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11 Dec 2025

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Understanding Pythagoras: Solve Lengths Easily

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Simran

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Pythagoras' theorem is a powerful mathematical tool that helps us... Show more

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Introduction to Pythagoras

Pythagoras' theorem is one of the most useful formulas you'll learn in mathematics. It works in any right-angled triangle and states that a² + b² = c², where c is the hypotenuse (the longest side opposite the right angle).

The formula helps us find missing sides in triangles when we know the other two sides. You'll often need to find square roots as part of solving these problems, so make sure your calculator is ready!

When tackling Pythagoras problems in exams, always start by identifying the right angle in the triangle and labeling which side is the hypotenuse.

Pro tip: Always draw a clear diagram if one isn't provided, and label the sides you know and the side you need to find.

Name:
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●
• Use black ink or ball-point pen.
●
Answer all Questions.
Answer the Questions in the spaces provided
- there may be m

Finding the Hypotenuse

When finding the hypotenuse (the longest side opposite the right angle), we use the formula a² + b² = c², where c is the hypotenuse. This is the most straightforward application of Pythagoras' theorem.

In question 1, we find the length of AC by using the measurements of the other two sides: 3.6 cm and 4.8 cm. We square these values (3.6² + 4.8² = 12.96 + 23.04 = 36), then take the square root to get 6 cm.

For question 2, we use the same approach but with different values. We know the hypotenuse BC is 17.55 cm and one side AC is 6.75 cm. To find AB, we rearrange the formula: AB² = BC² - AC² which gives us AB = 16.2 cm.

Remember: When finding the hypotenuse, you ADD the squares of the two shorter sides. When finding a shorter side, you SUBTRACT the known shorter side squared from the hypotenuse squared.

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• Use black ink or ball-point pen.
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Answer all Questions.
Answer the Questions in the spaces provided
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Finding a Shorter Side

Sometimes you need to find one of the shorter sides rather than the hypotenuse. In these cases, you rearrange the formula to a² = c² - b².

In question 3, we need to find BC when AC = 14 cm and AB = 6 cm. Since AC is the hypotenuse, we calculate BC² = AC² - AB² = 14² - 6² = 196 - 36 = 160. Taking the square root gives BC = 12.6 cm (to 1 decimal place).

For question 4, we find AC when AB = 4.2 cm and BC = 5.6 cm. Using a² + b² = c², we get 4.2² + 5.6² = AC². This simplifies to 17.64 + 31.36 = AC², so AC = 7 cm.

Watch out: Make sure you correctly identify which side is the hypotenuse before applying Pythagoras' theorem. The hypotenuse is always opposite the right angle and is the longest side.

Name:
Instructions
●
• Use black ink or ball-point pen.
●
Answer all Questions.
Answer the Questions in the spaces provided
- there may be m

Multi-Step Problems

More complex problems may require you to use Pythagoras' theorem multiple times to reach the final answer. These problems test your ability to break down complex shapes into manageable parts.

In question 5, we need to find the length AD in a shape with multiple right angles. First, we calculate the length y using Pythagoras: y² = 19² - 14² = 361 - 196 = 165, so y = √165 ≈ 12.85. Then we use y to find AD: AD² = 10² + y² = 100 + 165 = 265, giving AD = 16.3 m (3sf).

Question 6 follows a similar approach. We first find length y using Pythagoras: y² = 4² + 7² = 16 + 49 = 65. Then we solve for AB: AB² + 65 = 9.5², which gives AB = 5.02 cm (3sf).

Key insight: For complex shapes, identify right-angled triangles within the shape and solve them one at a time. Your answer from one calculation becomes input for the next.

Name:
Instructions
●
• Use black ink or ball-point pen.
●
Answer all Questions.
Answer the Questions in the spaces provided
- there may be m

Pythagoras in Geometric Shapes

Pythagoras' theorem is extremely useful when working with geometric shapes like rectangles, squares, and trapeziums. The diagonal of these shapes can be found using this theorem.

For rectangles, as in question 7, the diagonal AC forms a right-angled triangle with the sides of the rectangle. Using Pythagoras, AC² = 8² + 17² = 64 + 289 = 353, giving AC = 18.8 cm (1dp).

In question 8, we have a trapezium ABCD. We first find the height of the trapezium using Pythagoras: x² = 17² - 15² = 289 - 225 = 64, so x = 8 cm. Then we subtract this value from the total width to find BC: BC = 12 - 8 = 4 cm.

Handy tip: In regular shapes like squares and rectangles, the diagonals always form right-angled triangles with the sides, making Pythagoras' theorem perfect for finding their lengths.

Name:
Instructions
●
• Use black ink or ball-point pen.
●
Answer all Questions.
Answer the Questions in the spaces provided
- there may be m

More Complex Geometric Applications

Pythagoras' theorem helps us find crucial measurements in various geometric figures, like the height of triangles or distances in compound shapes.

In question 9, we need to find the perpendicular height of an isosceles triangle. Using Pythagoras, the distance from the base to the height is x² = 10² - 4² = 100 - 16 = 84, giving a height of 9.17 (3sf).

Question 10 involves finding the diagonal AC in a trapezium. We first calculate the length of one part using Pythagoras: x² = 13² - 5² = 169 - 25 = 144, so x = 12. Then we find AC using the second right-angled triangle: AC² = 8² + 12² = 64 + 144 = 208, giving AC = 14.4 (3sf).

Remember: When working with isosceles triangles, the perpendicular height bisects the base, creating two equal right-angled triangles—perfect for applying Pythagoras' theorem!

Name:
Instructions
●
• Use black ink or ball-point pen.
●
Answer all Questions.
Answer the Questions in the spaces provided
- there may be m

Real-World Applications

Pythagoras' theorem has many practical applications in real life, from navigation to construction problems.

In question 11, a ship sails 3.7 km North and then 2.4 km East. To find the direct distance between the starting and ending points, we use Pythagoras: x² = 2.4² + 3.7² = 5.76 + 13.69 = 19.45, giving a direct distance of 4.4 km (1dp).

Question 12 presents a classic ladder problem. With the ladder reaching 2.5 m (or 250 cm) up a wall and its base 70 cm from the wall, we can find the ladder's length using Pythagoras: x² = 70² + 250² = 4,900 + 62,500 = 67,400, giving a ladder length of 260 cm.

Real-life connection: Engineers and architects regularly use Pythagoras' theorem when designing structures, planning layouts, or calculating distances that can't be directly measured.

Name:
Instructions
●
• Use black ink or ball-point pen.
●
Answer all Questions.
Answer the Questions in the spaces provided
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Special Applications

Pythagoras' theorem can solve interesting problems involving special geometric relationships and real-world scenarios.

In question 13, we need to find the perimeter of a square when only given its diagonal (8 m). Since the diagonal creates a right-angled triangle with two equal sides, we use Pythagoras: x² + x² = 8², so 2x² = 64, making x = √32 ≈ 5.66 m. The perimeter is 4 times this length: 4 × 5.66 = 22.6 m (1dp).

Question 14 involves finding the dimensions of a TV with a 50-inch diagonal and a 4:3 aspect ratio. We can write the sides as 4x and 3x, then apply Pythagoras: (4x)² + (3x)² = 50². This gives us 16x² + 9x² = 2500, so 25x² = 2500, making x = 10. Therefore, the length is 4 × 10 = 40 inches and the width is 3 × 10 = 30 inches.

Cool fact: TV and monitor sizes are always measured diagonally! You can use Pythagoras' theorem to work out the actual screen dimensions when shopping for a new device.



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I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

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THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

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This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

 

Maths

105

11 Dec 2025

8 pages

Understanding Pythagoras: Solve Lengths Easily

user profile picture

Simran

@simran_task

Pythagoras' theorem is a powerful mathematical tool that helps us find unknown sides in right-angled triangles. This theorem states that the square of the hypotenuse (the longest side opposite the right angle) equals the sum of the squares of the... Show more

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Introduction to Pythagoras

Pythagoras' theorem is one of the most useful formulas you'll learn in mathematics. It works in any right-angled triangle and states that a² + b² = c², where c is the hypotenuse (the longest side opposite the right angle).

The formula helps us find missing sides in triangles when we know the other two sides. You'll often need to find square roots as part of solving these problems, so make sure your calculator is ready!

When tackling Pythagoras problems in exams, always start by identifying the right angle in the triangle and labeling which side is the hypotenuse.

Pro tip: Always draw a clear diagram if one isn't provided, and label the sides you know and the side you need to find.

Name:
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• Use black ink or ball-point pen.
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Finding the Hypotenuse

When finding the hypotenuse (the longest side opposite the right angle), we use the formula a² + b² = c², where c is the hypotenuse. This is the most straightforward application of Pythagoras' theorem.

In question 1, we find the length of AC by using the measurements of the other two sides: 3.6 cm and 4.8 cm. We square these values (3.6² + 4.8² = 12.96 + 23.04 = 36), then take the square root to get 6 cm.

For question 2, we use the same approach but with different values. We know the hypotenuse BC is 17.55 cm and one side AC is 6.75 cm. To find AB, we rearrange the formula: AB² = BC² - AC² which gives us AB = 16.2 cm.

Remember: When finding the hypotenuse, you ADD the squares of the two shorter sides. When finding a shorter side, you SUBTRACT the known shorter side squared from the hypotenuse squared.

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Finding a Shorter Side

Sometimes you need to find one of the shorter sides rather than the hypotenuse. In these cases, you rearrange the formula to a² = c² - b².

In question 3, we need to find BC when AC = 14 cm and AB = 6 cm. Since AC is the hypotenuse, we calculate BC² = AC² - AB² = 14² - 6² = 196 - 36 = 160. Taking the square root gives BC = 12.6 cm (to 1 decimal place).

For question 4, we find AC when AB = 4.2 cm and BC = 5.6 cm. Using a² + b² = c², we get 4.2² + 5.6² = AC². This simplifies to 17.64 + 31.36 = AC², so AC = 7 cm.

Watch out: Make sure you correctly identify which side is the hypotenuse before applying Pythagoras' theorem. The hypotenuse is always opposite the right angle and is the longest side.

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Multi-Step Problems

More complex problems may require you to use Pythagoras' theorem multiple times to reach the final answer. These problems test your ability to break down complex shapes into manageable parts.

In question 5, we need to find the length AD in a shape with multiple right angles. First, we calculate the length y using Pythagoras: y² = 19² - 14² = 361 - 196 = 165, so y = √165 ≈ 12.85. Then we use y to find AD: AD² = 10² + y² = 100 + 165 = 265, giving AD = 16.3 m (3sf).

Question 6 follows a similar approach. We first find length y using Pythagoras: y² = 4² + 7² = 16 + 49 = 65. Then we solve for AB: AB² + 65 = 9.5², which gives AB = 5.02 cm (3sf).

Key insight: For complex shapes, identify right-angled triangles within the shape and solve them one at a time. Your answer from one calculation becomes input for the next.

Name:
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• Use black ink or ball-point pen.
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Pythagoras in Geometric Shapes

Pythagoras' theorem is extremely useful when working with geometric shapes like rectangles, squares, and trapeziums. The diagonal of these shapes can be found using this theorem.

For rectangles, as in question 7, the diagonal AC forms a right-angled triangle with the sides of the rectangle. Using Pythagoras, AC² = 8² + 17² = 64 + 289 = 353, giving AC = 18.8 cm (1dp).

In question 8, we have a trapezium ABCD. We first find the height of the trapezium using Pythagoras: x² = 17² - 15² = 289 - 225 = 64, so x = 8 cm. Then we subtract this value from the total width to find BC: BC = 12 - 8 = 4 cm.

Handy tip: In regular shapes like squares and rectangles, the diagonals always form right-angled triangles with the sides, making Pythagoras' theorem perfect for finding their lengths.

Name:
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More Complex Geometric Applications

Pythagoras' theorem helps us find crucial measurements in various geometric figures, like the height of triangles or distances in compound shapes.

In question 9, we need to find the perpendicular height of an isosceles triangle. Using Pythagoras, the distance from the base to the height is x² = 10² - 4² = 100 - 16 = 84, giving a height of 9.17 (3sf).

Question 10 involves finding the diagonal AC in a trapezium. We first calculate the length of one part using Pythagoras: x² = 13² - 5² = 169 - 25 = 144, so x = 12. Then we find AC using the second right-angled triangle: AC² = 8² + 12² = 64 + 144 = 208, giving AC = 14.4 (3sf).

Remember: When working with isosceles triangles, the perpendicular height bisects the base, creating two equal right-angled triangles—perfect for applying Pythagoras' theorem!

Name:
Instructions
●
• Use black ink or ball-point pen.
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Real-World Applications

Pythagoras' theorem has many practical applications in real life, from navigation to construction problems.

In question 11, a ship sails 3.7 km North and then 2.4 km East. To find the direct distance between the starting and ending points, we use Pythagoras: x² = 2.4² + 3.7² = 5.76 + 13.69 = 19.45, giving a direct distance of 4.4 km (1dp).

Question 12 presents a classic ladder problem. With the ladder reaching 2.5 m (or 250 cm) up a wall and its base 70 cm from the wall, we can find the ladder's length using Pythagoras: x² = 70² + 250² = 4,900 + 62,500 = 67,400, giving a ladder length of 260 cm.

Real-life connection: Engineers and architects regularly use Pythagoras' theorem when designing structures, planning layouts, or calculating distances that can't be directly measured.

Name:
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●
• Use black ink or ball-point pen.
●
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Special Applications

Pythagoras' theorem can solve interesting problems involving special geometric relationships and real-world scenarios.

In question 13, we need to find the perimeter of a square when only given its diagonal (8 m). Since the diagonal creates a right-angled triangle with two equal sides, we use Pythagoras: x² + x² = 8², so 2x² = 64, making x = √32 ≈ 5.66 m. The perimeter is 4 times this length: 4 × 5.66 = 22.6 m (1dp).

Question 14 involves finding the dimensions of a TV with a 50-inch diagonal and a 4:3 aspect ratio. We can write the sides as 4x and 3x, then apply Pythagoras: (4x)² + (3x)² = 50². This gives us 16x² + 9x² = 2500, so 25x² = 2500, making x = 10. Therefore, the length is 4 × 10 = 40 inches and the width is 3 × 10 = 30 inches.

Cool fact: TV and monitor sizes are always measured diagonally! You can use Pythagoras' theorem to work out the actual screen dimensions when shopping for a new device.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.

Where can I download the Knowunity app?

You can download the app from Google Play Store and Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

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4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user