Ready to put your right-triangle know-how to the test? In this trigonometry side length quiz designed for students and math enthusiasts, you'll learn how to determine the unknown side lengths using sine, cosine, and tangent ratios. Whether you're eager to calculate unknown side lengths from class examples or looking for extra trigonometric ratios practice, our free challenge will sharpen your problem-solving skills. Dive in, solve triangle sides accurately, and see how you measure up. Start now with this trig ratios quiz or deepen your skills with a solving for side lengths of right triangles quiz - the perfect preparation for your next trigonometry test!
In a right triangle, the hypotenuse is 10 units and one acute angle is 30°. What is the length of the side opposite the 30° angle?
5
5?3
8
2
The sine of 30° is 0.5, so the opposite side equals 10 × sin(30°) = 5. For more on sine, visit Math is Fun: Trig Ratios.
A right triangle has a 45° acute angle and a hypotenuse of length 14. What is the length of the adjacent side to the 45° angle?
A ladder leans against a wall making a 70° angle with the ground. The foot of the ladder is 4 meters from the wall. How long is the ladder?
Approximately 12.3 m
Approximately 4.7 m
Approximately 11.7 m
Approximately 8.5 m
cos(70°) = adjacent/hypotenuse, so hypotenuse = 4 / cos(70°) ? 11.7 m. For ladder problems, see Trig Applications.
A plane ascends at a 15° incline and travels 2000 ft along its flight path. How much altitude has it gained?
Approximately 1500 ft
Approximately 800 ft
Approximately 300 ft
Approximately 517.6 ft
sin(15°) = opposite/hypotenuse, so altitude = 2000 × sin(15°) ? 517.6 ft. See Khan Academy: Trig.
From the top of a 50 m cliff, the angle of depression to a boat at sea level is 25°. How far is the boat from the base of the cliff?
Approximately 52.9 m
Approximately 107.3 m
Approximately 75.2 m
Approximately 120.5 m
tan(25°) = opposite/adjacent, so adjacent = 50 / tan(25°) ? 107.3 m. See Math is Fun.
A surveyor measures the angle of elevation to the top of a tree as 35° from a point 30 m away. What is the tree's height?
Approximately 25.8 m
Approximately 30.0 m
Approximately 10.5 m
Approximately 21.0 m
tan(35°) = height/30, so height ? 30 × tan(35°) ? 21.0 m. Read more at Khan Academy.
In a right triangle, the legs measure 9 and 4 units. What is the length of the hypotenuse?
Approximately 9.85
5
?65
13
By the Pythagorean theorem, hypotenuse = ?(9² + 4²) = ?97 ? 9.85. Learn more at Math is Fun: Pythagoras.
A right triangle has sides of lengths 8, 15, and 17. What is the measure of the smallest acute angle?
Approximately 30°
Approximately 61.9°
Approximately 28.1°
Approximately 45°
The smallest angle is opposite the shortest leg (8): ? = arcsin(8/17) ? 28.1°. For inverse trig functions, see Khan Academy.
In a right triangle, the angle at C is 58° and the hypotenuse is 25 units. What is the length of the side adjacent to that 58° angle?
Approximately 20.0
Approximately 13.25
Approximately 22.5
Approximately 10.5
cos(58°) = adjacent/25, so adjacent ? 25 × cos(58°) ? 13.25. See Math is Fun.
A right triangle has angles of 50° and 40° and a hypotenuse of 20 units. Find the side opposite the 40° angle.
Approximately 8.50
Approximately 12.86
Approximately 15.32
Approximately 9.64
sin(40°) = opposite/20, so opposite ? 20 × sin(40°) ? 12.86. Learn more at Khan Academy.
Two right triangles share a 30° angle. The larger triangle has a hypotenuse of 20 units, and the smaller has a hypotenuse of 10 units. What is the difference between their sides opposite the 30° angle?
0
5
15
10
Opposite = hypotenuse × sin(30°). So difference = (20×0.5) ? (10×0.5) = 10 ? 5 = 5. More at Math is Fun.
A right triangle has an acute angle of 22° with an adjacent side of 50 units. An altitude is drawn from the right angle to the hypotenuse. What is the length of that altitude (approximately)?
Approximately 53.9
Approximately 19.0
Approximately 20.2
50
Other leg = 50 tan(22°) ? 20.2, hypotenuse ? 50 / cos(22°) ? 53.9, altitude = (leg1×leg2)/hyp ? (50×20.2)/53.9 ? 19.0. See Wikipedia: Triangle Altitude.
A building casts a 30 m shadow when the sun's elevation angle is 38°. What is the height of the building?
Approximately 12.05 m
Approximately 19.44 m
Approximately 23.44 m
Approximately 30.00 m
tan(38°) = height/30, so height ? 30×tan(38°) ? 23.44 m. Read more at Math is Fun.
A surveyor's instrument is 2 m above ground. The angle of elevation to a building's top is 42°, and the angle of depression to its base is 7°. What is the building's height?
Approximately 18.30 m
Approximately 16.67 m
Approximately 12.40 m
Approximately 14.67 m
Horizontal distance = 2 / tan(7°) ? 16.29 m; height above instrument = 16.29 × tan(42°) ? 14.67 m; total = 14.67 + 2 ? 16.67 m. See Khan Academy.
In a right triangle, one leg is 12 and the hypotenuse exceeds this leg by 5 units. What are the lengths of all sides?
A right triangle has perimeter 60 and legs in the ratio 3 : 4. What is the length of the hypotenuse?
30
15
20
25
Let legs = 3k and 4k, hyp = 5k; total = 12k = 60, k = 5, so hypotenuse = 5k = 25. More at Wikipedia: Pythagorean Triple.
A 30°-60°-90° triangle has area 30?3. What is the length of its hypotenuse?
16
?240
15
14
In such a triangle, sides = h/2 and h?3/2; area = (h/2 × h?3/2)/2 = h²?3/8 = 30?3 ? h² = 240 ? h = ?240. See Wikipedia.
A ramp must reach a platform 1.5 m high and makes an 8° angle with the ground. What is the length of the ramp?
Approximately 10.78 m
Approximately 9.15 m
Approximately 5.43 m
Approximately 12.37 m
sin(8°) = opposite/hypotenuse, so ramp length = 1.5 / sin(8°) ? 10.78 m. Learn more at Math is Fun.
In a right triangle with angles of 15° and 75°, what is the length of the altitude to the hypotenuse expressed in terms of the hypotenuse length c?
c/4
c/?2
(?3 c)/4
c/2
Altitude = c·sin(15°)·sin(75°). Since sin(15°)·sin(75°) = 1/4, the altitude equals c/4. See Wikipedia: Triangle Altitude.
Given sin ? = 5/13 in a right triangle and hypotenuse = 26, what is the area of the triangle?
24
120
130
260
Opposite = 26×(5/13)=10, adjacent = ?(26²?10²)=24, area = ½×10×24 = 120. More at Khan Academy.
0
{"name":"In a right triangle, the hypotenuse is 10 units and one acute angle is 30°. What is the length of the side opposite the 30° angle?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"In a right triangle, the hypotenuse is 10 units and one acute angle is 30°. What is the length of the side opposite the 30° angle?, A right triangle has a 45° acute angle and a hypotenuse of length 14. What is the length of the adjacent side to the 45° angle?, In a right triangle, the angle at A is 60° and the side adjacent to A measures 5 units. What is the length of the side opposite angle A?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}
Score2/25
Easy0/7
Medium2/8
Hard0/8
Expert0/2
AI Study Notes
Email these to me
You can bookmark this page to review your notes in future, or fill out the email box below to email them to yourself.
Study Outcomes
Apply Trigonometric Ratios -
Use sine, cosine, and tangent to determine the unknown side lengths in right triangles with confidence.
Calculate Unknown Side Lengths -
Perform accurate computations of missing sides by interpreting given angles and known sides, reinforcing your trigonometric ratios practice.
Analyze Triangle Components -
Identify and label the hypotenuse, adjacent, and opposite sides from diagrams to solve triangle sides systematically.
Verify Solutions -
Use inverse trigonometric functions to check your results and ensure precise answers in the trigonometry side length quiz.
Enhance Problem-Solving Speed -
Develop efficient strategies to determine the unknown side lengths under timed conditions, boosting accuracy and confidence.
Cheat Sheet
Fundamental Trigonometric Ratios -
Understanding that sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, and tanθ = opposite/adjacent gives you the tools to determine the unknown side lengths of right triangles confidently. Reinforce these with the SOH-CAH-TOA mnemonic from reputable university texts to recall each ratio effortlessly under pressure. For example, if you know an angle and the opposite side, you can calculate the hypotenuse by hypotenuse = opposite / sinθ.
Inverse Trigonometric Functions -
When you have two sides and need to find an angle, use inverse functions like θ = arcsin(opposite/hypotenuse) or θ = arctan(opposite/adjacent) to solve triangle sides. Sources like MIT OpenCourseWare highlight that these functions are essential in trigonometric ratios practice for accurate angle determination. Once you've found the angle, plug it back into a primary ratio to calculate the remaining unknown side length.
Pythagorean Theorem Synergy -
The Pythagorean theorem (a² + b² = c²) remains a cornerstone for solving triangle sides and pairs perfectly with trig. After using a trig ratio to find one leg, apply c = √(a² + b²) to determine the hypotenuse or vice versa, as outlined in standard engineering mathematics resources. This dual-approach ensures you can calculate unknown side lengths even when only two measures are initially given.
Special Right Triangle Shortcuts -
Memorize the side ratios for 45°-45°-90° (1:1:√2) and 30°-60°-90° (1:√3:2) triangles to instantly solve triangle sides without a calculator, a trick emphasized by academic research on problem”solving speed. Knowing these patterns means you can quickly determine the unknown side lengths by scaling the ratios to any size triangle. It's a great way to boost accuracy and speed in trigonometry side length quizzes.
Real-World Angle of Elevation and Depression -
Applying tangent in real-world scenarios helps you calculate unknown side lengths like building heights or distances by using height = distance × tan(θ), a method supported by surveys in applied mathematics journals. Practice problems involving angles of elevation and depression sharpen your ability to connect trigonometric concepts to tangible tasks. This hands-on trigonometric ratios practice reinforces both understanding and practical skills.
