Surface Area Formulas and Volume Formulas of 3D Shapes


Surface area formulas and volume formulas appear time and again in calculations and homework problems. Pressure is a force per area and density is mass per volume. These are just two simple types of calculations that involve these formulas. This is a short list of common geometric shapes and their surface area formulas and volume formulas.

Sphere Surface Area Formula and Sphere Volume Formula

Sphere

A sphere is a solid figure where every point on the surface is equidistant from the center of the sphere. This distance is the radius, r, of the sphere.

Surface area = 4πr2

Volume = 43πr3

 

Prism Surface Area Formula and Prism Volume Formula

PrismA prism is a geometric shape consisting of a stack of identical base shapes stacked on top of each other to a depth d. This prism is a prism formed by a stack of triangles.

Surface Area of a Prism = 2 × (Area of the base shape) + (Perimeter of base shape) × (d)

Volume of a Prism = (Area of base shape) × d

To find the area and perimeter of the base shape, check out Area Formulas and Perimeter Formulas

 

Box Surface Area Formula and Box Volume Formula

Box

A box can be thought of a stack of rectangles L long and W wide piled on top of each other to a depth of D.

Surface Area of a Box = Sum of the areas of each face of the box, or

Surface Area of a Box = 2(L × W) + 2(L × D) + 2(W × D)

Volume of a Box = L × W × D

 

Cube Surface Area Formula and Cube Volume Formula

Cube with dimensions shownA cube is a special case box where all the sides are the same length.

Surface Area of a Cube = 6a2

Volume of a Cube = a3

 

Cylinder Surface Area Formula and Cylinder Volume Formula

Cylinder

 

A cylinder is a prism where the base shape is a circle.

Surface Area of a Cylinder = 2πr2 + 2πrh

Volume of a Cylinder = πr2h

 

Pyramid Surface Area Formula and Pyramid Volume Formula

Pyramid

A pyramid is a solid shape consisting of a polygon base and triangular faces meeting at a common point above the base. The pyramid shown here is a rectangular pyramid. There are two important measurements needed to calculate surface area and volume of a pyramid. The first is the height of the pyramid (h). This is the distance from the base to the point where the triangular faces meet. The second is the height of the individual face triangles (s).

Surface Area of a Pyramid = (sum of the areas of each face) + (area of the base)

Volume of a Pyramid = 13 A × h

For pyramids with identical face triangles

Surface Area of a Pyramid = (12 × Perimeter of base shape × s) + (Area of base shape)

Volume of a Pyramid = 13 A × h

If the base of the pyramid is a square (a = b), then

Surface area of a Square Pyramid = a2 + √3( a2 )

Volume of a Square Pyramid = √5(a3/6)

 

Surface Area Formula of a Cone and Volume Formula of a Cone
Cone

A cone is a pyramid with a circular base with radius r and height h. The side length s can be found using the Pythagorean Theorem.

s2 = r2 + h2
or
s = √( r2 + h2 )

Surface Area of a Cone = πr2 + πrs

Volume of a Cone = 13( πr2h )

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