2. The third angle is 90°, so it is a right triangle (and scalene) · 3. Equilateral and acute (each angle is 60°)

Next: Lesson 3, the Pythagorean relationship →

Lesson 3 · Pythagorean relationship

🔑 Key ideas

The hypotenuse is the longest side of a right triangle. It is opposite the right angle.
The other two sides, which make the right angle, are the legs.
The squares of the legs add up to the square of the hypotenuse: a² + b² = c².

a² + b² = c²

a and b are the legs, c is the hypotenuse. It only works for right triangles, and c must always be the side across from the right angle.

Finding the hypotenuse: c = √(a² + b²), add the squares.
Finding a leg: a = √(c² − b²), subtract the squares.

📘 Example 1 · The legs are 3 ft and 4 ft. Find the hypotenuse.

c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5 ft

📘 Example 2 · A 13-foot ladder, 3 feet from a wall

How far up the wall does it reach, to the nearest tenth of a foot? The wall, the ground and the ladder make a right triangle. The ladder is the hypotenuse.

3² + b² = 13², so 9 + b² = 169.
b² = 169 − 9 = 160
b = √160 ≈ 12.649…, which is 12.6 feet.

For the last step use the calculator: press 2nd x² for √, type 160 and press enter. If the display shows a square-root answer instead of a decimal, press the toggle key ◀▶ to change it.

💡 Look out for the whole-number triangles 3-4-5 and 5-12-13, and their multiples such as 6-8-10, 9-12-15 and 30-40-50. They come up often.

Two special right triangles

Angles	leg : leg : hypotenuse	Example
45°, 45°, 90°	x : x : x√2	legs 8 and 8 → hypotenuse 8√2 ≈ 11.3
30°, 60°, 90°	x : x√3 : 2x	short leg 5 → long leg 5√3 ≈ 8.7, hypotenuse 10

🧪 Try it: Pythagorean solver

Choose the side you want to find, then type the other two.

Find

leg a leg b

✏️ Try it

The legs of a right triangle are 9 yd and 12 yd. Find the hypotenuse.
The hypotenuse is 10 mm and one leg is 6 mm. Find the other leg.
Max hikes 300 yards due north, then 400 yards due east. How far is he from where he started, in a straight line?

Answers

1. √(81 + 144) = √225 = 15 yd · 2. √(100 − 36) = √64 = 8 mm · 3. √(90,000 + 160,000) = √250,000 = 500 yards (a 3-4-5 triangle, times 100)

Next: Lesson 4, perimeter and area →

Lesson 4 · Perimeter and area

🔑 Key ideas

Perimeter is the distance around a figure. Add the lengths of the sides.
Area is the space inside a flat figure, measured in square units (in², ft², cm², …).
The height is always measured at a right angle to the base, not along a slanted side.

Perimeter

Figure	Perimeter
Square	P = 4s (4 × side)
Rectangle	P = 2l + 2w (2 × length + 2 × width)
Triangle	P = a + b + c (add the three sides)
Any other figure	add the lengths of all its sides

📘 Example 1 · A rectangle 16 inches long and 9 inches wide

P = 2l + 2w = 2(16) + 2(9) = 32 + 18 = 50 inches

Area

The dashed lines are heights. Each one meets its base at a right angle.

Figure	Area	In words
Square	A = s²	side × side
Rectangle	A = lw	length × width
Parallelogram	A = bh	base × height
Triangle	A = bh	half of base × height
Trapezoid	A = h(b₁ + b₂)	add the parallel sides, multiply by the height, then halve

A triangle is half of a parallelogram with the same base and height, which is where the comes from. A trapezoid uses the average of its two parallel sides as its base.

📘 Example 2 · A parallelogram: base 12 cm, height 5 cm, slanted side 6 cm

For area you need the base and the height. Ignore the slanted side.
A = bh = 12 × 5 = 60 cm²
For perimeter you do need the slanted side: 12 + 6 + 12 + 6 = 36 cm.

📘 Example 3 · A trapezoid with parallel sides 6 m and 10 m, 4 m apart

A = × 4 × (6 + 10) = × 4 × 16 = 32 m²

💡 Perimeter is a length (ft). Area is square units (ft²). If the answer choices mix them up, the units tell you which is which.

✏️ Try it

A square measures 6 cm on a side. Find its perimeter and its area.
Find the area of a triangle with a base of 26 cm and a height of 15 cm.
A rectangle's length is three times its width, and its perimeter is 96 inches. How long is it?

Answers

1. P = 4 × 6 = 24 cm; A = 6² = 36 cm² · 2. ½ × 26 × 15 = 195 cm² · 3. Width w and length 3w: 2(3w) + 2w = 8w = 96, so w = 12 and the length is 36 inches

Next: Lesson 5, circles →

Lesson 5 · Circles

🔑 Key ideas

Pi (π) is the circumference of any circle divided by its diameter. On the test, use π ≈ 3.14.
Circumference (the distance around) = π × diameter.
Area = π × radius squared.

A circle is all the points that are the same distance from its centre. The radius goes from the centre to the edge. The diameter goes right across through the centre, so it is two radii long.

C = πd = 2πr

A = πr²

The digits of π never end, so every answer with π is an approximation. Using 3.14, as the test tells you to, keeps your answers in line with the answer choices.

📘 Example 1 · A plate 10.5 inches across has a gold rim

How long is the rim, to the nearest tenth? C = πd = 3.14 × 10.5 = 32.97 ≈ 33.0 inches

📘 Example 2 · A round panel with a radius of 4 meters

What area must be tiled? A = πr² = 3.14 × 4² = 3.14 × 16 = 50.24 m²

📘 Example 3 · Circumference when you know the radius

A circle has a radius of 5 cm. The diameter is 2 × 5 = 10 cm, so C = 3.14 × 10 = 31.4 cm. (Or 2 × 3.14 × 5, the same thing.)

💡 In πr², square the radius first, then multiply by 3.14. And check whether the question gives the radius or the diameter: if it gives the diameter, halve it before you find the area.

✏️ Try it

A round fountain is 12 feet across. How long is a tile border around its edge, to the nearest foot?
The radius of a circle is 6.5 cm. What is its diameter?
Find the area of a circle with a diameter of 10 inches.

Answers

1. 3.14 × 12 = 37.68 ≈ 38 feet · 2. 2 × 6.5 = 13 cm · 3. r = 5, so A = 3.14 × 25 = 78.5 in²

Next: Lessons 6 and 7, volume and surface area →

Lesson 6 · Volume

🔑 Key ideas

Volume is the space inside a solid. It is measured in cubic units: 6 cubic inches is written 6 in³.
For a solid with two matching ends (a box, a can), volume = area of the base × height.
A pyramid or cone holds one-third of the prism or cylinder with the same base and height.

Dashed lines are hidden edges and heights. The slant height s runs along the outside surface; the height h goes straight up through the middle.

Solid	Volume	In words
Rectangular prism (box)	V = lwh = Bh	length × width × height
Cube	V = s³	edge × edge × edge
Cylinder	V = πr²h	area of the circle × height
Square pyramid	V = Bh	a third of base area × height
Cone	V = πr²h	a third of the matching cylinder
Sphere	V = πr³	on the formula sheet

💡 Many volume formulas are an area formula with h on the end. Circle: πr². Cylinder: πr²h.

📘 Example 1 · Boxes and cubes

A box 6 ft long, 2 ft wide and 3 ft tall: V = 6 × 2 × 3 = 36 ft³.
A wooden block 2 inches on every edge: V = 2³ = 2 × 2 × 2 = 8 in³.

📘 Example 2 · A tank with radius 1.5 m and height 3 m

V = πr²h = 3.14 × 1.5² × 3 = 3.14 × 2.25 × 3 = 21.195 ≈ 21 m³

📘 Example 3 · Pyramids and cones

A square pyramid with base edges of 5 cm and a height of 6 cm. B = 5 × 5 = 25, so V = × 25 × 6 = 150 ÷ 3 = 50 cm³.
A cone with radius 2 in and height 9 in. V = × 3.14 × 2² × 9 = 113.04 ÷ 3 = 37.68 in³.

Multiply the other numbers first, then divide by 3, unless one of them divides by 3 easily (here 9 ÷ 3 = 3 would also work).

✏️ Try it

A crate is 5 feet on every edge. What is its volume?
A box has a volume of 80 ft³. Its base is 4 ft by 4 ft. How tall is it?
A cone is 4 inches tall, and its height is half the diameter of its base. Find its volume to the nearest cubic inch.

Answers

1. 5³ = 125 ft³ · 2. 80 ÷ (4 × 4) = 80 ÷ 16 = 5 ft · 3. The diameter is 8, so r = 4: ⅓ × 3.14 × 16 × 4 = 200.96 ÷ 3 ≈ 66.99 ≈ 67 in³

Lesson 7 · Surface area

🔑 Key ideas

Surface area is the total area of all the outside faces of a solid, in square units. Think wrapping paper, not filling.
A cube has 6 identical square faces: SA = 6s².
For other solids, add up the faces, or use the formula sheet.

Boxes and pyramids

📘 Example 1 · A cube with 3-inch edges

One face: 3 × 3 = 9 in². Six faces: 6 × 9 = 54 in².

📘 Example 2 · A box 6 ft long, 2 ft wide and 3 ft tall

It has three pairs of matching faces: top and bottom 6 × 2 = 12, front and back 6 × 3 = 18, two ends 2 × 3 = 6.
SA = 2(12) + 2(18) + 2(6) = 24 + 36 + 12 = 72 ft².
The sheet's form gives the same answer: p = 6 + 2 + 6 + 2 = 16 and B = 12, so ph + 2B = 16 × 3 + 24 = 72.

A pyramid's surface is its base plus its triangle sides: SA = ps + B. Here s is the slant height, measured from the top down the middle of a side face to the base edge.

📘 Example 3 · A pyramid: slant height 5 ft, base perimeter 14 ft, base area 12 ft²

SA = × 14 × 5 + 12 = 35 + 12 = 47 ft²

Cylinders, cones and spheres

Solid	Surface area	The parts
Cylinder	2πrh + 2πr²	the curved side (circumference × height) + two circle ends
Cone	πrs + πr²	the curved side + one circle base
Sphere	4πr²	four times a circle with the same radius

📘 Example 4 · Three curved solids

Cylinder, r = 4 in, h = 3 in: 2π(4)(3) + 2π(4²) = 24π + 32π = 56π ≈ 175.84 in².
Cone, r = 2 ft, slant height 5 ft: π(2)(5) + π(2²) = 10π + 4π = 14π ≈ 43.96 ft².
Sphere, r = 10 ft: 4π(10²) = 400π ≈ 1256 ft².

💡 Some answer choices are “in terms of π”, like 56π. Then leave π as it is and don't multiply by 3.14. If the choices are plain numbers, multiply by 3.14 at the end.

✏️ Try it

All the edges of a metal box are the same length, and its surface area is 150 in². How long is each edge?
A sphere has a diameter of 8. Find its surface area in terms of π.
A cylinder has a radius of 6 cm and a height of 10 cm. Find its surface area in terms of π.

Answers

1. 6s² = 150, so s² = 25 and s = 5 in · 2. r = 4: 4π × 16 = 64π · 3. 2π(6)(10) + 2π(36) = 120π + 72π = 192π cm²

Next: Lessons 8 and 9, combined figures and the calculator →

Lesson 8 · Combined figures

🔑 Key ideas

A combined figure is made of two or more simple figures.
Perimeter: add the lengths of all the outside sides. You may have to work out a missing one first.
Area or volume: break the figure into parts, find each part, then add (or subtract a piece that is cut out).

📘 Example 1 · An L-shaped room

Missing sides. The left side (12) equals x + 4, so x = 8 ft. The bottom (18) equals 10 + y, so y = 8 ft.
Perimeter: 10 + 8 + 8 + 4 + 18 + 12 = 60 ft.
Area: part 1 is 10 × 12 = 120 ft², part 2 is 8 × 4 = 32 ft². Total: 152 ft².

💡 There is usually more than one way to cut a figure up. Pick the easiest; the answer comes out the same. (Here, 18 × 12 minus the missing 8 × 8 corner is 216 − 64 = 152 too.)

📘 Example 2 · A rectangle with a half circle on one end

A rectangle 12 m long and 10 m wide has a half circle on one 10 m side.

The half circle has diameter 10, so radius 5. Its area is ½ × 3.14 × 5² = 39.25 m².
Area: 12 × 10 + 39.25 = 120 + 39.25 = 159.25 m².
Perimeter: the two long sides, the one straight end, and half a circumference: 12 + 12 + 10 + ½ × 3.14 × 10 = 34 + 15.7 = 49.7 m.

📘 Example 3 · A container: a cylinder with a cone on top

The cylinder has radius 1 m and height 2 m. The cone on top has the same radius and a height of 1.5 m.

Cylinder: 3.14 × 1² × 2 = 6.28 m³
Cone: × 3.14 × 1² × 1.5 = 1.57 m³
Total: 6.28 + 1.57 = 7.85 m³

📘 Example 4 · Subtracting a part

A circle with radius 2 in fits exactly inside a square, touching all four sides. What is the area of the square outside the circle?

The square is 4 in on a side, so 16 in². The circle is 3.14 × 2² = 12.56 in². The difference is 16 − 12.56 = 3.44 in².

✏️ Try it

A concrete slab 8 ft by 12 ft by 3 ft has a block 3 ft by 3 ft by 2 ft on each of its four corners. What is the total volume?
A rectangular piece 32 in wide and 20 in tall has a triangle cut out of its bottom edge. The triangle has a base of 20 in and a height of 12 in. What area is left?

Answers

1. Slab 8 × 12 × 3 = 288; blocks 4 × (3 × 3 × 2) = 72; total 360 ft³ · 2. 32 × 20 = 640, minus ½ × 20 × 12 = 120, leaves 520 in²

Lesson 9 · Geometry calculator skills

🔑 Key ideas

When a formula has many steps, the calculator follows the order of operations: powers first, then × and ÷, then + and −.
Check your work by estimating, or by entering the keys again to see if you get the same result.

Task	Keys on the TI-30XS MultiView™	Display shows
Volume of a pyramid: base 40 ft by 40 ft, height 81 ft	`clear` 40 `x²` `×` 81 `÷` 3 `enter`	43200 (ft³)
Hypotenuse with legs 10 in and 24 in	`clear` 10 `x²` `+` 24 `x²` `enter`, then `2nd` `x²` 676 `enter`	676, then 26 (in)
Area of a circle with radius 2.4 cm	`clear` `π` `×` 2.4 `x²` `enter`	18.09557368 ≈ 18.1 (cm²)
Volume of a cube with 3.5 ft edges	`clear` 3.5 `^` 3 `enter`	42.875 ≈ 43 (ft³)

x² squares a number. For any other power, use ^ and then the power.
The π key uses the calculator's more exact value of π, so its answers can differ a little from 3.14. With 3.14 the circle above is 18.0864, which also rounds to 18.1. If the choices are close together, use 3.14 as the test says.
If an answer appears with a √ or π in it, press ◀▶ (the toggle key) to see it as a decimal.
Watch the display as you type. If you make a mistake, use delete and type the right number.

🧪 Try it: shape calculator

Pick a shape and type its measurements. It shows the formula, the numbers put in, and the answer, using π ≈ 3.14.

Shape Units

✏️ Try it on your calculator

Find the volume of a cylinder with a radius of 1.5 m and a height of 5 m, to the nearest cubic meter.
Find the volume of a cone with a radius of 12 cm and a height of 20 cm, to the nearest cm³.
Find the hypotenuse of a right triangle with legs of 13 cm and 9 cm, to the nearest tenth.

Answers

1. 3.14 × 2.25 × 5 = 35.325 ≈ 35 m³ · 2. ⅓ × 3.14 × 144 × 20 = 3014.4 ≈ 3014 cm³ · 3. √(169 + 81) = √250 ≈ 15.8 cm

Next: summary and quiz →

Chapter summary

Four-sided figures are named by their sides and angles; their angles add up to 360°. A trapezoid has exactly one pair of parallel sides.
Triangles are named by sides (equilateral, isosceles, scalene) and by angles (right, acute, obtuse); their angles add up to 180°.
In a right triangle, a² + b² = c², where c is the hypotenuse. Add the squares to find c; subtract them to find a leg.
Perimeter is the distance around (units). Area is the space inside (square units). Use the height at a right angle to the base.
For circles, C = πd and A = πr², with π ≈ 3.14. Check whether you were given the radius or the diameter.
Volume (cubic units) is base area × height for prisms and cylinders, and one-third of that for pyramids and cones. Surface area (square units) adds up all the outside faces.
Break combined figures into simple parts. On the calculator, use x², ^, π and 2nd x² for √, and check your answer.

📄 Practice questions

The book's own practice sets for Lessons 1–9 and its chapter practice questions are in the printable assignment for this chapter. You may use the formula sheet on the Start here tab.

Mini-quiz

1. A four-sided figure has exactly one pair of parallel sides. What is it?

2. Two angles of a triangle measure 55° and 100°. What is the third angle?

3. A right triangle has legs of 6 cm and 8 cm. How long is the hypotenuse?

4. A 13-foot ladder leans on a wall with its foot 5 feet from the wall. How high does it reach?

5. A rectangle is 16 inches long and 9 inches wide. What is its perimeter?

6. A triangle has a base of 10 m and a height of 6 m. What is its area?

7. Using π ≈ 3.14, what is the circumference of a circle with a radius of 5 cm?

8. Using π ≈ 3.14, what is the area of a circle with a radius of 4 m?

9. A box is 5 ft long, 4 ft wide and 3 ft tall. What is its volume?

10. A cone and a cylinder have the same base and height. How much does the cone hold?

11. A cube has edges of 3 inches. What is its surface area?

12. An L-shaped floor is a 10 ft by 6 ft rectangle joined to a 4 ft by 3 ft one. What is its area?

Source: GED® Test Prep Mathematics, Chapter 7 (Geometry). Explanations, figures, the Try it questions' answers, the tools and the quiz are Guru Jazzy's own. GED® is a registered trademark of the American Council on Education; this page is not made or endorsed by GED Testing Service.