How to find the area of a prism

Square Pyramid Shape

h = height
s = slant height
a = side length
e = lateral edge length
r = a/2
V = volume
Stot = total surface area
Slat = lateral surface area
Sbot = bottom surface area

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Online calculator to calculate the surface area of geometric solids including a capsule, cone, frustum, cube, cylinder, hemisphere, pyramid, rectangular prism, sphere, spherical cap, and triangular prism

Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft2 or ft3. For example, if you are starting with mm and you know r and h in mm, your calculations will result with V in mm3 and S in mm2.

Below are the standard formulas for surface area.

Surface Area Formulas:

Capsule Surface Area
Volume = πr2((4/3)r + a)
Surface Area = 2πr(2r + a)
Circular Cone Surface Area
Volume = (1/3)πr2h
Lateral Surface Area = πrs = πr√(r2 + h2)
Base Surface Area = πr2
Total Surface Area
= L + B = πrs + πr2 = πr(s + r) = πr(r + √(r2 + h2))
Circular Cylinder Surface Area
Volume = πr2h
Top Surface Area = πr2
Bottom Surface Area = πr2
Total Surface Area
= L + T + B = 2πrh + 2(πr2) = 2πr(h+r)
Conical Frustum Surface Area
Volume = (1/3)πh (r12 + r22 + (r1 * r2))
Lateral Surface Area
= π(r1 + r2)s = π(r1 + r2)√((r1 - r2)2 + h2)
Top Surface Area = πr12
Base Surface Area = πr22
Total Surface Area
= π(r12 + r22 + (r1 * r2) * s)
= π[ r12 + r22 + (r1 * r2) * √((r1 - r2)2 + h2) ]
Cube Surface Area
Volume = a3
Surface Area = 6a2
Hemisphere Surface Area
Volume = (2/3)πr3
Curved Surface Area = 2πr2
Base Surface Area = πr2
Total Surface Area= (2πr2) + (πr2) = 3πr2
Pyramid Surface Area
Volume = (1/3)a2h
Lateral Surface Area = a√(a2 + 4h2)
Base Surface Area = a2

Total Surface Area
= L + B = a2 + a√(a2 + 4h2))
= a(a + √(a2 + 4h2))
Rectangular Prism Surface Area
Volume = lwh
Surface Area = 2(lw + lh + wh)
Sphere Surface Area

Volume = (4/3)πr3
Surface Area = 4πr2
Spherical Cap Surface Area
Volume = (1/3)πh2(3R - h)
Surface Area = 2πRh
Triangular Prism Surface Area
Top Surface Area of a Triangular Prism Formula
\[ A_{top} = \dfrac{1}{4} \sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)} \]
\[ A_{top} = \dfrac{1}{4} \sqrt{\begin{aligned}(a+&b+c)(b+c-a)\\&\times(c+a-b)(a+b-c)\end{aligned}} \]
Bottom Surface Area of a Triangular Prism Formula
\[ A_{bot} = \dfrac{1}{4} \sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)} \]
\[ A_{bot} = \dfrac{1}{4} \sqrt{\begin{aligned}(a+&b+c)(b+c-a)\\&\times(c+a-b)(a+b-c)\end{aligned}} \]

Lateral Surface Area of a Triangular Prism Formula
\[ A_{lat} = h (a+b+c) \]
Total Surface Area of a Triangular Prism Formula
\[ A_{tot} = A_{top} + A_{bot} + A_{lat} \]

Finding the Surface Area of a Triangular Prism

Step 1: Determine the values of {eq}b_1 {/eq}, {eq}b_2 {/eq}, and {eq}b_3 {/eq}, the sides of the triangular base. Also determine the value of {eq}h {/eq}, the height of the triangular base and {eq}l {/eq}, the length of the prism (the length between the bases).

Step 2: Calculate the area of a triangular base using the formula for the area of a triangle: {eq}A = \dfrac{1}{2}bh {/eq}, where {eq}b {/eq} is one of {eq}b_1 {/eq}, {eq}b_2 {/eq}, or {eq}b_3 {/eq} (whichever one is perpendicular to {eq}h {/eq}). Multiply this result by 2, since there are two triangular bases with this area.

Step 3: Find the area of the rectangular sides by multiplying the perimeter of a base triangle by the length of the prism: {eq}A = (b_1 + b_2 + b_3)l {/eq}.

Step 4: Add the results from step 3 and step 4 together. This is the surface area of the triangular prism.

Finding the Surface Area of a Triangular Prism: Vocabulary and Equations

Surface area: The surface area of a three-dimensional figure is the sum of the areas of all of its faces. This can be imagined by cutting the figure into two-dimensional faces and calculating the area of each piece.

Triangular prism: A triangular prism is a three-dimensional figure with two triangular bases connected by rectangular sides. A triangular prism has five faces - two of which are triangles and three of which are rectangles.

Surface area of a triangular prism: The surface area of a triangular prism is given by {eq}A = bh + (b_1 + b_2 + b_3)l \text{ units}^2 {/eq} where {eq}b {/eq} is the base of a triangular face, {eq}h {/eq} is the height of a triangular face, {eq}b_1 {/eq}, {eq}b_2 {/eq}, and {eq}b_3 {/eq} are the sides of the triangular base, and {eq}l {/eq} is the length of the prism.

We will use these steps, definitions, and equations to find the surface area of a triangular prism in the following two examples.

Finding the Surface Area of a Triangular Prism: Example Problem 1

Find the surface area of the triangular prism shown in the figure.

Figure for Example 1

The sides of the triangular base are:

{eq}b_1 = 3\text{ in.} {/eq}

{eq}b_2 = 4\text{ in.} {/eq}

{eq}b_3 = 5\text{ in.} {/eq}

The triangular base is a right triangle, and so the height is one of our sides. We will use {eq}h = 3\text{ in.} {/eq}.

The length of the prism is {eq}l = 6\text{ in.} {/eq}.

The area of a triangular base is calculated as {eq}A = \dfrac{1}{2}bh {/eq} where

{eq}h = 3\text{ in.} {/eq}

{eq}b = 4\text{ in.} {/eq} (this is {eq}b_2 {/eq} which is the side perpendicular to the height of the base).

So the area of one triangular base is:

{eq}\begin{align} A {}& = \dfrac{1}{2}bh\\ & = \dfrac{1}{2}(4\text{ in.})(3\text{ in.})\\ & = 6\text{ in.}^2 \end{align} {/eq}

Multiplying this by 2, since we have two triangular bases, we have that the area of the bases is:

{eq}\begin{align} A {}&= 2\cdot 6\text{ in.}^2\\ & = 12\text{ in.}^2 \end{align} {/eq}

Step 3: Find the area of the rectangular sides by multiplying the perimeter of a base triangle by the length of the prism: {eq}A = (b_1 + b_2 + b_3)l {/eq}.

To find the area of the rectangular sides, we need to multiply the perimeter of the base by the length of the prism.

{eq}\begin{align} A {}& = (b_1 + b_2 + b_3)l\\ & = (3\text{ in.} + 4\text{ in.} + 5\text{ in.})6\text{ in.}\\ & = 12\text{ in.}\cdot 6\text{ in.}\\ & = 72\text{ in.}^2 \end{align} {/eq}

Step 4: Add the results from step 3 and step 4 together. This is the surface area of the triangular prism.

Adding the results from steps 2 and 3, we get the surface area of the triangular prism.

{eq}A = 12\text{ in.}^2 + 72\text{ in.}^2\\ A = 84\text{ in.}^2 {/eq}

Therefore, the surface area of the triangular prism is {eq}84\text{ in.}^2 {/eq}.

Finding the Surface Area of a Triangular Prism: Example Problem 2

Find the surface area of the triangular prism shown in the figure.

Figure for Example 2

{eq}b_1 = 20\text{cm.} {/eq}

{eq}b_2 = 29\text{ cm.} {/eq}

{eq}b_3 = 29\text{ cm.} {/eq}

{eq}h = 21\text{ cm.} {/eq}

{eq}l = 31\text{ cm.} {/eq}

The area of a triangular base is:

{eq}\begin{align} A {}& = \dfrac{1}{2}bh\\ & = 210\text{ cm.}^2 \end{align} {/eq}

Multiplying this by 2, we have that the area of the bases is:

{eq}A = 420\text{ cm.}^2 {/eq}

Step 3: Find the area of the rectangular sides by multiplying the perimeter of a base triangle by the length of the prism: {eq}A = (b_1 + b_2 + b_3)l {/eq}.

{eq}\begin{align} A {}& = (b_1 + b_2 + b_3)l\\ & = 78\text{ cm.}\cdot 31\text{ cm.}\\ & = 2,418\text{ cm.}^2 \end{align} {/eq}

Step 4: Add the results from step 3 and step 4 together. This is the surface area of the triangular prism.

Adding the results,

{eq}A = 420\text{ cm.}^2 + 2,418\text{ cm.}^2\\ A = 2,838\text{ cm.}^2 {/eq}

Therefore, the surface area of the triangular prism is {eq}2,838\text{ cm.}^2 {/eq}.

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What is the formula for any prism?

The formula for the volume of a prism is given by the product of the area of the base and height of the prism. Thus, the volume of a prism can be given as V = B × H where V is the volume, B base area, and H height of the prism.

How do you find the base area of a rectangular prism?

One edge of the rectangle is labeled length and the other is labeled width. To find the area of the rectangle, just multiply the two edges together. Area (bottom edge) = length times width = lw. Going back to our example, the area of the bottom face is 4 inches x 3 inches = 12 square inches.