8.G.B.7: Using the Pythagorean Theorem to Find Missing Sides in 2D and 3D
In plain English: 8.G.B.7 is the Common Core grade 8 math standard that asks students to use the Pythagorean Theorem, a² + b² = c², to find an unknown side of a right triangle. Students solve real-world and math problems in two dimensions, such as ladders and TV screens, and in three dimensions, such as the longest diagonal of a box. It is taught in Grade 8 Math.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Understand and apply the Pythagorean Theorem. Also written as 8.G.7 · Official standard
In 8.G.B.6, students explained why the Pythagorean Theorem is true. This lesson puts it to work. In a right triangle (a triangle with one 90° angle), the two sides that form the right angle are the legs, and the side across from the right angle is the hypotenuse, always the longest side. If the legs are a and b and the hypotenuse is c, then a² + b² = c². Knowing any two sides, students can find the third: add the squares of the legs to find the hypotenuse, or subtract the square of the known leg from the square of the hypotenuse to find a leg. Then they take a square root (the number that, multiplied by itself, gives the square), often rounded to the nearest tenth.
The standard asks for problems in two and three dimensions. In two dimensions, students find the height a ladder reaches, the diagonal of a TV screen or the length of a ramp. In three dimensions, they find a right triangle hidden inside a solid (a three-dimensional shape): the space diagonal of a box (the segment from a bottom corner to the opposite top corner, through the inside), the height of a pyramid (a solid with a flat base and triangular faces that meet at a top point), or the longest pencil that fits in a cup. Many 3D problems take two steps, with two right triangles, and students check every answer against the situation: a hypotenuse must be longer than each leg.
Learning Objectives
By the end of this lesson, students will be able to:
Identify the legs and the hypotenuse of a right triangle in a picture or a word problem
Use a² + b² = c² to find the hypotenuse or an unknown leg, rounding square roots to the nearest tenth when needed
Solve real-world problems in two dimensions, such as ladders, ramps and screen sizes
Find a right triangle inside a three-dimensional figure and use it to find a space diagonal, a height or a slant height (the height of a slanted face)
Check whether an answer makes sense in the situation, for example that the hypotenuse is the longest side
Prior Knowledge Required
Students should already be comfortable with:
The Pythagorean Theorem a² + b² = c² and why it is true 8.G.B.6
Square roots: √p is the positive number whose square is p, and perfect squares such as 144 = 12² 8.EE.A.2
Estimating a square root that is not a whole number, for example √40 is between 6 and 7 8.NS.A.2
Squaring numbers, including decimals, as a whole-number exponent 6.EE.A.1
Rectangular prisms (box shapes) and their length, width and height 5.MD.C.5
Circles: the radius (from the center to the edge) and the diameter (across the circle through the center, twice the radius) 7.G.B.4
Hand out centimeter grid paper and rulers. Students work alone for 5 minutes, then compare with a partner.
Warm-Up Prompt
"Draw a right triangle with legs of 6 cm and 8 cm along the grid lines. Before you measure, guess the length of the third side. Then measure it. Was 6 + 8 = 14 a good guess? What do you get for 6² + 8²?"
The third side measures 10 cm, not 14 cm: the straight path is shorter than going along both legs. 6² + 8² = 36 + 64 = 100, and 10² = 100. Remind students that this is the Pythagorean Theorem from 8.G.B.6, and ask: "If you knew the 10 and the 6, how could you get back to the 8?" Keep the question open for Direct Instruction.
Direct Instruction20 minutes
Build a one-page reference with the class. Students copy each step next to a small sketch.
Name the sides: the legs (a and b) make the right angle. The hypotenuse (c) is across from the right angle and is always the longest side. Mark it first in every problem.
Missing hypotenuse: add the squares of the legs, then take the square root: c = √(a² + b²).
Missing leg: subtract the square of the known leg from the square of the hypotenuse, then take the square root: a = √(c² - b²).
Square roots that are not whole numbers: use a calculator and round to the nearest tenth, and estimate first. For example, √240 is between 15 and 16, because 15² = 225 and 16² = 256.
Real-world problems: sketch the situation, find the right angle (a wall meets the ground, a screen corner, the corner of a field), and label the known sides with units.
Three dimensions: look for a right triangle inside the solid. In a box, the floor diagonal (corner to opposite corner across the bottom) and a vertical edge meet at a right angle, so they are the legs of a triangle whose hypotenuse is the space diagonal.
Check the answer: a hypotenuse must be longer than each leg but shorter than the two legs added together. A leg must be shorter than the hypotenuse.
Two dimensions: find the hypotenuse (TV screen)
TVs are sold by the length of the screen diagonal. A TV screen is 48 inches wide and 27 inches tall. How long is its diagonal, and what size would the store call it?
Equation: The width and height are the legs, and the diagonal is the hypotenuse. c² = 48² + 27² = 2,304 + 729 = 3,033, so c = √3,033 ≈ 55.1 inches. The store calls it a 55-inch TV. Check: 55.1 is longer than 48 and shorter than 48 + 27 = 75.
Two dimensions: find a leg (ladder, Diagram 1)
A 16-foot ladder leans against a wall with its base 4 feet from the wall. How high up the wall does the ladder reach?
Equation: The wall and the ground meet at a right angle. The ladder is the hypotenuse (16 ft), and the ground distance is a leg (4 ft). h² + 4² = 16², so h² = 256 - 16 = 240 and h = √240 ≈ 15.5 feet. A common ladder safety guideline puts the base 1 foot out for every 4 feet of height, and 15.5 ÷ 4 ≈ 3.9 fits it.
Two dimensions: a mathematical problem (isosceles triangle)
An isosceles triangle (two equal sides) has two sides of 13 cm and a base of 10 cm. Find its height and its area.
Equation: The height from the top meets the base at a right angle in the middle, which splits the triangle into two right triangles with hypotenuse 13 cm and one leg 5 cm. h² = 13² - 5² = 169 - 25 = 144, so h = 12 cm. Area = (1/2)(10)(12) = 60 cm².
Three dimensions: space diagonal of a box (Diagram 2)
A shipping box is 24 inches long, 18 inches wide and 16 inches tall. Can a 32-inch youth baseball bat fit inside, lying flat on the bottom? What if it is tilted from a bottom corner to the opposite top corner?
Equation: Step 1, the floor diagonal: d² = 24² + 18² = 576 + 324 = 900, so d = 30 in. The bat does not fit flat, because 32 > 30. Step 2, the space diagonal: the floor diagonal and the 16-in height are the legs, so s² = 30² + 16² = 900 + 256 = 1,156 and s = 34 in. Tilted, the bat fits, because 32 < 34.
Three dimensions: height of a square pyramid
A square pyramid has a base 14 m on each side. Its slant height (the height of each triangular face, from the middle of a base edge up to the top) is 25 m. How tall is the pyramid?
Equation: Inside the pyramid, a right triangle joins the top, the center of the base and the middle of a base edge. Its legs are the height h and half the base, 7 m, and its hypotenuse is the slant height, 25 m. h² = 25² - 7² = 625 - 49 = 576, so h = 24 m.
For Example 4, show Diagram 2 and trace the two right triangles with a finger: first across the floor, then up the back corner. Ask students to point to the right angle in each one before they write an equation. Stress that the 30-inch floor diagonal is a hypotenuse in Step 1 and a leg in Step 2.
Guided Practice15 minutes
Pairs solve each problem. One partner sketches and labels the right triangle, and the other writes the equation; they switch after each problem.
Guided practice problems with answers
Problem
Answer
A moving truck ramp is 10 ft long, and the back of the truck is 3 ft above the street. How far from the truck does the ramp touch the street?
The ramp is the hypotenuse: √(10² - 3²) = √91 ≈ 9.5 ft
The bases on a baseball diamond form a square with 90-ft sides. How far does the catcher throw from home plate to second base?
Two sides of the square are the legs: √(90² + 90²) = √16,200 ≈ 127.3 ft
A desk drawer is 20 in long, 15 in wide and 6 in deep. Find the diagonal across its bottom and the longest straight length that fits inside.
Floor diagonal √(400 + 225) = 25 in; space diagonal √(25² + 6²) = √661 ≈ 25.7 in
A right triangle has a hypotenuse of 41 m and one leg of 9 m. Find the other leg.
√(41² - 9²) = √(1,681 - 81) = √1,600 = 40 m
Listen for students who add the squares when the hypotenuse is given. Ask: "Which side is across from the right angle? Is it the one you are looking for?"
Independent Practice15 minutes
Students work alone, then check with a partner. Each answer must include a labeled sketch.
Independent practice problems with answers
Problem
Answer
A kite is on 100 ft of string, pulled straight. The kite is directly above a point 60 ft from the person holding the string. How high is the kite?
√(100² - 60²) = √6,400 = 80 ft
A laptop screen is 12 in wide and 7.5 in tall. What is its diagonal?
√(144 + 56.25) = √200.25 ≈ 14.2 in, sold as a 14-inch laptop
A fish tank is 30 in long, 12 in wide and 18 in tall. Find the diagonal across its bottom and its space diagonal.
Floor diagonal √1,044 ≈ 32.3 in; space diagonal √(1,044 + 324) = √1,368 ≈ 37.0 in
A box is 3 cm long, 4 cm wide and 12 cm tall. Find its space diagonal without a calculator.
Floor diagonal √(9 + 16) = 5 cm; space diagonal √(25 + 144) = 13 cm
Ava says a right triangle with legs 11 and 60 has a hypotenuse of 71. Is she right?
No. She added the legs. √(121 + 3,600) = √3,721 = 61
Closure5-10 minutes
Exit ticket: (1) A tablet screen is 10 in wide and 7 in tall. What is its diagonal, to the nearest tenth? (√149 ≈ 12.2 in.) (2) A box is 10 in long, 10 in wide and 5 in tall. Find its floor diagonal and its space diagonal. (√200 ≈ 14.1 in, and √225 = 15 in.) (3) In one sentence, explain how you decide whether to add or subtract the squares.
Differentiation Strategies
For Struggling Students
Give a table with columns "leg a", "leg b", "hypotenuse c" and have students write each known number in its column before computing
Start with whole-number answers (6-8-10, 5-12-13) so students can check the idea before they meet decimals
Hand out a sketch with the right angle already marked for every word problem, and ask students to shade the hypotenuse
For Advanced Students
Work backward: a box has a bottom 6 cm by 8 cm and a space diagonal of 20 cm. How tall is it? (The floor diagonal is 10 cm, so the height is √(400 - 100) = √300 ≈ 17.3 cm.)
Have students find the longest pole that fits in a room of their choice, measured with a tape measure, and explain each right triangle they used
Extension (beyond this standard): the length from one corner of a box to the opposite corner is √(l² + w² + h²). Ask students to explain, with the two steps, why this shortcut works
Assessment Guidance
What to Look For
A complete answer starts with a sketch that marks the right angle and labels the hypotenuse, writes the equation before any arithmetic, and gives the answer with units and a sensible rounding. In 3D problems, look for two separate right triangles. Watch for students who add the legs instead of their squares, who forget the square root, who add squares when a leg is missing, who use the full base of a pyramid or the diameter of a cone (a solid with a circular base that narrows to a point) where the right triangle uses half of it, and who use the radius of a cylinder (a can shape) where the longest-object triangle uses the full diameter.
02
Classroom Activities
3 Activities
1
Measure, Predict, Check
15 minPairs
Pairs measure the length and width of three flat rectangles, predict each diagonal with the Pythagorean Theorem, and then measure the diagonal to see how close the prediction is.
The Objects
A sheet of letter paper (8.5 in by 11 in)
A 3 x 5 index card (3 in by 5 in)
A 3 x 3 sticky note (3 in by 3 in)
One rectangle of your choice from the classroom: a book cover, a desk top or a window pane
Procedure
Measure the length and the width to the nearest eighth of an inch and write them down
Predict the diagonal: square both sides, add, and take the square root, rounded to the nearest tenth
Measure the diagonal with a ruler and record the difference between your prediction and your measurement
Answer Key
Letter paper: √(8.5² + 11²) = √193.25 ≈ 13.9 in
Index card: √(3² + 5²) = √34 ≈ 5.8 in
Sticky note: √(3² + 3²) = √18 ≈ 4.2 in
Measured diagonals are usually within about 1/8 in of the prediction
Discussion Questions
For each object, the diagonal was longer than each side but shorter than the two sides added together. Why must that always happen?
The sticky note is a square. How does its diagonal compare with its side length?
Why do real measurements differ a little from the prediction?
Modification for Distance Learning
Students measure a phone, a tablet or a picture frame at home and post a photo of their labeled sketch with the prediction and the measurement.
2
Leg or Hypotenuse? Card Sort
20 minPairs
Each pair gets 8 situation cards. First they sort the cards into two piles, "find the hypotenuse" and "find a leg", without computing anything. Then they sketch each situation and solve it.
The 8 Cards
Card A: A 20-ft ladder leans against a house with its base 5 ft from the wall. How high does it reach?
Card B: A park is a rectangle 400 m by 300 m. How long is the straight path from one corner to the opposite corner?
Card C: A phone screen has a 6.1-in diagonal and is 2.6 in wide. How tall is it?
Card D: A zip line drops 22 m over a horizontal distance of 120 m. How long is the cable, pulled straight?
Card E: A wire runs from the top of a 5-m flagpole to a stake 2 m from its base. How long is the wire?
Card F: A 65-inch TV screen is 32 in tall. How wide is it?
Card G: A swimmer crosses a pool 25 m long and 10 m wide from corner to opposite corner. How far does she swim?
Card H: A wheelchair ramp is 30 ft long and rises 2 ft. What horizontal distance does it cover?
Answer Key
Find the hypotenuse: B, D, E, G. Find a leg: A, C, F, H.
Card A: Leg: √(400 - 25) = √375 ≈ 19.4 ft
Card B: Hypotenuse: √(160,000 + 90,000) = 500 m, which is 200 m shorter than walking along two sides
Which words in a card told you that the missing side is the hypotenuse?
On Card H the horizontal distance is only 0.1 ft shorter than the ramp. Why is a gentle ramp almost as long as its horizontal distance?
On Card C, what answer do you get if you add the squares by mistake, and how could you tell right away that it is wrong?
Challenge Variation
Each pair writes two new cards from their own life, one of each kind, and trades them with another pair.
3
The Longest String in the Box
20 minGroups of 3
Each group gets an empty shoebox with the lid off. They predict the longest straight string that fits inside, from a bottom corner to the opposite top corner, then test the prediction with string.
Procedure
Measure the inside length, width and height of the box in centimeters
Compute the floor diagonal, then the space diagonal, rounded to the nearest tenth
Tape one end of a string in a bottom corner, stretch it to the opposite top corner, cut it and measure it
Record the prediction, the measurement and the difference
Sample Box
A shoebox measuring 33 cm by 19 cm by 12 cm inside has a floor diagonal of √(1,089 + 361) = √1,450 ≈ 38.1 cm and a space diagonal of √(1,450 + 144) = √1,594 ≈ 39.9 cm.
Discussion Questions
Which of the three box measurements affects the space diagonal the most, and why?
In the sample box, the space diagonal is less than 2 cm longer than the floor diagonal. What does that tell you about the height of the box?
Could a 40-cm wooden stick lie inside the sample box with the lid closed?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: How High Does the Ladder Reach?
A 16-ft ladder with its base 4 ft from a wall, drawn to scale (20 px per foot). The wall and the ground form the right angle, so the ladder is the hypotenuse and the height h is a leg: h² + 4² = 16², h = √240 ≈ 15.5 ft.
Diagram 2: Two Right Triangles Inside a Box
A box 24 in long, 18 in wide and 16 in tall. The depth is drawn slanted and shortened, as in most 3D sketches, so lengths on the slant are not to scale. The dashed floor diagonal (30 in) is the hypotenuse of the bottom right triangle and then a leg of the upright right triangle, whose hypotenuse is the space diagonal (34 in).
04
Homework Assignment
~30 min
8.G.B.7 Homework: Finding Unknown Sides in Right Triangles
Directions: For each problem, draw a sketch, mark the right angle, and label the hypotenuse. Write the equation before you compute, and round to the nearest tenth when the answer is not a whole number. Include units.
Part 1: Two Dimensions (Problems 1-3)
A high school soccer field is a rectangle 110 yd long and 70 yd wide. (a) How long is the diagonal from one corner flag to the opposite corner flag? (b) How much shorter is that than walking along two sides of the field?
A right triangle has a hypotenuse of 73 cm and one leg of 48 cm. (a) Find the other leg. (b) Find the area and the perimeter of the triangle.
In the US, building guidelines for wheelchair ramps allow a rise of at most 1 inch for every 12 inches of horizontal length. A porch is 21 inches above the ground. (a) What is the shortest horizontal length the ramp can have? (b) How long is the sloped surface of that ramp, in inches and in feet?
Part 2: Three Dimensions (Problems 4-6)
A common carry-on suitcase size is 22 in by 14 in by 9 in. (a) Find the diagonal across the bottom of the suitcase (the 22-in by 14-in side). (b) Find the space diagonal. (c) Can a 27-in folded camera tripod lie flat on the bottom? Can it fit if it is tilted from corner to corner?
A camping tent is shaped like a square pyramid. Its base is 7 ft by 7 ft and its center pole is 6 ft tall. (a) Find the slant height, from the middle of a base edge to the top. (b) Find the distance from the center of the floor to a corner of the floor. (c) Find the length of an edge from a floor corner to the top.
A pencil cup is a cylinder with an inside diameter of 8 cm and a height of 11 cm. (a) Draw the right triangle that gives the longest straight object that fits completely inside. (b) Find that length. (c) Which fits completely inside: a new 19-cm pencil, a used 12-cm pencil, or both?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sketch and Labels
Right angle marked, hypotenuse labeled, all known sides with units
Sketch with some labels missing
No sketch
Choosing the Operation
Adds squares for a hypotenuse and subtracts for a leg every time
One problem with the wrong operation
Adds or subtracts the sides themselves
3D Right Triangles
Finds and uses both right triangles in Problems 4-6
Finds one triangle but not the second
No right triangle identified
Accuracy and Sense
All answers correct, rounded to the tenth, and checked against the situation
One or two computing or rounding errors
Many errors
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Use a calculator where you need one, and round to the nearest tenth when an answer is not a whole number. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A right triangle has sides of 9 cm, 12 cm and 15 cm. Which equation shows the Pythagorean Theorem for this triangle?
Answer: A
The hypotenuse is the longest side, 15 cm, so it goes alone on one side: 81 + 144 = 225. Choices B and C put a shorter side in the place of the hypotenuse, and 81 + 225 = 306 is not 144. Choice D adds the sides instead of their squares, and 9 + 12 = 21, not 15.
Question 2 of 20 · Multiple Choice
The legs of a right triangle are 21 m and 28 m. How long is the hypotenuse?
Answer: B
c² = 21² + 28² = 441 + 784 = 1,225, so c = √1,225 = 35 m. Choice A adds the legs without squaring. Choice C stops before the square root. Choice D subtracts the squares, √(784 - 441) = √343 ≈ 18.5, which is the method for a missing leg.
Question 3 of 20 · Multiple Choice
A right triangle has a hypotenuse of 18 ft and one leg of 7 ft. To the nearest tenth, how long is the other leg?
Answer: D
The missing side is a leg, so subtract: b² = 18² - 7² = 324 - 49 = 275, and b = √275 ≈ 16.6 ft. Choice A subtracts the sides, 18 - 7. Choice B adds the squares, √373 ≈ 19.3, which would make the leg longer than the hypotenuse. Choice C forgets the square root.
Question 4 of 20 · Multiple Choice
A 12-ft ladder leans against a wall with its base 3 ft from the wall. To the nearest tenth, how high up the wall does it reach?
Answer: C
The ladder is the hypotenuse: h² = 12² - 3² = 144 - 9 = 135, so h = √135 ≈ 11.6 ft. Choice A adds the squares, √153 ≈ 12.4, which is longer than the ladder itself. Choice B subtracts the lengths, 12 - 3. Choice D adds them.
Question 5 of 20 · Multiple Choice
A TV screen is 35 in wide and 20 in tall. To the nearest tenth, how long is its diagonal?
Answer: D
The diagonal is the hypotenuse: √(35² + 20²) = √(1,225 + 400) = √1,625 ≈ 40.3 in, so it is sold as a 40-inch TV. Choice A adds the sides. Choice B subtracts the squares, √825 ≈ 28.7. Choice C forgets the square root.
Question 6 of 20 · Multiple Choice
A gift box is 6 in long, 3 in wide and 2 in tall. How long is its space diagonal, from a bottom corner to the opposite top corner?
Answer: A
Floor diagonal: d² = 6² + 3² = 45. Space diagonal: s² = 45 + 2² = 49, so s = 7 in. Choice B adds the three lengths. Choice C stops at the floor diagonal, √45 ≈ 6.7. Choice D forgets the square root.
Question 7 of 20 · Multiple Choice
The space diagonal of a box goes from a bottom corner to the opposite top corner. It is the hypotenuse of which right triangle?
Answer: C
The floor diagonal lies on the bottom, and the vertical edge at its end rises straight up from it, so they meet at a right angle. The space diagonal joins their far ends. Choice B describes the floor triangle, whose hypotenuse is the floor diagonal. Choice A pairs two segments on the bottom that do not meet at a right angle. Choice D is the triangle on a side face, whose hypotenuse is the diagonal of that face.
Question 8 of 20 · Multiple Choice
A drinking glass is 14 cm tall, and its inside diameter is 6 cm. To the nearest tenth, what is the longest straight stirring stick that fits completely inside?
Answer: B
Slice the glass straight down through its center: the height and the diameter are the legs of a right triangle. √(14² + 6²) = √(196 + 36) = √232 ≈ 15.2 cm. Choice A adds the lengths. Choice C subtracts the squares, √160. Choice D uses the radius, 3 cm, instead of the diameter: √205 ≈ 14.3.
Question 9 of 20 · Multiple Choice
A square pyramid has a base 20 m on each side and a height of 12 m. To the nearest tenth, what is its slant height, from the middle of a base edge to the top?
Answer: A
The right triangle inside uses the height and half the base: √(12² + 10²) = √244 ≈ 15.6 m. Choice B uses the full base, √(12² + 20²) = √544 ≈ 23.3. Choice C subtracts the squares, √(144 - 100) = √44 ≈ 6.6. Choice D forgets the square root.
Question 10 of 20 · Multiple Choice
A cone-shaped party hat has a radius of 3 in and a slant height of 8 in. To the nearest tenth, how tall is the hat?
Answer: B
The height and the radius are the legs, and the slant height is the hypotenuse: h = √(8² - 3²) = √55 ≈ 7.4 in. Choice A adds the squares, √73 ≈ 8.5, which is taller than the slanted side. Choice C subtracts the lengths. Choice D adds them.
Question 11 of 20 · Multiple Choice
A rectangle is 39 cm wide, and its diagonal is 89 cm. What is the area of the rectangle?
Answer: C
The length is a leg: √(89² - 39²) = √6,400 = 80 cm, so the area is 39 × 80 = 3,120 cm². Choice A multiplies the width by the diagonal. Choice B uses 89 - 39 = 50 as the length. Choice D finds the area of the triangle, half the rectangle.
Question 12 of 20 · Multiple Choice
The bases on a softball field form a square with 60-ft sides. To the nearest tenth, how far is it from home plate to second base?
Answer: D
Home to first and first to second are the legs of a right triangle: √(60² + 60²) = √7,200 ≈ 84.9 ft. Choice A adds the two sides. Choice B assumes the diagonal equals a side. Choice C forgets the square root.
Question 13 of 20 · Multiple Choice
Marco needs the missing leg of a right triangle with a leg of 16 in and a hypotenuse of 65 in. He writes √(16² + 65²) ≈ 66.9 in. What is his mistake?
Answer: A
A leg is shorter than the hypotenuse, so 66.9 in cannot be right. For a missing leg, subtract: √(65² - 16²) = √3,969 = 63 in. Choice B subtracts the sides without squaring. Choice C forgets the square root. Choice D accepts a leg longer than the hypotenuse.
Question 14 of 20 · Multiple Choice
A classroom is 30 ft long, 24 ft wide and 10 ft tall. To the nearest tenth, what is the distance from a floor corner to the opposite ceiling corner?
Answer: B
Floor diagonal: d² = 30² + 24² = 1,476. Space diagonal: √(1,476 + 10²) = √1,576 ≈ 39.7 ft. Choice A stops at the floor diagonal, √1,476 ≈ 38.4. Choice C adds the three lengths. Choice D uses only the length and the height, √1,000 ≈ 31.6.
Question 15 of 20 · Short Answer
A drone flies 120 m east from its launch point and then 50 m north. How far is it from the launch point, in a straight line?
The east and north paths meet at a right angle, so they are the legs: √(120² + 50²) = √(14,400 + 2,500) = √16,900 = 130 m.
Question 16 of 20 · Short Answer
A 15-m wire runs from the top of a pole to a stake in the ground 6 m from the base of the pole. To the nearest tenth, how tall is the pole?
The wire is the hypotenuse: h = √(15² - 6²) = √(225 - 36) = √189 ≈ 13.7 m.
Question 17 of 20 · Short Answer
A storage bin is a cube 15 in on each side. Find the diagonal across its bottom and its space diagonal, to the nearest tenth.
Floor diagonal: √(15² + 15²) = √450 ≈ 21.2 in. Space diagonal: √(450 + 15²) = √675 ≈ 26.0 in.
Question 18 of 20 · Short Answer
A box has a bottom 16 in by 12 in and a space diagonal of 25 in. How tall is the box?
Floor diagonal: √(16² + 12²) = √400 = 20 in. The floor diagonal and the height are the legs of the triangle with hypotenuse 25 in, so h = √(25² - 20²) = √225 = 15 in.
Question 19 of 20 · Short Answer
An equilateral triangle (all sides equal) has sides of 10 cm. Find its height to the nearest tenth, and explain which right triangle you used.
The height meets the base at its midpoint, so it makes a right triangle with hypotenuse 10 cm and a leg of 5 cm. h = √(10² - 5²) = √75 ≈ 8.7 cm.
Question 20 of 20 · Short Answer
A doorway is 3 ft wide and 6.5 ft tall. Can a thin round tabletop 7 ft across fit through it if it is tilted? Explain with a calculation.
The longest straight line in the doorway is its diagonal: √(3² + 6.5²) = √51.25 ≈ 7.2 ft. Yes: 7 ft is less than 7.2 ft, so the tabletop fits when it is tilted along the diagonal, if it is thin enough.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.G.B.7 mean?
8.G.B.7 means students use the Pythagorean Theorem, a² + b² = c², to find a missing side of a right triangle. The problems come from real life (ladders, screens, ramps, boxes) and from pure math (triangles and solids), in both flat figures and 3D figures.
Is 8.G.B.7 taught in grade 8 or in high school geometry?
It is a grade 8 standard, usually taught in Grade 8 Math right after students explain why the theorem works (8.G.B.6). High school Geometry builds on it with trigonometry in HSG.SRT.C.8, so students who master it in grade 8 arrive ready.
How do I know which side is the hypotenuse?
The hypotenuse is the side across from the right angle, and it is always the longest side. In a word problem, find where the right angle is (a wall meets the ground, the corner of a screen), and the hypotenuse is the slanted side that does not touch that corner, such as the ladder or the diagonal.
When do you add the squares and when do you subtract them?
Add them when you are looking for the hypotenuse, and subtract when you are looking for a leg. For a leg, start with the square of the hypotenuse and subtract the square of the known leg. A quick check: a leg must come out shorter than the hypotenuse.
How do you use the Pythagorean Theorem in 3D?
Look for a right triangle inside the solid. In a box, first find the diagonal across the bottom, then use it with the height as the two legs of a second right triangle, whose hypotenuse is the space diagonal. In a pyramid or a cone, the height, half the base (or the radius) and the slanted side form the triangle.
Why are so many answers decimals?
Because the square root of a whole number that is not a perfect square is irrational: it cannot be written as a fraction, and its decimal never ends or repeats (8.NS.A.2). Students round to the nearest tenth, or to what makes sense for the situation, and should estimate first: a square root of 240 must be a little less than 16, because 16² = 256.
Does the Pythagorean Theorem work for every triangle?
No. It works only for right triangles. For a triangle without a right angle, a² + b² does not equal c². In problems, students must first find or create a right angle, for example by drawing a height from the top of a triangle to its base.
What are Pythagorean triples, and do students need them?
They are sets of three whole numbers that fit a² + b² = c², such as 3, 4, 5. Any multiple works too, like 6, 8, 10. Students do not need to memorize them, but recognizing them speeds up mental checks.
How is 8.G.B.7 different from 8.G.B.6 and 8.G.B.8?
8.G.B.6 asks students to explain a proof of the theorem and its converse (the reverse statement: if a² + b² = c², the triangle has a right angle). 8.G.B.7 applies the theorem to find unknown sides in 2D and 3D problems. 8.G.B.8 uses it to find the distance between two points on a coordinate grid.
How can parents help with 8.G.B.7 at home?
Measure the width and height of a TV or a laptop screen, and ask your child to predict the diagonal before measuring it. Ask them to point to the right angle and the hypotenuse. A common mistake to watch for is adding the two sides instead of their squares.
07
Related Standards
6 standards
These standards connect to 8.G.B.7: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.G.B.6Prerequisite
Explain a proof of the Pythagorean Theorem and its converse
