Projection Calculator
Right Triangle Projection p Calculator
Use this calculator when you know leg a and the hypotenuse c, and you need the projection segment p on the hypotenuse. It focuses on one projection theorem only, so the inputs stay simple.
Calculate Right Triangle Projection p
This calculator finds Projection p using .
Enter inputs to calculate Projection p.
Projection p
Result-
Solution Steps
Formula:
How This Right Triangle Projection p Calculator Works
Projection p is the part of the hypotenuse connected with leg a. In many geometry problems, this small segment is the missing piece needed before finding altitude, the other projection, or checking similar-triangle work.
Enter positive values for leg a and hypotenuse c. Since c is always the hypotenuse, c must be greater than a.
Known values
Leg a and hypotenuse c
Finds
Projection p, the hypotenuse segment paired with leg a
Main formula
p = a² / c
Best for
Finding the p segment without solving the full triangle
Right Triangle Projection p Formula
The projection p is found by squaring leg a and dividing by the full hypotenuse c. This comes from the right-triangle projection theorem.
Use this formula only when a is the leg connected to projection p. If your known leg is b, use the projection q calculator instead.
Triangle Diagram: Projection p on the Hypotenuse
The diagram marks the altitude h dropped to the hypotenuse c. That altitude divides c into projection p beside leg a and projection q beside leg b.
Right triangle with leg a, leg b, hypotenuse c, altitude h, and hypotenuse projections p and q.
Diagram Key
a = leg paired with p
Leg a is the known leg used in this calculator.
b = other leg
Leg b is shown for orientation, but it is not entered here.
c = full hypotenuse
c is the longest side and must be greater than leg a.
p = projection result
p is the segment on the hypotenuse associated with leg a.
q = other projection
q is the remaining hypotenuse segment.
h = altitude
h divides the hypotenuse into p and q.
- Projection p lies on the hypotenuse, not on leg a.
- The full hypotenuse is c, and the two pieces satisfy c = p + q.
- For this calculator, leg a and hypotenuse c are the known values.
How to Use This Calculator
- Identify leg a, the leg connected with projection p.
- Find the hypotenuse c, which is the longest side of the right triangle.
- Enter leg a in the first field.
- Enter hypotenuse c in the second field.
- Calculate the result and read projection p in the output box.
- Check that p is smaller than c, because p is only one part of the hypotenuse.
Worked Example: Find Projection p From a = 6 and c = 10
Suppose leg a = 6 and hypotenuse c = 10.
Projection p is 3.6 units. That means the altitude touches the hypotenuse at a point 3.6 units along the side associated with leg a.
What the Result Means
The result is a length on the hypotenuse, not a new side outside the triangle.
A smaller p means the foot of the altitude is closer to the end of the hypotenuse near leg a. A larger p means that endpoint takes up more of the hypotenuse.
If you also know q, the two projection segments should add to c.
When to Use This Calculator
This method is useful when your known values match this projection relation and you want a direct result.
Common situations where this calculator helps:
- You know leg a and hypotenuse c, but not projection p.
- You are working through a projection theorem problem in geometry.
- You need p before using h² = p × q or c = p + q.
- You want to check whether a triangle diagram has the projection labels placed correctly.
Why This Formula Works
Dropping altitude h to the hypotenuse creates smaller right triangles that are similar to the original triangle.
From those similar triangles, leg a is the geometric mean between the full hypotenuse c and projection p. That gives a² = c × p, and solving for p gives p = a² / c.
Common Mistakes
Projection formulas are short, but it is easy to use the wrong segment or stop one step early. Check these points before trusting the result.
- Using leg b instead of leg a.
- Forgetting to square a before dividing.
- Dividing c by a² instead of dividing a² by c.
- Treating c as a leg even though c is the hypotenuse.
- Entering a value for a that is greater than or equal to c.
Additional Example: Projection p With Decimal Result
Suppose leg a = 7 and hypotenuse c = 14.
- p = 7² / 14
- p = 49 / 14
- p = 3.5 units
- Projection p takes 3.5 units of the hypotenuse.
Frequently Asked Questions
Answers to common questions about right-triangle projection calculations.
01 What is the formula for projection p? expand_more
The formula is p = a² / c. Square leg a, then divide by the hypotenuse c.
02 What does projection p mean in a right triangle? expand_more
Projection p is the part of the hypotenuse associated with leg a. It is created when the altitude from the right angle meets the hypotenuse.
03 Can projection p be longer than the hypotenuse? expand_more
No. Projection p is only one segment of the hypotenuse, so it must be smaller than c.
04 Should I use a or b to find p? expand_more
Use leg a to find p. If you know leg b instead, use q = b² / c.
05 Why does the calculator require c to be greater than a? expand_more
Because c is the hypotenuse and must be longer than either leg in a valid right triangle.