Triangular Prism Surface Area Calculator

A triangular prism has five faces: two matching triangles at the ends and three rectangles joining them. Tents, wedge-shaped ramps, gable roofs and glass prisms all have this form. Every rectangle is as long as the prism and as wide as one side of the triangle, so the calculation needs nothing more than the triangle's measurements and the length.

Some textbooks call the length the height of the prism. It is the distance between the two triangles, and it is a different thing from the height of the triangle itself.

Formula

A = 2 × E + (a + b + c) × l

A

the total surface area

E

the area of one triangular end

a, b, c

the three sides of that triangle

l

the length of the prism, between the two ends

The lateral area is (a + b + c) × l.

Surface area from three sides and the length

When all three sides are known, Heron's formula gives the end area without any height or angle. A prism whose ends have sides of 5, 6 and 7, and whose length is 10, has an end area of √216 ≈ 14.70. The rectangles add up to (5 + 6 + 7) × 10 = 180, so the total is 180 + 2 × √216 ≈ 209.39.

The three sides must actually make a triangle, with each one shorter than the other two combined. Sides of 2, 3 and 10 cannot close up, and the calculator says so rather than giving a number.

• E = √( s(s − a)(s − b)(s − c) )
• s = (a + b + c) ÷ 2

Surface area of a right triangular prism

If the end is a right triangle, two legs are enough. The hypotenuse comes from the Pythagorean theorem and the end area is half the product of the legs. Legs of 3 and 4 with a length of 10 give a hypotenuse of 5, an end area of 6 and rectangles totaling 12 × 10 = 120. The surface is 120 + 2 × 6 = 132.

A wedge-shaped ramp is the usual example. A ramp 4 ft long, 1 ft high and 3 ft wide has a total surface of about 31.37 ft². Only its sloping top needs decking, and that is one rectangle: √17 × 3 ≈ 12.37 ft².

• c = √(a² + b²)
• E = a × b ÷ 2

Lateral surface area

The lateral area covers the three rectangles and leaves out the triangular ends. Unfolded, they form a single long rectangle whose width is the perimeter of the triangle, which is why the formula is perimeter times length. A prism with a right-triangle end of 3, 4 and 5 and a length of 10 has a lateral area of 12 × 10 = 120.

• Lateral area = (a + b + c) × l

Working out a ridge tent

Real jobs rarely want every face. Take a ridge tent with ends of 2 m, 2 m and 2.4 m, the last being the width across the floor, and a length of 3 m. Each end is 1.92 m² and the lateral area is 6.4 × 3 = 19.2 m², for a total of 23.04 m². The floor is the 2.4 × 3 = 7.2 m² rectangle, so without it the fabric comes to 15.84 m². Add a margin for seams and hems.

Surface area and volume

A prism's volume is the end area times the length, and it does not depend on the rectangles at all. The 3, 4, 5 prism holds 6 × 10 = 60 cubic units, and the 5, 6, 7 prism √216 × 10 ≈ 146.97. The volume calculator handles other solids, and the triangle area calculator works out the end by other routes, such as two sides and an angle.

• V = E × l

Frequently asked questions