Subject: Compulsory Mathematics

A triangular prism is a prism composed of two triangular bases and three rectangular sides. A prism with the triangle as the base is called triangular prism. In case of triangular prism, two congruent and parallel triangles

A prism with the triangle as the base is called triangular prism. In the case of a triangular prism, two congruent and parallel triangles ABC and EFG are called the base of the prism. Area of each triangle is called base area or area of the base.

The lateral faces (AEFB, AEGC and BCGF), are rectangles formed by joining corresponding vertices of the bases. The intersection of lateral faces is lateral edges.

The length or height ( AE, BF, CG ) is the perpendicular distance between the bases. The lateral surface is the total area of the lateral faces (The length times the perimeter of base) and volume is equal to the product of base area and its length or height.

So,

Area of triangular base = Area of ΔABC or Area of ΔEFG

Lateral (Curved) Surface Area (L.S.A) = Perimeter of triangular base × length

Total Surface Area (T.S.A) = 2 × Area of triangular base + L.S.A.

Volume of triangular prism = Area of triangular base × length (or height of prism)

Area of triangular base = Area of ΔABC or Area of ΔEFG

Lateral (Curved) Surface Area (L.S.A) = Perimeter of triangular base × length

Total Surface Area (T.S.A) = 2 × Area of triangular base + L.S.A.

Volume of triangular prism = Area of triangular base × length (or height of prism)

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Find the volume of the prism given in the adjoining diagram.

**Solution:**

Here,

\begin{align*}\text{base area of prism (A)} &= \frac{1}{2} \times base \times height \\ &= \frac{1}{2} \times 6 \times 8 \\ &= 24 \: cm^2 \end{align*}

height of prism (h) = 30 cm

By formula,

\( Volume\: of \: prism (V) = A\times h = 24 \times 30 = 720\:cm^2 \: \: _{Ans}\)

In the given solid tringular prism, AB = 3 cm, B'C' = 5 cm, and CC' = 20 cm. If the area of the rectangular surfaces of the prism is 240 sq. cm, find the measurement of AC.

**Solution:**

BC = B'C' = 5 cm

\begin{align*} \text{Perimeter of the base triangle} &= AB+BC+AC\\ &= 3cm+5cm+AC \\ &= 8cm+AC \\ Height \: of\: the \: prism (h) &= 20\: cm \\ Rectangular \: surface \: area \: of\:prism &=ph \\ or, 240cm^2 &= (8cm + AC) .20cm \\ or, 8cm + AC &= \frac{240cm^2}{20cm}\\ or, 8cm+AC &= 12cm \\ or, AC &= 12cm-8cm \\ \therefore AC &= 4 cm \: _{Ans}\end{align*}

In the given triangular prism, PQ = 6cm, PR = 7cm, QR = 5cm and RC = 18cm. Find the volume of the prism.

**Solution:**

\begin{align*} 2s &= PQ+PR+QR \\or, 2s &= (6+7+5)cm \\or,2s &= 18 cm \\ or, s&=\frac{18}{2}\\ \therefore s &= 9cm \end{align*}

Now,

\begin{align*} Area \: of \: \Delta PQR &= \sqrt{s(s-a)(s-b)(s-c)}\\ &= \sqrt{9 (9-6)(9-7)(9-5)}cm^2\\ &= \sqrt{9\times3\times2\times4}cm^2 \\ &= \sqrt{216}cm^2 \\ \: \\ Volume \: of \: prism &= A \times height \\ &= \sqrt{216}\times18 \: cm^3 \\ &= 264.54 \: cm^3 \end{align*}

The volume of a prism having its base a right angled triangle is 864 cubic cm. If the lengths of the sides of the right angled triangle containing the right angle are 8 cm and 9 cm, calculate the height of the prism.

**Solution:**

V = Volume of prisms = 864 cm^{3}

\(A = \text{Area of rt. angled triangle }= \frac{1}{2}\times 8 \times 9 = 36cm^2\)

H = height of prism = ?

By formula, we have

\begin{align*} V &= A \times h \\ or, 864cm^3 &= 36cm^2 \times h \\ or, h &= \frac{864cm^3}{36cm^2} \\ \therefore h &= 24cm _{ans} \end{align*}

Find the lateral surface area of the given triangular prism.

**Solution:**

Here, AE =10cm, AF = BC = 8cm

\(EF = \sqrt{AE^2 - AF^2 } = \sqrt{10^2 - 8^2} = \sqrt{36} = 6 cm\)

\(\text{Perimeter of base triangle} = 10cm + 8cm+6cm = 24 cm \)

height (h)= 20cm

\begin{align*} \text{Lateral surface area } \: &= P \times h \\ &= 24 cm \times 20 cm \\ &= 480 cm^2 \: \: \: _{Ans}\end{align*}

In the given triangular prism AB = BC = CA = \(2\sqrt{3}cm\) and \(CK = 4 \sqrt{3}cm \). Find trhe area of rectangular surfaces at the prism.

**Solution:**

Here,

\(P = AB + BC + CA \\ \: \: \: = 2\sqrt{3} + 2\sqrt{3} + 2\sqrt{3}\\ \: \: \: = 6\sqrt{3} cm \)

\begin{align*} \text{Area of rectangular surface}&= P \times CK \\ &= 6\sqrt{3}\times 4\sqrt{3}\\ &= 72cm^2 \: _{Ans} \end{align*}

Find the total surface area of the given triangular prism, where AB= 12 cm, BC = 5 cm, CC^{1} = 30cm and ∠B = 90° .

**Solution:**

\begin{align*} A&=Area \: of \: base \\ &=area \: of \: rt. \: angled \: \Delta ABC \\ &= \frac{1}{2}\times BC \times AB \\ &= \frac{1}{2}\times 5cm \times 12 cm \\ &= 30 \: cm^2 \\ In \: ABC, \\ AC^2 &=AB^2+BC^2\\ or, AC&=\sqrt{(12cm)^2 + (5cm)^2}\\ or, AC &= \sqrt{144cm^2+25cm^2} \\ \therefore AC &= \sqrt{169cm^2} = 13cm \\ \: \: \\ P = Perimeter \: of \: \Delta &= AB+BC+AC \\ &= 12cm+5cm+13cm\\ &= 30 cm \\ \: \\ S= Lateral\:surface\:area&= P\times h\\ &= 30cm\times 30cm\\ &= 900cm^2 \\ \: \\ Total \: surface \: area&= 2A + S \\ &=2 \times 30cm^2 + 900cm^2\\ &= 960cm^2 \: \: _{Ans} \end{align*}

Find the total surface area of a prism of which the height is 12 cm. and the base is a square of a side is 6cm.

**Solution:**

\begin{align*} The \:area (A) \: of \: the \: base &= l^2\\ &= (6cm)^2 \\&=36cm^2 \\ Perimeter (P)\: of \: of \: the \: base &= 4l \\ &=4 \times 6 \\ &=24cm \\ The \: height(h) of \: the\:prism&=12 \\ Here, \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ Total\:surface\:area &= 2 \times area \: of \: base + L.S.A \\ &= 2A + Ph\\ &= 2\times36+24\times12 \\ &= 72cm^2+288cm^2 \\ &= 360cm^2 \end{align*}

What height of the prism shown in the figure makes its volume 48cm^{2}.

**Solution: **

Let h be the height of the prism.

Here, Volume = 48cm^{3}

Area of base triangle (A) =?

\begin{align*} A &= \frac{1}{2} \times 4 \times 3 \\ &= 6 \: cm^2 \\ By \: formula, \\ Volume &= A \times h \\48^3 &= 6cm^2 \times h \\ or, h &= \frac{48cm^3}{6cm^2} \\ \therefore h &= 8 cm \: \: _{Ans} \end{align*}

The given figure is of a equivalent triangular prism. If the length of the base is 6 cm and volume 162 cubic cm, find the height.

**Solution:**

Length of the side of the base (a) = 6 cm

\begin{align*}Area \: of \: base\: triangle\: (A)&= \frac{\sqrt{3}}{4}a^2 \\ &= \frac{\sqrt{3}}{4}6^2 \\ &= 9\sqrt{3} \: cm^2 \\ Volume(V) &= 162 \: cm^3\\ height \: of \: the \: prism \: (h)&= ? \\ We \: know \: that, \: \: \:\:\:\:\:\:\:\:\:\: \\ V&=A \times h \\ 162 \:cm^3&=9\sqrt{3} \times h \\ or, h &= \frac{162}{9\sqrt{3}} \\ or, h &= 6\sqrt{3} \\ \therefore h &= 10.39 \: cm \end{align*}

In the given solid triangular prism, if AB = 3 cm, AC = 4 cm, B'C' = 5 cm and the area of the rectangular surfaces is 240 sq. cm, find the measurement of CC'.

**Solution:**

\begin{align*} Perimeter \: of \: base \: triangle \: (P) &= AB+BC+AC \\ &= 3cm+5cm+4cm\\&= 12cm\\ Height \: (h) \:\:= CC' &= ? \\ By, formula, area \: of \: rectangular \: faces &= Ph \\ or, 240 \: cm^2 &= 12cm\times h \\ or,h&=\frac{240cm^2}{12cm}\\ \therefore h &= 20cm \: _{Ans} \end{align*}

If the area of rectangular surfaces and the height of a triangular prism are 660 cm^{2} and 22 cm respectively, find the perimeter of its base.

**Solution:**

Perimeter of base (P) = ?

Rectangular surface area (S) = 600 cm^{2}

Height of prism (h) = 32 cm

By formula,

\begin{align*} S &=Ph \\ or, 660&=P\times 22 \\ or, P &= \frac{660}{22}\\ \therefore P &= 30 \: cm \: _{Ans} \end{align*}

The volume of given solid prism is 450 \(cm^3,\) if AB = 12cm and CD = 15 cm, find the length of AC.

**Solution:**

Here, V = 450 cm^{3} , h = CD = 15 cm, AC = ?

By formula,

\begin{align*} V&=Ah\\ or, A &= \frac{V}{h}\\ &= \frac{450}{15}\\ \therefore A &= 30 \: cm^2 \\ Again \: by \: the \: formula, \\ A&=\frac{1}{2}AB \times BC \\ or, 30 &= \frac{1}{2}\times 12 \times BC \\ or, 30 &= 6BC \\ or, BC &= \frac{30}{6}\\ \therefore BC &= 5 \: cm \\ From \: right \: angled \: &triangle \: ABC, \\ AC^2 &= AB^2 + BC^2 \\ or, AC^2&= 12^2 + 5^2 \\ or, AC&=\sqrt {169} \\ \therefore AC &= 13 \: cm \: _{Ans} \end{align*}

\(In \: the \: given\: prism, \angle ABC = 90°, AB=12cm \: and \: AC = 20cm.\\ \text{If the volume of the prism is 1920 cubic cm, find the height of the prism.}\)

**Solution:**

Here,

\begin{align*} BC &= \sqrt{AC^2-AB^2}\\ &=\sqrt{(20cm)^2 - (12cm)^2}\\ &=\sqrt{400-144}cm \\ &= \sqrt{256}cm \\ &= 16cm \\ \: \\ \text{Area of right angled} \: &triangle \: of \: base \: (A) = \frac{1}{2}\times AB \times BC \\ &= \frac{1}{2} \times 12 \times 16 \\ &= 6cm \times 16cm \\ &= 96cm^2 \\ \: \\ Suppose, height \: of \: the \: prism &= h \\ Then, volume \: of \: the \: prism &= Ah \\ or, 1920cm^3 &= 96cm^2\times h \\ or, h &= \frac{1920cm^3}{96cm^2}\\ \therefore h &= 20 cm \: _{Ans} \: cm \end{align*}

© 2019-20 Kullabs. All Rights Reserved.