Of the base radius \(r\), the sphere radius \(R\) and the height \(h\) of the spherical cap (the dome shape cut from a sphere), enter only the two you know. The third is worked out, and then the area of the curved surface (\(2\pi Rh\)), the area of the base circle and their total are calculated.
Enter only two values (leave the remaining one blank). Lengths must be numbers greater than 0 (decimals are fine), all in the same unit (for example, all in cm). The area comes out in that unit squared (cm² for cm). If you enter \(r\) and \(R\), there may be two answers for the height \(h\) (a shallow cap and a deep cap).
Result and figure
Enter the two values you know out of the three in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formulas and figures
-
Curved surface area of the cap
-
Area of the base (the cut circle)
-
Total surface area (curved surface + base)
-
How to find the base radius \(r\) (when you know \(R\) and \(h\))
-
How to find the height \(h\) (when you know \(r\) and \(R\); two answers)
-
How to find the sphere radius \(R\) (when you know \(r\) and \(h\))
-
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
-
How to calculate it in Google Sheets
-
How to calculate it in Python
-
How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
-
DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Find the surface area of a spherical cap (the dome-like piece you get when you slice a sphere with one flat cut) on the spot
- Just enter the two you know out of the base radius \(r\), the sphere radius \(R\) and the height \(h\). The third is worked out automatically
- Shows the area of the curved surface (\(2\pi Rh\)), the area of the base (the cut circle) and their total, all with steps
- The resulting cap is also drawn in 3D (drag to rotate it)
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
The area comes out in the square of the length unit you enter (in² if you enter inches, ft² if you enter feet). This page is for the shape you get by cutting a sphere straight with a flat plane (so the cut is a circle). Dome roofs, bowls, lenses and floats sticking out of the water all have this shape.
What is this calculation used for?
Estimating the area of a planetarium or dome roof (architecture)
The round roofs of planetariums, observatories and domed stadiums are spherical caps. You can estimate how much roofing material or paint you need from the curved surface area \(2\pi Rh\).
For a dome 48 ft across (a base radius of 24 ft) and 12 ft tall, the original sphere has a radius of \(R = (12^2 + 24^2) \div (2 \times 12) = 30\) ft, and the roof area is \(2\pi \times 30 \times 12 \approx 2{,}262\,\mathrm{ft^2}\).
Measuring the curve of a lens (radius of curvature) (optics and eyeglasses)
The surface of an eyeglass lens or a contact lens is a spherical cap, and how curved it is is given by the radius of the original sphere (the radius of curvature). A spherometer, a tool used by lens makers, measures the width \(2r\) of the lens surface and how high it rises, \(h\), and then finds the radius of the original sphere with Formula 6 (\(R = (h^2 + r^2) \div 2h\)).
For example, if the surface rises 0.9 mm over a radius of 6 mm, the radius of curvature is \((0.9^2 + 6^2) \div 1.8 \approx 20.5\) mm (lens measurements are usually given in millimeters, even in the US).
Estimating how much coating a dome cake needs (cooking and baking)
When you pour chocolate or a mirror glaze over a dome cake (a hemisphere), the amount you need is proportional to the surface area. For a 6-inch hemisphere cake, \(R = h = 3\) in, so the curved surface area is \(2\pi \times 3 \times 3 \approx 56.5\,\mathrm{in^2}\).
It also helps with estimates such as "if I double the size of the mold, how much more coating do I need?" (about 4 times, because it is an area).
The area of the part of a buoy above the water (sea and boats)
When a round buoy floats, the part above the water is a spherical cap. If a buoy with a radius of 6 in sticks out 4 in above the water, the area exposed to the air is \(2\pi \times 6 \times 4 \approx 151\,\mathrm{in^2}\).
Subtract this from the surface area of the whole sphere, \(4\pi R^2 \approx 452\,\mathrm{in^2}\), and you find that the area under the water (the part that needs antifouling paint) is about \(302\,\mathrm{in^2}\).
How large an area a satellite signal covers (space and communications)
The area on the Earth that a satellite signal reaches (its footprint) can be treated as part of the Earth's surface, that is, a spherical cap. The covered area can be estimated as \(2\pi Rh\), using the Earth's radius \(R \approx 3{,}959\) mi and the depth \(h\) of the covered region (a real design also looks at how the signal strength is spread).
Planning a communications network, such as "how much wider does the coverage get if the satellite goes higher?", builds on the idea of this formula.
Formulas and figures
Curved surface area of the cap
Figure
Standard notation
(the usual math form)
\(S_1\)
\(=\)
\(2\)
\(\times\)
\(\pi\)
\(\times\)
\(R\)
\(\times\)
\(h\)
In words
(symbols replaced with words)
⑤
\(S_1\): curved surface area
\(=\)
④
the constant \(2\)
\(\times\)
③
\(\pi\): pi
\(\times\)
①
\(R\): sphere radius
\(\times\)
②
\(h\): cap height
The formula in words
①
Multiply the
\(R\): sphere radius
②
by the
\(h\): cap height
,
③
multiply by
\(\pi\): pi
,
④
and then by
the constant \(2\)
to get the
⑤
\(S_1\): curved surface area
Quick example
Cut a sphere with a radius of 10 in so that the cap is 5 in tall, measured from the top. The area of the curved surface of this cap is
\(S_1\): curved surface area
\(=\)
the constant \(2\)
\(\times\)
\(\pi\): pi
\(\times\)
sphere radius (10 in)
\(\times\)
height (5 in)
\(2 \times \pi \times 10 \times 5 = 100\pi \approx 314.2\,\mathrm{in^2}\)
Key idea
Surprisingly, the base radius \(r\) does not appear in this formula. The curved surface area of a cap depends only on the radius \(R\) of the original sphere and the height \(h\). What is more, \(2\pi Rh\) is exactly the lateral area of a cylinder with radius \(R\) and height \(h\). If you slice a sphere into layers of equal thickness, each layer has the same curved area, whether it is near the edge or in the middle. This beautiful fact was discovered by the ancient Greek mathematician Archimedes.
Put in \(h = 2R\) (the whole sphere) and you get \(2\pi R \times 2R = 4\pi R^2\), which matches the formula for the surface area of a sphere exactly.
Area of the base (the cut circle)
Figure
Standard notation
(the usual math form)
\(S_2\)
\(=\)
\(\pi\)
\(\times\)
\(r\)
\(2\)
In words
(symbols replaced with words)
④
\(S_2\): base circle area
\(=\)
③
\(\pi\): pi
\(\times\)
①
\(r\): base radius
②
squared (the number times itself)
The formula in words
①
Take the
\(r\): base radius
,
②
find its
square (the number times itself)
,
③
and multiply by
\(\pi\): pi
to get the
④
\(S_2\): base circle area
Quick example
For a cap whose base (the cut circle) has a radius of 3 in, the area of the base is
\(S_2\): base circle area
\(=\)
\(\pi\): pi
\(\times\)
base radius (3 in)
squared
\(\pi \times 3^{2} = 9\pi \approx 28.3\,\mathrm{in^2}\)
Key idea
This is simply the usual "area of a circle = pi × radius squared". Wherever you cut a sphere with a flat plane, the cut is always a circle, so the area of the base comes from the circle area formula.
Total surface area (curved surface + base)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(2\pi R h\)
\(+\)
\(\pi r^2\)
In words
(symbols replaced with words)
③
\(S\): total surface area
\(=\)
①
curved area \(2\pi Rh\)
\(+\)
②
base area \(\pi r^2\)
The formula in words
①
Add the
curved surface area \(2\pi Rh\) (Formula 1)
②
and the
base circle area \(\pi r^2\) (Formula 2)
to get the
③
\(S\): total surface area
Quick example
For a cap cut from a sphere with a radius of 10 in, with a height of 5 in (the base radius is \(\sqrt{75} \approx 8.66\) in), the total surface area is
\(S\): total surface area
\(=\)
curved area (\(100\pi\) in²)
\(+\)
base area (\(75\pi\) in²)
\(2\pi \times 10 \times 5 + \pi \times (\sqrt{75})^{2} = 100\pi + 75\pi\)
\(100\pi + 75\pi = 175\pi \approx 549.8\,\mathrm{in^2}\)
Key idea
The "total surface area" is the area of the whole outside of the solid cap cut from the sphere (the curved surface plus the cut circle).
When you only want the curved part, such as the area of a dome roof, use \(S_1 = 2\pi Rh\) from Formula 1 (the "Curved surface area of the cap" in the result).
How to find the base radius \(r\) (when you know \(R\) and \(h\))
Figure
Standard notation
(the usual math form)
\(r\)
\(=\)
\(\sqrt{2Rh - h^{2}}\)
In words
(symbols replaced with words)
②
\(r\): base radius
\(=\)
①
square root of \(2Rh - h^2\)
The formula in words
①
Find the
square root of \(2Rh - h^2\) (2 times the sphere radius times the height, minus the height squared)
to get the
②
\(r\): base radius
Quick example
For a cap cut from a sphere with a radius of 10 in, with a height of 5 in, the radius of the base (the cut circle) is
\(r\): base radius
\(=\)
square root of \(2 \times 10 \times 5 - 5^2\) (= 75)
\(r = \sqrt{2 \times 10 \times 5 - 5^{2}} = \sqrt{100 - 25}\)
\(r = \sqrt{75} \approx 8.66\,\mathrm{in}\)
Key idea
Connect three points: the center of the sphere, the center of the base circle and a point on the edge of the cut. They form a right triangle. Apply the Pythagorean theorem to it and you get \(r^2 = R^2 - (R-h)^2 = 2Rh - h^2\), which gives this formula.
The expression under the root can be rewritten as \(2Rh - h^2 = h(2R - h)\). So as long as the height \(h\) is no more than the diameter \(2R\), it is 0 or more, and the square root can always be taken.
How to find the height \(h\) (when you know \(r\) and \(R\); two answers)
Figure
Standard notation
(the usual math form)
\(h\)
\(=\)
\(R\)
\(\pm\)
\(\sqrt{R^{2} - r^{2}}\)
In words
(symbols replaced with words)
③
\(h\): cap height
\(=\)
①
\(R\): sphere radius
\(\pm\)
②
square root of \(R^2 - r^2\)
The formula in words
①
To the
\(R\): sphere radius
,
②
add or subtract the
square root of \(R^2 - r^2\) (the sphere radius squared minus the base radius squared)
(the two cases of ±). This gives the
③
\(h\): cap height
Quick example
When you cut a sphere with a radius of 2 ft so that the base has a radius of 1 ft, the height of the cap is
\(h\): cap height
\(=\)
sphere radius (2 ft)
\(\pm\)
square root of \(2^2 - 1^2\) (= 3)
\(h = 2 \pm \sqrt{2^{2} - 1^{2}} = 2 \pm \sqrt{3} \approx 2 \pm 1.732\)
\(h = 2 - \sqrt{3} \approx 0.268, \quad h = 2 + \sqrt{3} \approx 3.732\)
Key idea
There are two answers because the same size of cut circle can make a shallow cap (the small piece on the top side) or a deep cap (the large piece that remains). The two heights always add up to the diameter of the sphere, \(2R\) (in the example, \(0.268 + 3.732 = 4 = 2R\)).
When \(r = R\) (the cut goes through the widest part of the sphere), \(\sqrt{R^2 - r^2} = 0\), so there is only one answer, \(h = R\): exactly a hemisphere.
How to find the sphere radius \(R\) (when you know \(r\) and \(h\))
Figure
Standard notation
(the usual math form)
\(R\)
\(=\)
\((\)
\(h^{2}\)
\(+\)
\(r^{2}\)
\()\)
\(\div\)
\((\)
\(2h\)
\()\)
In words
(symbols replaced with words)
④
\(R\): sphere radius
\(=\)
\((\)
①
height \(h\) squared
\(+\)
②
base radius \(r\) squared
\()\)
\(\div\)
\((\)
③
twice the height, \(2h\)
\()\)
The formula in words
①
Add the
height \(h\) squared
②
and the
base radius \(r\) squared
,
③
and divide by
twice the height, \(2h\)
to get the
④
\(R\): sphere radius
Quick example
For a dome shape (a spherical cap) with a base radius of 3 in and a height of 2 in, the radius of the original sphere is
\(R\): sphere radius
\(=\)
\((\)
height squared (\(2^2 = 4\))
\(+\)
radius squared (\(3^2 = 9\))
\()\)
\(\div\)
\((\)
twice the height (\(4\))
\()\)
\(R = \dfrac{2^{2} + 3^{2}}{2 \times 2} = \dfrac{4 + 9}{4} = \dfrac{13}{4} = 3.25\,\mathrm{in}\)
Key idea
This formula rebuilds the size of the original sphere from a shape that is only part of a ball, such as a piece of a plate, a lens or a dome. Just measure two things, the radius \(r\) of the cut and how high the surface rises from it, \(h\), and you know the radius of the original sphere.
The spherometer, a tool that measures the radius of curvature of a lens or mirror surface, uses exactly this formula.
The curved surface area of a spherical cap is simply "2 × pi (π) × sphere radius R × height h" (the key point is that the base radius r does not appear). For the total surface area including the cut circle, just add the circle area πr². If you know any two of r, R and h, you can calculate the third.
Symbols and terms
Symbols
| \(S\) | ess | A common symbol for area, from the first letter of "surface". On this page it stands for the surface area of the spherical cap. |
| \(R\) | capital R | The radius of the original sphere (the whole sphere before the cut), from the first letter of "radius". It is written as a capital letter to tell it apart from lowercase \(r\). |
| \(r\) | lowercase r | The radius of the base (the cut circle). It is also a radius, but a different length from capital \(R\) (the sphere radius), so be careful. |
| \(h\) | aitch | The height of the cap, the distance from the cut to the top of the curved surface. From the first letter of "height". |
| \(\pi\) | pi | The ratio of a circle's circumference to its diameter. It is about 3.14159… and its digits never end. In school, 3.14 is often used. |
| \(r^2\) | r squared | The number \(r\) multiplied by itself (\(r \times r\)). The small 2 at the upper right is an exponent. (Example: \(3^2 = 3 \times 3 = 9\)) |
| \(\sqrt{\phantom{0}}\) | square root (radical sign) | The symbol for the number of 0 or more that gives the original number when squared. (Example: \(\sqrt{75}\) is the number whose square is 75, about 8.66) |
| \(\pm\) | plus or minus | A symbol that says both the "add" case and the "subtract" case are answers. In Formula 5, it shows that the height \(h\) has two answers. |
| \(\mathrm{in^2}\) | square inches | A unit of area. A square with 1-inch sides has an area of 1 in². This calculator gives the area in the square of the length unit you enter (ft² if you enter feet). |
Terms
| spherical cap | The piece you get when you slice a sphere with one flat cut, shaped like a dome roof, a bowl or a contact lens. The name is used both for the curved surface alone and for the solid piece. |
| solid cap | The solid piece cut off from a sphere by one flat plane. Its surface is made of the curved surface and the base (the cut circle), and the "total surface area" on this page is the area of its whole surface. |
| hemisphere | Half of a sphere, cut by a plane through its center. It is a special spherical cap whose height \(h\) equals the sphere radius \(R\) (\(h = R\)). |
| surface area | The area of the whole outside of a solid. It is a different quantity from volume (how much space is inside), and it is what you need for things like painting a surface or wrapping an object. |
| curved surface | A surface that is rounded, not flat. The rounded part of a spherical cap is a curved surface, while the cut circle is flat. |
| Pythagorean theorem | The theorem that in a right triangle, "the sum of the squares of the two legs = the square of the hypotenuse". Formulas 4 to 6, which work backward, come from applying it to the right triangle in a cross section of the sphere. |
| square root | A number that gives the original number when squared. The symbol is \(\sqrt{\phantom{0}}\). It appears in Formulas 4 and 5 to undo squaring. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Area of a circle and pi (Grade 7) |
|
| Exponents (Grade 6) |
|
| Surface area of a sphere (high school geometry) |
|
| Square roots (Grade 8) |
|
| The Pythagorean theorem (Grade 8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the curved surface area
| Sphere radius R | 10 |
| Cap height h | 5 |
| Curved surface area | =2*PI()*B1*B2 |
Table to find the base area (the cut circle)
| Base radius r | 3 |
| Base circle area | =PI()*B1^2 |
Table to find the total surface area (from R and h)
| Sphere radius R | 10 |
| Cap height h | 5 |
| Base radius r (calculated) | =SQRT(2*B1*B2-B2^2) |
| Total surface area | =2*PI()*B1*B2+PI()*B3^2 |
Table to find the base radius r
| Sphere radius R | 10 |
| Cap height h | 5 |
| Base radius r | =SQRT(2*B1*B2-B2^2) |
Table to find the height h (two answers)
| Base radius r | 1 |
| Sphere radius R | 2 |
| Height h (solution 1, shallow cap) | =B2-SQRT(B2^2-B1^2) |
| Height h (solution 2, deep cap) | =B2+SQRT(B2^2-B1^2) |
Table to find the sphere radius R
| Base radius r | 3 |
| Cap height h | 2 |
| Sphere radius R | =(B2^2+B1^2)/(2*B2) |
After pasting, column A holds the item names and column B holds the numbers. The upper rows are your inputs, and the formula in the last row calculates automatically from them.
"PI()" is a function that returns pi (π, 3.14159…) and "SQRT()" is a function that finds a square root. "*" is multiplication, "/" is division and "^" is a power (squared).
For example, B3 in the first table shows about 314.16, and B4 in the third table shows about 549.78. The fifth table is an example with two answers for the height (about 0.268 and about 3.732). Just replace the input numbers with your own.
"PI()" is a function that returns pi (π, 3.14159…) and "SQRT()" is a function that finds a square root. "*" is multiplication, "/" is division and "^" is a power (squared).
For example, B3 in the first table shows about 314.16, and B4 in the third table shows about 549.78. The fifth table is an example with two answers for the height (about 0.268 and about 3.732). Just replace the input numbers with your own.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the curved surface area
| Sphere radius R | 10 |
| Cap height h | 5 |
| Curved surface area | =2*PI()*B1*B2 |
Table to find the base area (the cut circle)
| Base radius r | 3 |
| Base circle area | =PI()*B1^2 |
Table to find the total surface area (from R and h)
| Sphere radius R | 10 |
| Cap height h | 5 |
| Base radius r (calculated) | =SQRT(2*B1*B2-B2^2) |
| Total surface area | =2*PI()*B1*B2+PI()*B3^2 |
Table to find the base radius r
| Sphere radius R | 10 |
| Cap height h | 5 |
| Base radius r | =SQRT(2*B1*B2-B2^2) |
Table to find the height h (two answers)
| Base radius r | 1 |
| Sphere radius R | 2 |
| Height h (solution 1, shallow cap) | =B2-SQRT(B2^2-B1^2) |
| Height h (solution 2, deep cap) | =B2+SQRT(B2^2-B1^2) |
Table to find the sphere radius R
| Base radius r | 3 |
| Cap height h | 2 |
| Sphere radius R | =(B2^2+B1^2)/(2*B2) |
The same formulas as in Excel (including the PI() and SQRT() functions) work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with your own.
How to calculate it in Python
import math
ball_radius = 10 # sphere radius R (in inches in this example)
cap_height = 5 # cap height h
base_radius = math.sqrt(2 * ball_radius * cap_height - cap_height ** 2) # base radius r (worked out)
cap_area = 2 * math.pi * ball_radius * cap_height # curved surface area of the cap
base_area = math.pi * base_radius ** 2 # base circle area
total_area = cap_area + base_area # total surface area
print(f"Base radius r: {base_radius} in")
print(f"Curved surface area: {cap_area} in2")
print(f"Base circle area: {base_area} in2")
print(f"Total surface area: {total_area} in2")
Runs with the standard library only. "math.pi" is pi (π), "math.sqrt()" is the square root and "**" is a power (squared). Change the sphere radius and the height at the top and run it (this example uses inches). To find the sphere radius from r and h instead, replace that line with "ball_radius = (cap_height ** 2 + base_radius ** 2) / (2 * cap_height)".
How to write it in LaTeX and other math languages (copy and paste)
Curved surface area of the cap
S₁ = 2πRh
S_1 = 2\pi R h
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mn>1</mn></msub>
<mo>=</mo>
<mn>2</mn>
<mi>π</mi>
<mi>R</mi>
<mi>h</mi>
</mrow>
</math>
S_1 = 2 pi R h
2*Pi*R*h
S1 := 2*Pi*R*h;
S1 = 2*pi*R*h;
S_1 = 2πRh
Area of the base (the cut circle)
S₂ = πr²
S_2 = \pi r^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mn>2</mn></msub>
<mo>=</mo>
<mi>π</mi>
<msup><mi>r</mi><mn>2</mn></msup>
</mrow>
</math>
S_2 = pi r^2
Pi*r^2
S2 := Pi*r^2;
S2 = pi*r^2;
S_2 = πr^2
Total surface area (curved surface + base)
S = 2πRh + πr²
S = 2\pi R h + \pi r^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mn>2</mn><mi>π</mi><mi>R</mi><mi>h</mi>
<mo>+</mo>
<mi>π</mi><msup><mi>r</mi><mn>2</mn></msup>
</mrow>
</math>
S = 2 pi R h + pi r^2
2*Pi*R*h + Pi*r^2
S := 2*Pi*R*h + Pi*r^2;
S = 2*pi*R*h + pi*r^2;
S = 2πRh + πr^2
How to find the base radius \(r\) (when you know \(R\) and \(h\))
r = √(2Rh − h²)
r = \sqrt{2Rh - h^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>r</mi>
<mo>=</mo>
<msqrt>
<mrow>
<mn>2</mn><mi>R</mi><mi>h</mi>
<mo>−</mo>
<msup><mi>h</mi><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
r = sqrt(2Rh - h^2)
Sqrt[2*R*h - h^2]
r := sqrt(2*R*h - h^2);
r = sqrt(2*R*h - h^2);
r = √(2Rh − h^2)
How to find the height \(h\) (when you know \(r\) and \(R\); two answers)
h = R ± √(R² − r²)
h = R \pm \sqrt{R^{2} - r^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>h</mi>
<mo>=</mo>
<mi>R</mi>
<mo>±</mo>
<msqrt>
<mrow>
<msup><mi>R</mi><mn>2</mn></msup>
<mo>−</mo>
<msup><mi>r</mi><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
h = R +- sqrt(R^2 - r^2)
Solve[r^2 + (R - h)^2 == R^2, h]
solve(r^2 + (R - h)^2 = R^2, h);
h = [R - sqrt(R^2 - r^2), R + sqrt(R^2 - r^2)];
h = R ± √(R^2 − r^2)
How to find the sphere radius \(R\) (when you know \(r\) and \(h\))
R = (h² + r²) ÷ (2h)
R = \dfrac{h^{2} + r^{2}}{2h}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>R</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msup><mi>h</mi><mn>2</mn></msup>
<mo>+</mo>
<msup><mi>r</mi><mn>2</mn></msup>
</mrow>
<mrow><mn>2</mn><mi>h</mi></mrow>
</mfrac>
</mrow>
</math>
R = (h^2 + r^2) / (2h)
(h^2 + r^2)/(2*h)
R := (h^2 + r^2)/(2*h);
R = (h^2 + r^2)/(2*h);
R = (h^2 + r^2)/(2h)
How to have ChatGPT do the calculation
You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). A sphere with a radius of 10 in is cut by a plane 5 in down from the top to make a spherical cap (a dome shape). Find each of the following: 1. The radius r of the base (the cut circle) (r = √(2Rh − h²)) 2. The area of the curved surface of the cap (S₁ = 2πRh) 3. The area of the base circle (S₂ = πr²) 4. The total surface area (S = S₁ + S₂) Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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Related Features
Sphere Volume Calculator (V = 4/3 × π × r³)
Finds the volume of the whole sphere, before any part is cut off, from its radius
Circle Calculator (Radius, Diameter, Circumference and Area)
For the base of the cap (the cut circle), finds the radius, circumference and area all at once
Pythagorean Theorem Calculator (Hypotenuse and Legs of a Right Triangle)
Finds the sides of a right triangle, the idea behind the relation between r, R and h of a cap
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