Hypotenuse Calculator
The Hypotenuse Calculator is an online tool that helps you calculate the hypotenuse of any right triangle—or even solve for missing sides, angles, perimeter, area, or altitude—based on a variety of input combinations. Whether you know the legs, hypotenuse, area, angle, or even the perimeter, this calculator adapts to your input and delivers accurate results instantly.
This calculator uses right triangle formulas to find the hypotenuse (c), side a, side b, area (A), perimeter (P), altitude (h), and angles α and β, depending on what values you enter. It’s designed for students, teachers, engineers, and anyone needing fast and accurate right triangle calculations.
Accepted Input Combinations
You can use a wide variety of input pairs to calculate the missing values in a right triangle. Below are the supported input combinations and what the calculator can solve for:
1. Two Sides Given
- a & b (Legs)
- a & c (Leg & Hypotenuse)
- b & c (Leg & Hypotenuse)
2. Area and One Side
- Area & a
- Area & b
- Area & c
3. Angle and One Side
- a & ∠α
- a & ∠β
- b & ∠α
- b & ∠β
- c & ∠α
- c & ∠β
4. Perimeter and One Side
- Perimeter & a
- Perimeter & b
- Perimeter & c
5. Perimeter and Area
- Perimeter & Area
6. Altitude and One Side
- a & Altitude (h)
- b & Altitude (h)
- c & Altitude (h)
7. Altitude and Area or Perimeter
- Area & Altitude (h)
- Perimeter & Altitude (h)
Outputs Provided by the Calculator
Based on your selected input, the Area of a Right Triangle Calculator will compute and display:
- Side a (Perpendicular)
- Side b (Base)
- Hypotenuse (c)
- Area (A)
- Perimeter (P)
- Altitude (h)
- Angle α (opposite side a)
- Angle β (opposite side b)
Note: The Hypotenuse Calculator helps to calculate any unknown value in a right-angled triangle—be it the perpendicular (a), base (B), hypotenuse (C), area (A), perimeter (P), angle α (Alpha), angle β (Beta), or altitude (H). By simply providing any two known values, the calculator intelligently applies geometric and trigonometric formulas to solve for the rest. It supports multiple units for length—meters (m), centimeters (cm), millimeters (mm), yards (yd), feet (ft), and inches (in)—as well as angle units in both degrees and radians.
How to Use the Hypotenuse Calculator
This calculator allows you to find the hypotenuse of a right triangle or solve for an unknown side or angle or perimeter or altitude or angle α or angle β using various combinations of inputs. Follow these steps to get started:
1. Select Your Input Type: Use the dropdown menu labeled “Given” to choose one of the following options:
- a & b (Legs)
- a & c (Leg & Hypotenuse)
- b & c (Leg & Hypotenuse)
- Area & a
- Area & b
- Area & c
- a & ∠α
- a & ∠β
- b & ∠α
- b & ∠β
- c & ∠α
- c & ∠β
- Perimeter & a
- Perimeter & b
- Perimeter & c
- Perimeter & Area
- a & Altitude (h)
- b & Altitude (h)
- c & Altitude (h)
- Area & Altitude (h)
- Perimeter & Altitude (h)
2. Enter Known Values: In this calculator, you can calculate key properties of a right triangle such as the perpendicular (a), base (b), hypotenuse (c), area, perimeter, angle α, angle β, and altitude (h). The calculator works by allowing you to enter any two known values, and it will automatically compute the remaining quantities with accurate results.
3. Choose Decimal Precision:
- Select how many decimal places you want the result to be rounded to (e.g., 3)
- Select Units:
- Length Units: Meters (m), Centimeters (cm), Millimeters (mm), Yards (yd), Feet (ft), Inches (in)
- Angle Units: Degrees (°), Radians (rad)
4. Click “Calculate”: The calculator will compute the hypotenuse and all other unknown.
5. Hide Steps: Toggle this option to hide or show detailed calculation steps.
6. Reset: Clear all inputs to start a new calculation.
Hypotenuse Calculator
Important Formulas Used in This Hypotenuse Calculator
The Hypotenuse Calculator uses a variety of geometric and trigonometric formulas to calculate all the missing values of a right triangle based on the two inputs you provide. Below are the key formulas this calculator applies:
Area of a Right Triangle
$A = \frac{1}{2} \times a \times b$
This formula calculates the area of a right-angled triangle where $a$ and $b$ are the two perpendicular sides (base and height).
Pythagorean Theorem
$c = \sqrt{a^2 + b^2}$
This determines the hypotenuse $c$ in a right triangle when the two legs $a$ and $b$ are known.
Perimeter of Triangle
$P = a + b + c$
The perimeter is the sum of all three sides of the triangle: $a$, $b$, and hypotenuse $c$.
Altitude from Hypotenuse
$h = \frac{a \times b}{c}$
This formula gives the height ($h$) dropped from the right angle to the hypotenuse $c$, based on the two perpendicular sides $a$ and $b$.
Angle $\alpha$ (opposite side $a$)
$\alpha = \arcsin\left(\frac{a}{c}\right)$ or $\alpha = \arctan\left(\frac{a}{b}\right)$
These equations calculate angle $\alpha$ using either sine or tangent, depending on which sides are known.
Angle $\beta$ (opposite side $b$)
$\beta = 90^\circ – \alpha$
Since the triangle is right-angled, the two other angles must add up to $90^\circ$. Subtracting $\alpha$ gives angle $\beta$.
Side $a$ (from Area and $b$)
$a = \frac{2A}{b}$
If the area $A$ and one side $b$ are known, this formula solves for side $a$.
Side $b$ (from Area and $a$)
$b = \frac{2A}{a}$
This calculates side $b$ when the area and side $a$ are given.
Derivation of the Formula
\( c = \sqrt{ \frac{a^2}{1 – \frac{h^2}{a^2}} } \)
Step 1: Use the formula for the altitude from the right angle to the hypotenuse:
\( h = \frac{a \cdot b}{c} \)
Step 2: Use the Pythagorean theorem to express \( b \) in terms of \( c \) and \( a \):
\( b^2 = c^2 – a^2 \Rightarrow b = \sqrt{c^2 – a^2} \)
Step 3: Substitute this into the altitude formula:
\( h = \frac{a \cdot \sqrt{c^2 – a^2}}{c} \)
Step 4: Square both sides to eliminate the square root:
$ h^2 $$= \left( \frac{a \cdot \sqrt{c^2 – a^2}}{c} \right)^2 $$= \frac{a^2 (c^2 – a^2)}{c^2} $
Step 5: Multiply both sides by \( c^2 \):
\( h^2 c^2 = a^2(c^2 – a^2) \)
Step 6: Expand and rearrange:
\( h^2 c^2 = a^2 c^2 – a^4 \)
$ h^2 c^2 – a^2 c^2 = -a^4 $
$\Rightarrow c^2(h^2 – a^2) = -a^4$
Multiply both sides by -1:
\( c^2(a^2 – h^2) = a^4 \)
Step 7: Solve for \( c \):
\( c^2 = \frac{a^4}{a^2 – h^2} \Rightarrow c = \sqrt{ \frac{a^4}{a^2 – h^2} } \)
Now simplify:
\( c = \sqrt{ \frac{a^2}{1 – \frac{h^2}{a^2}} } \)
Final Formula:
$ \boxed{c = \sqrt{ \frac{a^2}{1 – \frac{h^2}{a^2}} } } $
Similer:
$ \boxed{c = \sqrt{ \frac{b^2}{1 – \frac{h^2}{b^2}} } } $
Derivation of the Formula
\( c = \frac{P^2}{2(P + h)} \)
Let the sides of a right triangle be:
- \( a \), \( b \): the legs
- \( c \): the hypotenuse
- \( h \): the altitude to hypotenuse
- \( P \): the perimeter, \( P = a + b + c \)
Step 1: Express sum of legs
From the perimeter formula:
$a + b = P – c \tag{1}$
Step 2: Use area identity involving altitude
The altitude to the hypotenuse is given by:
$h = \frac{ab}{c} \Rightarrow ab = ch \tag{2}$
Step 3: Use identity for squared sum
$(a + b)^2 = a^2 + b^2 + 2ab$
From Pythagoras: \( a^2 + b^2 = c^2 \)
So:
$(a + b)^2 = c^2 + 2ab \tag{3}$
Step 4: Substitute known expressions
From (1): \( a + b = P – c \)
From (2): \( ab = ch \)
So substitute into (3):
$(P – c)^2$$ = c^2 + 2ch$
Step 5: Expand and simplify
$P^2 – 2Pc + c^2 $$= c^2 + 2ch$
Subtract \( c^2 \) from both sides:
$P^2 – 2Pc = 2ch$
Step 6: Solve for \( c \)
$P^2 = 2c(P + h)$
$\Rightarrow \boxed{c = \frac{P^2}{2(P + h)}}$
Hypotenuse Calculator Example.1 : Find All Remaining Parts of a Right Triangle Given \(a = 3\) and \(b = 4\)
Given:
\( a = 3 \) units (one leg)
\( b = 4 \) units (other leg)
Find: hypotenuse \( c \), area \( A \), perimeter \( P \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Find the hypotenuse \( c \) using Pythagorean theorem
$c = \sqrt{a^2 + b^2} $$= \sqrt{3^2 + 4^2} $$= \sqrt{9 + 16} $$= \sqrt{25} = 5$
Step 2: Find the area \( A \)
$A = \frac{1}{2} \times a \times b $$= \frac{1}{2} \times 3 \times 4 = 6$
Step 3: Find the perimeter \( P \)
$P = a + b + c $$= 3 + 4 + 5 = 12$
Step 4: Find the altitude \( h \) from the right angle to hypotenuse
$h = \frac{a \times b}{c} $$= \frac{3 \times 4}{5} $$= \frac{12}{5} = 2.4$
Step 5: Find angle \( \alpha \) (opposite side \( b \)) using tan⁻¹
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) $$= \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
Step 6: Find angle \( \beta \) (opposite side \( a \))
$\beta = 90^\circ – \alpha $$= 90^\circ – 53.13^\circ $$= 36.87^\circ$
Hypotenuse Calculator Example.2 : Find All Remaining Parts of a Right Triangle Given \(a = 3\) and \(c = 5\)
Given:
\( a = 3 \) units (one leg)
\( c = 5 \) units (hypotenuse)
Find:
other leg \( b \), area \( A \), perimeter \( P \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Find the other leg \( b \) using Pythagorean theorem
$b = \sqrt{c^2 – a^2} $$= \sqrt{5^2 – 3^2} $$= \sqrt{25 – 9} $$= \sqrt{16} = 4$
Step 2: Find the area \( A \)
$A = \frac{1}{2} \times a \times b $$= \frac{1}{2} \times 3 \times 4 $$= 6$
Step 3: Find the perimeter \( P \)
$P = a + b + c $$= 3 + 4 + 5 = 12$
Step 4: Find the altitude \( h \) from the right angle to hypotenuse
$h = \frac{a \times b}{c} $$= \frac{3 \times 4}{5} $$= \frac{12}{5} = 2.4$
Step 5: Find angle \( \alpha \) (opposite side \( b \)) using \(\tan^{-1}\)
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) $$= \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
Step 6: Find angle \( \beta \) (opposite side \( a \))
$\beta = 90^\circ – \alpha $$= 90^\circ – 53.13^\circ $$= 36.87^\circ$
Hypotenuse Calculator Example.3 : Find All Remaining Parts of a Right Triangle Given \(b = 4\) and \(c = 5\)
Given:
\( b = 4 \) units (one leg)
\( c = 5 \) units (hypotenuse)
Find:
other leg \( a \), area \( A \), perimeter \( P \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Find the other leg \( a \) using Pythagorean theorem
$a = \sqrt{c^2 – b^2} $$= \sqrt{5^2 – 4^2} $$= \sqrt{25 – 16} $$= \sqrt{9} = 3$
Step 2: Find the area \( A \)
$A = \frac{1}{2} \times a \times b $$= \frac{1}{2} \times 3 \times 4 = 6$
Step 3: Find the perimeter \( P \)
$P = a + b + c $$= 3 + 4 + 5 = 12$
Step 4: Find the altitude \( h \) from the right angle to hypotenuse
$h = \frac{a \times b}{c} $$= \frac{3 \times 4}{5} $$= \frac{12}{5} = 2.4$
Step 5: Find angle \( \alpha \) (opposite side \( b \)) using \(\tan^{-1}\)
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) $$= \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
Step 6: Find angle \( \beta \) (opposite side \( a \))
$\beta = 90^\circ – \alpha$$ = 90^\circ – 53.13^\circ$$ = 36.87^\circ$
Hypotenuse Calculator Example.4 : Find All Remaining Parts of a Right Triangle Given Area \( A = 6 \) and side \( a = 3 \)
Given:
\( A = 6 \) square units (area)
\( a = 3 \) units (one leg)
Find:
other leg \( b \), hypotenuse \( c \), perimeter \( P \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Find leg \( b \) using area formula
$A = \frac{1}{2} \times a \times b \Rightarrow b = \frac{2A}{a} = \frac{2 \times 6}{3} = \frac{12}{3} = 4$
Step 2: Find the hypotenuse \( c \)
$c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
Step 3: Find the perimeter \( P \)
$P = a + b + c = 3 + 4 + 5 = 12$
Step 4: Find the altitude \( h \)
$h = \frac{a \times b}{c} = \frac{3 \times 4}{5} = \frac{12}{5} = 2.4$
Step 5: Find angle \( \alpha \) (opposite side \( b \))
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
Step 6: Find angle \( \beta \) (opposite side \( a \))
$\beta = 90^\circ – \alpha = 36.87^\circ$
Hypotenuse Calculator Example.5 : Find All Remaining Parts of a Right Triangle Given Area \( A = 6 \) and side \( b = 4 \)
Given:
\( A = 6 \) square units (area)
\( b = 4 \) units (one leg)
Find:
other leg \( a \), hypotenuse \( c \), perimeter \( P \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Find leg \( a \) using area formula
$A = \frac{1}{2} \times a \times b \Rightarrow a $$= \frac{2A}{b} = \frac{2 \times 6}{4} $$= \frac{12}{4} = 3$
Step 2: Find the hypotenuse \( c \)
$c = \sqrt{a^2 + b^2} $$= \sqrt{3^2 + 4^2} $$= \sqrt{9 + 16} $$= \sqrt{25} = 5$
Step 3: Find the perimeter \( P \)
$P = a + b + c $$= 3 + 4 + 5 = 12$
Step 4: Find the altitude \( h \)
$h = \frac{a \times b}{c} $$= \frac{3 \times 4}{5} $$= \frac{12}{5} = 2.4$
Step 5: Find angle \( \alpha \) (opposite side \( b \))
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) $$= \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
Step 6: Find angle \( \beta \) (opposite side \( a \))
$\beta = 90^\circ – \alpha $$= 36.87^\circ$
Hypotenuse Calculator Example.6 : Find All Remaining Parts of a Right Triangle Given Area \( A = 6 \) and hypotenuse \( c = 5 \)
Given:
\( A = 6 \) square units (area)
\( c = 5 \) units (hypotenuse)
Find:
legs \( a \) and \( b \), perimeter \( P \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Use identity \( A = \frac{a \times b}{2} \) and \( a^2 + b^2 = c^2 \)
We solve:
$a \times b $$= 12 \quad \text{(from area)}$
$a^2 + b^2 $$= 25 \quad \text{(from Pythagoras)}$
Solving these equations gives \( a = 3 \), \( b = 4 \)
Step 2: Find the perimeter \( P \)
$P = a + b + c $$= 3 + 4 + 5 = 12$
Step 3: Find the altitude \( h \)
$h = \frac{a \times b}{c} $$= \frac{3 \times 4}{5} = 2.4$
Step 4: Find angle \( \alpha \) (opposite side \( b \))
$\alpha$$ = \tan^{-1}\left(\frac{b}{a}\right) $$= \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
Step 5: Find angle \( \beta \) (opposite side \( a \))
$\beta = 90^\circ – \alpha $$= 36.87^\circ$
Hypotenuse Calculator Example.7 : Find All Remaining Parts of a Right Triangle Given side \( a = 3 \) and angle \( \alpha = 36.87^\circ \)
Given:
\( a = 3 \) units (one leg)
\( \alpha = 36.87^\circ \) (angle opposite side \( a \))
Find:
other leg \( b \), hypotenuse \( c \), area \( A \), perimeter \( P \), altitude \( h \), angle \( \beta \)
Step 1: Find angle \( \beta \)
$\beta = 90^\circ – \alpha $$= 90^\circ – 36.87^\circ $$= 53.13^\circ$
Step 2: Find side \( b \) using tangent
$\tan(\beta) = \frac{b}{a} \Rightarrow b $$= a \cdot \tan(\beta) $$= 3 \cdot \tan(53.13^\circ) \approx 3 \cdot 1.333 = 4$
Step 3: Find hypotenuse \( c \)
$c = \sqrt{a^2 + b^2} $$= \sqrt{3^2 + 4^2} $$= \sqrt{25} = 5$
Step 4: Find area \( A \)
$A = \frac{1}{2} \cdot a \cdot b $$= \frac{1}{2} \cdot 3 \cdot 4 = 6$
Step 5: Find perimeter \( P \)
$P = a + b + c $$= 3 + 4 + 5 $$= 12$
Step 6: Find altitude \( h \)
$h = \frac{a \cdot b}{c} $$= \frac{12}{5} = 2.4$
Hypotenuse Calculator Example.8 : Find All Remaining Parts of a Right Triangle Given side \( a = 3 \) and angle \( \beta = 53.13^\circ \)
Given:
\( a = 3 \) units (adjacent to angle \( \beta \))
\( \beta = 53.13^\circ \)
Find:
other leg \( b \), hypotenuse \( c \), area \( A \), perimeter \( P \), altitude \( h \), angle \( \alpha \)
Step 1: Find angle \( \alpha \)
$\alpha = 90^\circ – \beta $$= 90^\circ – 53.13^\circ $$= 36.87^\circ$
Step 2: Find side \( b \)
$\tan(\beta) = \frac{b}{a} \Rightarrow b $$= a \cdot \tan(53.13^\circ) $$= 3 \cdot 1.333 = 4$
Step 3: Find hypotenuse \( c \)
$c = \sqrt{a^2 + b^2} $$= \sqrt{9 + 16} = 5$
Step 4: Find area \( A \)
$A = \frac{1}{2} \cdot a \cdot b = 6$
Step 5: Find perimeter \( P \)
$P = 3 + 4 + 5 = 12$
Step 6: Find altitude \( h \)
$h = \frac{3 \cdot 4}{5} = 2.4$
Hypotenuse Calculator Example.9 : Find All Remaining Parts of a Right Triangle Given side \( b = 4 \) and angle \( \alpha = 36.87^\circ \)
Given:
\( b = 4 \) units (opposite angle \( \alpha \))
\( \alpha = 36.87^\circ \)
Find:
side \( a \), hypotenuse \( c \), area \( A \), perimeter \( P \), altitude \( h \), angle \( \beta \)
Step 1: Find angle \( \beta \)
$\beta = 90^\circ – \alpha $$= 53.13^\circ$
Step 2: Find side \( a \)
$\tan(\alpha) = \frac{b}{a} \Rightarrow a $$= \frac{b}{\tan(\alpha)} $$= \frac{4}{0.75} = 3$
Step 3: Find hypotenuse \( c \)
$c = \sqrt{a^2 + b^2} $$= 5$
Step 4: Find area \( A \)
$A = \frac{1}{2} \cdot a \cdot b $$= 6$
Step 5: Find perimeter \( P \)
$P = 3 + 4 + 5 $$= 12$
Step 6: Find altitude \( h \)
$h = 2.4$
Hypotenuse Calculator Example.10 : Find All Remaining Parts of a Right Triangle Given side \( b = 4 \) and angle \( \beta = 53.13^\circ \)
Given:
\( b = 4 \) units
\( \beta = 53.13^\circ \)
Find:
side \( a \), hypotenuse \( c \), area \( A \), perimeter \( P \), altitude \( h \), angle \( \alpha \)
Step 1: Find angle \( \alpha \)
$\alpha = 90^\circ – \beta $$= 36.87^\circ$
Step 2: Find side \( a \)
$\tan(\alpha) = \frac{b}{a} \Rightarrow a $$= \frac{b}{\tan(\alpha)} $$= \frac{4}{0.75} = 3$
Step 3: Find hypotenuse \( c \)
$c = \sqrt{a^2 + b^2}$$ = 5$
Step 4: Find area \( A \)
$A = \frac{1}{2} \cdot 3 \cdot 4 $$= 6$
Step 5: Find perimeter \( P \)
$P = 3 + 4 + 5 = 12$
Step 6: Find altitude \( h \)
$h = 2.4$
Hypotenuse Calculator Example.11 : Find All Remaining Parts of a Right Triangle Given hypotenuse \( c = 5 \) and angle \( \alpha = 36.87^\circ \)
Given:
\( c = 5 \) units
\( \alpha = 36.87^\circ \)
Find:
sides \( a \), \( b \), area \( A \), perimeter \( P \), altitude \( h \), angle \( \beta \)
Step 1: Find angle \( \beta \)
$\beta = 90^\circ – \alpha $$= 53.13^\circ$
Step 2: Use sine to find \( a \)
$\sin(\alpha) = \frac{a}{c} \Rightarrow a $$= c \cdot \sin(\alpha) $$= 5 \cdot 0.6 = 3$
Step 3: Use cosine to find \( b \)
$\cos(\alpha) = \frac{b}{c} \Rightarrow b $$= 5 \cdot 0.8 = 4$
Step 4: Find area \( A \)
$A = \frac{1}{2} \cdot 3 \cdot 4 $$= 6$
Step 5: Find perimeter \( P \)
$P = 3 + 4 + 5 = 12$
Step 6: Find altitude \( h \)
$h = \frac{3 \cdot 4}{5} = 2.4$
Hypotenuse Calculator Example.12 : Find All Remaining Parts of a Right Triangle Given hypotenuse \( c = 5 \) and angle \( \beta = 53.13^\circ \)
Given:
\( c = 5 \) units
\( \beta = 53.13^\circ \)
Find:
sides \( a \), \( b \), area \( A \), perimeter \( P \), altitude \( h \), angle \( \alpha \)
Step 1: Find angle \( \alpha \)
$\alpha = 90^\circ – \beta = 36.87^\circ$
Step 2: Use sine to find \( b \)
$\sin(\beta) = \frac{b}{c} \Rightarrow b $$= 5 \cdot \sin(53.13^\circ) = 5 \cdot 0.8 = 4$
Step 3: Use cosine to find \( a \)
$\cos(\beta) = \frac{a}{c} \Rightarrow a $$= 5 \cdot 0.6 = 3$
Step 4: Find area \( A \)
$A = \frac{1}{2} \cdot 3 \cdot 4 $$= 6$
Step 5: Find perimeter \( P \)
$P = 3 + 4 + 5 = 12$
Step 6: Find altitude \( h \)
$h = \frac{3 \cdot 4}{5} = 2.4$
Hypotenuse Calculator Example.13 : Find All Remaining Parts of a Right Triangle Given Perimeter \( P = 12 \) and side \( a = 3 \)
Given:
\( P = 12 \) units (perimeter)
\( a = 3 \) units (one leg)
Find:
other leg \( b \), hypotenuse \( c \), area \( A \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Use identity \( a + b + c = P \) and \( a^2 + b^2 = c^2 \)
Solve system:
- \( a + b + \sqrt{a^2 + b^2} = 12 \)
- Substitute \( a = 3 \), solve numerically to get \( b = 4 \), \( c = 5 \)
Step 2: Find area \( A \)
$ A = \frac{1}{2} \times a \times b $$= \frac{1}{2} \times 3 \times 4 = 6 $ square units
Step 3: Find altitude \( h \)
$ h = \frac{a \times b}{c} $$= \frac{3 \times 4}{5} = 2.4 $units
Step 4: Find angles
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) $$= \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ $
\( \beta = 90^\circ – \alpha = 36.87^\circ \)
Hypotenuse Calculator Example.14 : Find All Remaining Parts of a Right Triangle Given Perimeter \( P = 12 \) and side \( b = 4 \)
Given:
\( P = 12 \) units (perimeter)
\( b = 4 \) units (one leg)
Find:
other leg \( a \), hypotenuse \( c \), area \( A \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Use system \( b + a + \sqrt{a^2 + b^2} = 12 \)
Substitute \( b = 4 \), solve:
- \( a + 4 + \sqrt{a^2 + 16} = 12 \)
- Numerically solving gives \( a = 3 \), \( c = 5 \)
Step 2: Find area \( A \)
$ A = \frac{1}{2} \times a \times b $$= \frac{1}{2} \times 3 \times 4 = 6 $square units
Step 3: Find altitude \( h \)
\( h = \frac{3 \times 4}{5} = 2.4 \) units
Step 4: Find angles
$ \alpha = \tan^{-1}\left(\frac{b}{a}\right) \approx 53.13^\circ$
$ \beta = 90^\circ – \alpha = 36.87^\circ$
Hypotenuse Calculator Example.15 : Find All Remaining Parts of a Right Triangle Given Perimeter \( P = 12 \) and hypotenuse \( c = 5 \)
Given:
\( P = 12 \) units (perimeter)
\( c = 5 \) units (hypotenuse)
Find:
legs \( a \) and \( b \), area \( A \), altitude \( h \), angles \( \alpha \) and \( \beta \)
Step 1: Use system \( a + b + 5 = 12 \Rightarrow a + b = 7 \) and \( a^2 + b^2 = 25 \)
Solve:
- Substitute \( b = 7 – a \) into Pythagoras
- Solve: \( a^2 + (7 – a)^2 = 25 \Rightarrow a = 3 \), \( b = 4 \)
Step 2: Area
$ A = \frac{1}{2} \times a \times b$$ = \frac{1}{2} \times 3 \times 4 = 6$
Step 3: Altitude
$ h = \frac{3 \times 4}{5} = 2.4$
Step 4: Angles
$\alpha = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ $
$\beta = 90^\circ – \alpha = 36.87^\circ$
Hypotenuse Calculator Example.16 : Find All Remaining Parts of a Right Triangle Given Perimeter \( P = 12 \) and Area \( A = 6 \)
Given:
\( P = 12 \) units (perimeter)
\( A = 6 \) square units
Find:
sides \( a \), \( b \), \( c \), altitude \( h \), angles \( \alpha \), \( \beta \)
Step 1: Use system:
- \( a + b + \sqrt{a^2 + b^2} = 12 \)
- \( \frac{1}{2} \times a \times b = 6 \Rightarrow ab = 12 \)
Solving this system gives: \( a = 3 \), \( b = 4 \), \( c = 5 \)
Step 2: Altitude
$ h = \frac{ab}{c} = \frac{12}{5} = 2.4$
Step 3: Angles
\( \alpha = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \)
\( \beta = 90^\circ – \alpha = 36.87^\circ \)
Hypotenuse Calculator Example.17 : Find All Remaining Parts of a Right Triangle Given Side \( a = 3 \) and Altitude \( h = 2.4 \)
Given:
\( a = 3 \) units (leg)
\( h = 2.4 \) units (altitude to hypotenuse)
Find:
other side \( b \), hypotenuse \( c \), area \( A \), perimeter \( P \), angles \( \alpha \), \( \beta \)
Step 1: Use formula to find hypotenuse \( c \)
$c = \sqrt{\frac{a^2}{1 – \frac{h^2}{a^2}}} $$= \sqrt{\frac{9}{1 – \frac{5.76}{9}}} $$= \sqrt{\frac{9}{1 – 0.64}} $$= \sqrt{\frac{9}{0.36}} = \sqrt{25} = 5$
Step 2: Find side \( b \) using Pythagoras
$b $$= \sqrt{c^2 – a^2} = \sqrt{25 – 9} $$= \sqrt{16} = 4$
Step 3: Find Area \( A \)
$A = \frac{1}{2} \times a \times b $$= \frac{1}{2} \times 3 \times 4 = 6$
Step 4: Find Perimeter \( P \)
$P = a + b + c = 3 + 4 + 5 = 12$
Step 5: Find Angles
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) $$= \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
$\beta = 90^\circ – \alpha = 36.87^\circ$
Hypotenuse Calculator Example.18 : Find All Remaining Parts of a Right Triangle Given Side \( b = 4 \) and Altitude \( h = 2.4 \)
Given:
\( b = 4 \) units (leg)
\( h = 2.4 \) units (altitude to hypotenuse)
Find:
other side \( a \), hypotenuse \( c \), area \( A \), perimeter \( P \), angles \( \alpha \), \( \beta \)
Step 1: Use formula to find hypotenuse \( c \)
$c = \sqrt{\frac{b^2}{1 – \frac{h^2}{b^2}}} $$= \sqrt{\frac{16}{1 – \frac{5.76}{16}}} $$= \sqrt{\frac{16}{1 – 0.36}} $$= \sqrt{\frac{16}{0.64}} $$= \sqrt{25} = 5$
Step 2: Find side \( a \) using Pythagoras
$a = \sqrt{c^2 – b^2} = \sqrt{25 – 16} = \sqrt{9} = 3$
Step 3: Find Area \( A \)
$A = \frac{1}{2} \times a \times b = \frac{1}{2} \times 3 \times 4 = 6$
Step 4: Find Perimeter \( P \)
$P = a + b + c = 3 + 4 + 5 = 12$
Step 5: Find Angles
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
$\beta = 90^\circ – \alpha = 36.87^\circ$
Hypotenuse Calculator Example.19 : Find All Remaining Parts of a Right Triangle Given Hypotenuse \( c = 5 \) and Altitude \( h = 2.4 \)
Given:
\( c = 5 \) units
\( h = 2.4 \) units
Find:
sides \( a \), \( b \), area \( A \), perimeter \( P \), angles \( \alpha \), \( \beta \)
Step 1: Use $ A = \frac{1}{2} \times c \times h $$= \frac{1}{2} \times 5 \times 2.4 = 6$
Then use \( ab = 12 \), and \( a^2 + b^2 = 25 \)
Solving gives \( a = 3 \), \( b = 4 \)
Step 2: Perimeter
\( P = 3 + 4 + 5 = 12 \)
Step 3: Angles
\( \alpha = 53.13^\circ \), \( \beta = 36.87^\circ \)
Hypotenuse Calculator Example.20 : Find All Remaining Parts of a Right Triangle Given Area \( A = 6 \) and Altitude \( h = 2.4 \)
Given:
\( A = 6 \) square units
\( h = 2.4 \) units
Find:
sides \( a \), \( b \), \( c \), perimeter \( P \), angles \( \alpha \), \( \beta \)
Step 1: Use \( A = \frac{1}{2} \times c \times h \)
$\Rightarrow c = \frac{2A}{h} = \frac{12}{2.4} = 5 $
Now, \( ab = 12 \), and \( a^2 + b^2 = 25 \) → solving gives \( a = 3 \), \( b = 4 \)
Step 2: Perimeter
\( P = 3 + 4 + 5 = 12 \)
Step 3: Angles
\( \alpha = 53.13^\circ \), \( \beta = 36.87^\circ \)
Hypotenuse Calculator Example.21 : Find All Remaining Parts of a Right Triangle Given Altitude \( h = 2.4 \) and Perimeter \( P = 12 \)
Given:
Altitude \( h = 2.4 \) meters (from right angle to hypotenuse)
Perimeter \( P = 12 \) meters
Find:
Sides \( a \), \( b \), hypotenuse \( c \), area \( A \), angles \( \alpha \), \( \beta \)
Step 1: Use formula to find hypotenuse \( c \)
We use the formula:
$c = \frac{P^2}{2(P + h)}$$ = \frac{12^2}{2(12 + 2.4)} $$= \frac{144}{2 \times 14.4} $$= \frac{144}{28.8} = 5 $
Step 2: Use perimeter relation to find sum of legs
Since \( P = a + b + c \), and \( c = 5 \):
\( a + b = 12 – 5 = 7 \)
Let \( a = x \), then \( b = 7 – x \)
Step 3: Use the Pythagorean Theorem
\( a^2 + b^2 = c^2 = 25 \)
Substitute \( b = 7 – x \):
\( x^2 + (7 – x)^2 = 25 \)
\( x^2 + 49 – 14x + x^2 = 25 \)
\( 2x^2 – 14x + 49 = 25 \)
\( 2x^2 – 14x + 24 = 0 \)
Solve the quadratic equation:
\( x = 3 \Rightarrow a = 3 \), \( b = 4 \)
Step 4: Verify altitude
Use: $h = \frac{a \cdot b}{c} $$= \frac{3 \cdot 4}{5} $$= \frac{12}{5} = 2.4 $
Step 5: Calculate Area
$ A = \frac{1}{2} \cdot a \cdot b $$= \frac{1}{2} \cdot 3 \cdot 4 $$= 6 \, \text{sq. meters} $
Step 6: Calculate Angles
$\alpha = \tan^{-1} \left( \frac{b}{a} \right) $$= \tan^{-1} \left( \frac{4}{3} \right) \approx 53.13^\circ$
$\beta = 90^\circ – \alpha = 36.87^\circ $
Frequently Asked Questions (FAQs) on Hypotenuse Calculator
1. What is a Hypotenuse Calculator?
A Hypotenuse Calculator is an online tool used to calculate the longest side (hypotenuse) of a right-angled triangle using the Pythagorean Theorem. Just input the lengths of the other two sides (legs), and the calculator instantly computes the hypotenuse.
2. Which formula does this calculator use?
It uses the Pythagorean Theorem:
\[ c = \sqrt{a^2 + b^2} \]
Where:
\( a \) and \( b \) are the legs of the triangle,
\( c \) is the hypotenuse.
3. What units does the calculator support?
The calculator is unit-agnostic—it works with any unit (cm, m, inches, etc.) as long as both input sides use the same unit. The result will be in the same unit.
4. Can it solve for other sides or angles?
Some advanced hypotenuse calculators can also compute:
- Missing leg (if one leg and the hypotenuse are known)
- Angles (using trigonometric ratios)
- Area and perimeter of the triangle
- Altitude of the triangle
5. Is this calculator only for right-angled triangles?
Yes. The hypotenuse only exists in right-angled triangles—it is always opposite the right angle. For other triangle types, use a general triangle calculator.
6. Can I use decimals or fractions?
Yes, the calculator accepts both decimal and fractional inputs (e.g., 3.5 or 7/2) for higher precision.
7. Is it useful for real-life applications?
Absolutely! It’s widely used in fields like construction, architecture, carpentry, navigation, and any task involving right-angled measurements.
8. What if I enter negative values?
Negative values are not valid for side lengths. The calculator will display an error or prompt you to enter a positive number.
9. Can I use it on my phone or tablet?
Yes, most hypotenuse calculators are fully mobile-responsive and work seamlessly on smartphones, tablets, and any browser.
10. Do I need to download anything to use it?
No downloads are required. It’s a web-based tool—just visit the site, input your values, and get instant results.
Final Thoughts: The Hypotenuse Calculator helps to calculate any unknown value in a right-angled triangle—be it the perpendicular (a), base (B), hypotenuse (C), area (A), perimeter (P), angle α (Alpha), angle β (Beta), or altitude (H). By simply providing any two known values, the calculator intelligently applies geometric and trigonometric formulas to solve for the rest. It supports multiple units for length—meters (m), centimeters (cm), millimeters (mm), yards (yd), feet (ft), and inches (in)—as well as angle units in both degrees and radians. Whether you’re a student, teacher, engineer, or anyone working with triangles, this tool ensures fast, accurate, and unit-flexible results.
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