Area of a Right Triangle Calculator
The Area of a Right Triangle Calculator is an online tool that helps you calculate the area of any right triangle—or even solve for missing sides, angles, perimeter, 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 area (A), side a, side b, hypotenuse (c), 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)
Find: Area, Perimeter, Missing Side, Both Angles, Altitude
2. Area and One Side
- Area & a
- Area & b
- Area & c
Find: Other Sides, Altitude, Both Angles, Perimeter
3. Angle and One Side
- a & ∠α
- a & ∠β
- b & ∠α
- b & ∠β
- c & ∠α
- c & ∠β
Find: Area, Other Sides, Perimeter, Altitude, Other Angles
4. Perimeter and One Side
- Perimeter & a
- Perimeter & b
- Perimeter & c
Find: Area, Other Sides, Angles, Altitude
5. Perimeter and Area
- Perimeter & Area
Find: All Sides, Both Angles, Altitude, Hypotenuse
6. Altitude and One Side
- a & Altitude (h)
- b & Altitude (h)
- c & Altitude (h)
Find: Area, Other Sides, Perimeter, Both Angles
7. Altitude and Area or Perimeter
- Area & Altitude (h)
- Perimeter & Altitude (h)
Find: Sides, Angles, Hypotenuse, Area, Perimeter
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 Area of a Right Triangle Calculator is help 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 Area of a Right Triangle Calculator
This calculator allows you to find the area 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 area 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.
Area of a Right Triangle Calculator
Important Formulas Used in This Calculator
The Area of a Right Triangle 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 \( c \)
$P^2 = 2c(P + h)$
$\Rightarrow \boxed{c = \frac{P^2}{2(P + h)}}$
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$
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$
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$
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 \( a \))
$\beta = 90^\circ – \alpha = 36.87^\circ$
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$
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$
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$
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 \( h \)
$h = \frac{3 \cdot 4}{5} = 2.4$
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 \( h \)
$h = 2.4$
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 \( h \)
$h = 2.4$
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 \( h \)
$h = \frac{3 \cdot 4}{5} = 2.4$
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 \( h \)
$h = \frac{3 \cdot 4}{5} = 2.4$
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 $
\( \beta = 90^\circ – \alpha = 36.87^\circ \)
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$
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 $
$\beta = 90^\circ – \alpha = 36.87^\circ$
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
\( \alpha = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \)
\( \beta = 90^\circ – \alpha = 36.87^\circ \)
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$
$\beta = 90^\circ – \alpha = 36.87^\circ$
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
$\alpha = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$
$\beta = 90^\circ – \alpha = 36.87^\circ$
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
\( \alpha = 53.13^\circ \), \( \beta = 36.87^\circ \)
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
\( \alpha = 53.13^\circ \), \( \beta = 36.87^\circ \)
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 Area of a Right Triangle Calculator
Q1. What does the Area of a Right Triangle Calculator do?
The Area of a Right Triangle Calculator calculates the area of a right-angled triangle when you provide any two of the following values: base, height, or hypotenuse. It uses the standard formula \( A = \frac{1}{2} \times \text{base} \times \text{height} \).
Q2. What inputs do I need to use this calculator?
You need to enter any two known parts of the triangle—such as base and height, or one leg and the hypotenuse—depending on what is known. The calculator will then compute the area and any missing sides if required.
Q3. Can I calculate the area if I only know the hypotenuse and one side?
Yes. If you know the hypotenuse and one leg, the calculator uses the Pythagorean Theorem to determine the other leg. Then it calculates the area using the right triangle area formula.
Q4. Is this calculator only for right triangles?
Yes, this calculator is specifically designed for right-angled triangles. If you’re working with other types of triangles—like scalene, isosceles, or obtuse—you should use a general triangle area calculator such as the Heron’s Formula Calculator.
Q5. Can I use this calculator for solving academic problems or homework?
Absolutely! This tool is helpful for students, educators, and professionals who need to solve or verify problems involving the area of right-angled triangles quickly and accurately.
Q6. What is the formula used by this calculator?
The primary formula used is:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
When required, it also uses the Pythagorean Theorem:
$ a^2 + b^2 = c^2 $
to compute unknown side lengths.
Q7. What units does the calculator support?
The calculator supports any consistent units. You can input values in meters, feet, inches, etc. The resulting area will be given in corresponding square units (e.g., m², ft²).
Q8. Can I find the perimeter too using this calculator?
While the main focus is on area calculation, if enough data is provided, the calculator can also determine other properties such as missing side lengths and the perimeter.
Final Thoughts: The Area of a Right Triangle Calculator is help 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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